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Since B is due in part to bound currents (which we don't yet know), we cannot compute it directly. However, this is one of those symmetrical cases in which we can get H from the free current alone, using Ampère's law in the form of Eq. 6.20: $$\mathbf{H} = nI \,\hat{\mathbf{z}}$$ (Fig. 6.22). According to E...	{ "Header 1": "Solution", "token_count": 1061, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In a linear medium, the alignment of atomic dipoles is maintained by a magnetic field imposed from the outside. Ferromagnets—which are emphatically not linear11—require no external fields to sustain the magnetization; the alignment is "frozen in." Like paramagnetism, ferromagnetism involves the magnetic dipoles assoc...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 2047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
![](_page_310_Picture_2.jpeg) FIGURE 6.30 (b) Show that the interaction energy of two magnetic dipoles separated by a displacement $\mathbf{r}$ is given by $$U = \frac{\mu_0}{4\pi} \frac{1}{r^3} [\mathbf{m}_1 \cdot \mathbf{m}_2 - 3(\mathbf{m}_1 \cdot \hat{\mathbf{r}})(\mathbf{m}_2 \cdot \hat{\mathbf{r}})]...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 1986, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Show that the magnetic field inside the sphere (0 < r < R) is $$\frac{\mu}{4\pi} \left\{ \frac{1}{r^3} [3(\mathbf{m} \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{m}] + \frac{2(\mu_0 - \mu)\mathbf{m}}{(2\mu_0 + \mu)R^3} \right\}.$$ What is the field outside the sphere? ! Problem 6.29 You are asked to...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 460, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 7.1 ■ ELECTROMOTIVE FORCE #### 7.1.1 ■ Ohm's Law To make a current flow, you have to push on the charges. How fast they move, in response to a given push, depends on the nature of the material. For most substances, the current density J is proportional to the force per unit charge, f: $$\mathbf{J} = \s...	{ "Header 1": "Electrodynamics", "token_count": 1312, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The field between the cylinders is $$\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 s} \hat{\mathbf{s}},$$ where λ is the charge per unit length on the inner cylinder. The current is therefore $$I = \int \mathbf{J} \cdot d\mathbf{a} = \sigma \int \mathbf{E} \cdot d\mathbf{a} = \frac{\sigma}{\epsilon_0} \lambda L.$$ ...	{ "Header 1": "Solution", "token_count": 2043, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7.4b), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.) ...	{ "Header 1": "Solution", "token_count": 753, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
If you think about a typical electric circuit—a battery hooked up to a light bulb, say (Fig. 7.7)—a perplexing question arises: In practice, the current is the same all the way around the loop; why is this the case, when the only obvious driving force is inside the battery? Off hand, you might expect a large current ...	{ "Header 1": "7.1.2 Electromotive Force", "token_count": 1761, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section, I listed several possible sources of electromotive force, batteries being the most familiar. But I did not mention the commonest one of all: the generator. Generators exploit motional emfs, which arise when you move a wire through a magnetic field. Figure 7.10 suggests a primitive model f...	{ "Header 1": "7.1.3 Motional emf", "token_count": 2031, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The flux rule assumes you have a single wire loop—it can move, rotate, stretch, or distort (continuously), but beware of switches, sliding contacts, or extended conductors allowing a variety of current paths. A standard "flux rule paradox" involves the circuit in Figure 7.14. When the switch is thrown (from a to b)...	{ "Header 1": "7.1.3 Motional emf", "token_count": 221, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The speed of a point on the disk at a distance s from the axis is v = ωs, so the force per unit charge is fmag = v × B = ωs Bsˆ. The emf is therefore $$\mathcal{E} = \int_0^a f_{\text{mag}} \, ds = \omega B \int_0^a s \, ds = \frac{\omega B a^2}{2},$$ and the current is $$I = \frac{\mathcal{...	{ "Header 1": "Solution", "token_count": 1251, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In 1831 Michael Faraday reported on a series of experiments, including three that (with some violence to history) can be characterized as follows: Experiment 1. He pulled a loop of wire to the right through a magnetic field (Fig. 7.21a). A current flowed in the loop. Experiment 2. He moved the magnet to t...	{ "Header 1": "7.2.1 Faraday's Law", "token_count": 426, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
It is this induced<sup>8</sup> electric field that accounts for the emf in Experiment 2.<sup>9</sup> Indeed, if (as Faraday found empirically) the emf is again equal to the rate of change of the flux, $$\mathcal{E} = \oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi}{dt},\tag{7.14}$$ then E is related to the change...	{ "Header 1": "A changing magnetic field induces an electric field.", "token_count": 674, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$\mathcal{E} = -\frac{d\Phi}{dt} \tag{7.17}$$ #### will appear in the loop. Many people call this "Faraday's law." Maybe I'm overly fastidious, but I find this confusing. There are really two totally different mechanisms underlying Eq. 7.17, and to identify them both as "Faraday's law" is a little like saying ...	{ "Header 1": "Whenever (and for whatever reason) the magnetic flux through a loop changes, an emf", "token_count": 729, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The induced current will flow in such a direction that the flux it produces tends to cancel the change. (As the front end of the magnet in Ex. 7.5 enters the ring, the flux increases, so the current in the ring must generate a field to the right—it therefore flows clockwise.) Notice that it is the change in flu...	{ "Header 1": "Nature abhors a change in flux.", "token_count": 369, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Before you turned on the current, the flux through the ring was zero. Afterward a flux appeared (upward, in the diagram), and the emf generated in the ring led to a current (in the ring) which, according to Lenz's law, was in such a direction that its field tended to cancel this new flux. This means that the cu...	{ "Header 1": "Solution", "token_count": 2025, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
To convince you of this, I deliberately set things up so that the magnetic field is zero at the location of the charge. The experimenter may tell you she never put in any electric field—all she did was switch off the magnetic field. But when she did that, an electric field automatically appeared, and it's this el...	{ "Header 1": "Solution", "token_count": 2038, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Some of the field lines pass <sup>&</sup>lt;sup>16</sup>This paradox was suggested by Tom Colbert. Refer to Problem 2.55. ![](_page_340_Picture_2.jpeg) FIGURE 7.30 FIGURE 7.31 through loop 2; let <sup>2</sup> be the flux of B<sup>1</sup> through 2. You might have a tough time actually *calculating...	{ "Header 1": "Solution", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Example 7.12. Suppose a current I is flowing around a loop, when someone suddenly cuts the wire. The current drops "instantaneously" to zero. This generates a whopping back emf, for although I may be small, d I/dt is enormous. (That's why you sometimes draw a spark when you unplug an iron or toaster elect...	{ "Header 1": "Solution", "token_count": 1401, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
It takes a certain amount of energy to start a current flowing in a circuit. I'm not talking about the energy delivered to the resistors and converted into heat—that is irretrievably lost, as far as the circuit is concerned, and can be large or small, depending on how long you let the current run. What I am concerned w...	{ "Header 1": "7.2.4 Energy in Magnetic Fields", "token_count": 1777, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
According to Ampère's law, the field between the cylinders is $$\mathbf{B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol{\phi}}.$$ Elsewhere, the field is zero. Thus, the energy per unit volume is $$\frac{1}{2\mu_0} \left( \frac{\mu_0 I}{2\pi s} \right)^2 = \frac{\mu_0 I^2}{8\pi^2 s^2}.$$ The energy in a cylindrica...	{ "Header 1": "Solution", "token_count": 1319, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
So far, we have encountered the following laws, specifying the divergence and curl of electric and magnetic fields: (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ (Gauss's law), (ii) $$\nabla \cdot \mathbf{B} = 0$$ (no name), (iii) $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$...	{ "Header 1": "7.3.1 Electrodynamics Before Maxwell", "token_count": 896, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The problem is on the right side of Eq. 7.36, which should be zero, but isn't. Applying the continuity equation (5.29) and Gauss's law, the offending term can be rewritten: $$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial t} (\epsilon_0 \nabla \cdot \mathbf{E}) = -\nabla \...	{ "Header 1": "7.3.2 ■ How Maxwell Fixed Ampère's Law", "token_count": 493, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
<sup>&</sup>lt;sup>20</sup>For the history of this subject, see A. M. Bork, Am. J. Phys. 31, 854 (1963). <sup>&</sup>lt;sup>21</sup>See footnote 8 (page 313) for commentary on the word "induce." The same issue arises here: Should a changing electric field be regarded as an independent source of magnetic field (al...	{ "Header 1": "A changing electric field induces a magnetic field.", "token_count": 737, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The inner one (radius a) carries a charge Q(t), and the outer one (radius b) an opposite charge -Q(t). The space between them is filled with Ohmic material of conductivity $\sigma$ , so a radial current flows: $$\mathbf{J} = \sigma \mathbf{E} = \sigma \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \,\hat{\mathbf{r}}; \quad ...	{ "Header 1": "A changing electric field induces a magnetic field.", "Header 2": "Example 7.14. Imagine two concentric metal spherical shells (Fig. 7.44).", "token_count": 1148, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section we put the finishing touches on Maxwell's equations: (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ (Gauss's law), (ii) $\nabla \cdot \mathbf{B} = 0$ (no name), (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ (Faraday's law), (iv) $\nabla \times \mathbf...	{ "Header 1": "7.3.3 ■ Maxwell's Equations", "token_count": 932, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
There is a pleasing symmetry to Maxwell's equations; it is particularly striking in free space, where ρ and J vanish: $$\nabla \cdot \mathbf{E} = 0, \qquad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},$$ $$\nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \fra...	{ "Header 1": "7.3.4 Magnetic Charge", "token_count": 1354, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Maxwell's equations in the form 7.40 are complete and correct as they stand. However, when you are working with materials that are subject to electric and magnetic polarization there is a more convenient way to write them. For inside polarized matter there will be accumulations of "bound" charge and current, over whi...	{ "Header 1": "7.3.5 ■ Maxwell's Equations in Matter", "token_count": 1904, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In general, the fields E, B, D, and H will be discontinuous at a boundary between two different media, or at a surface that carries a charge density $\sigma$ or a current density K. The explicit form of these discontinuities can be deduced from Maxwell's equations (7.56), in their integral form ...	{ "Header 1": "7.3.6 ■ Boundary Conditions", "token_count": 1942, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
[Answer: (0V0/πa)tan(φ/2)] Problem 7.43 The magnetic field outside a long straight wire carrying a steady current I is $$\mathbf{B} = \frac{\mu_0}{2\pi} \frac{I}{s} \,\hat{\boldsymbol{\phi}}.$$ The electric field inside the wire is uniform: $$\mathbf{E} = \frac{I\rho}{\pi a^2} \,\hat{\mathbf{z}},$...	{ "Header 1": "7.3.6 ■ Boundary Conditions", "token_count": 1826, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that $$\mathbf{E} = -\frac{\partial \mathbf{A}}{\partial t},\tag{7.66}$$ for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. (b) A spherical shell of radius R carries a uniform surface charge σ. It spins about ...	{ "Header 1": "Problem 7.49", "token_count": 1364, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
- (a) Use the Neumann formula (Eq. 7.23) to calculate the mutual inductance of the configuration in Fig. 7.37, assuming a is very small (a b, a z). Compare your answer to Prob. 7.22. - (b) For the general case (not assuming a is small), show that $$M = \frac{\mu_0 \pi \beta}{2} \sqrt{ab\beta} \left( 1 + \fr...	{ "Header 1": "Problem 7.56", "token_count": 2047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The current travels down one strip and back along the other. In each case, it spreads out uniformly over the surface of the ribbon. - (a) Find the capacitance per unit length, C. - (b) Find the inductance per unit length, $\mathcal{L}$ . - (c) What is the product $\mathcal{LC}$ , numerically? [ $\mathcal{L}$ and ...	{ "Header 1": "Problem 7.56", "token_count": 1364, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
All of our cards are now on the table, and in a sense my job is done. In the first seven chapters we assembled electrodynamics piece by piece, and now, with Maxwell's equations in their final form, the theory is complete. There are no more laws to be learned, no further generalizations to be considered, and (with perha...	{ "Header 1": "Intermission", "token_count": 250, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In this chapter we study conservation of energy, momentum, and angular momentum, in electrodynamics. But I want to begin by reviewing the conservation of charge, because it is the paradigm for all conservation laws. What precisely does conservation of charge tell us? That the total charge in the universe is constant?...	{ "Header 1": "8.1.1 The Continuity Equation", "token_count": 568, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In Chapter 2, we found that the work necessary to assemble a static charge distribution (against the Coulomb repulsion of like charges) is (Eq. 2.45) $$W_{\rm e} = \frac{\epsilon_0}{2} \int E^2 d\tau,$$ where E is the resulting electric field. Likewise, the work required to get currents going (against the back ...	{ "Header 1": "8.1.2 Poynting's Theorem", "token_count": 1885, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{8.11}$$ What if no work is done on the charges in V—what if, for example, we are in a region of empty space, where there is no charge? In that case dW/dt = 0, so $$\int \frac{\partial u}{\partial t} d\tau = -\oint \mathbf{S} \cdot d\mathbf{a} = -\int (\mathbf{\nabla} \cdot \mathbf{S}) d\tau,$$ and hence ...	{ "Header 1": "8.1.2 Poynting's Theorem", "token_count": 898, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Imagine a point charge q traveling in along the x axis at a constant speed v. Because it is moving, its electric field is not given by Coulomb's law; nevertheless, $\mathbf{E}$ still points radially outward from the instantaneous position of the charge (Fig. 8.2a), as we'll see in Chapter 10. Since, moreover, a mov...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 2015, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Thus $$T_{xx} = \frac{1}{2} \epsilon_0 \left( E_x^2 - E_y^2 - E_z^2 \right) + \frac{1}{2\mu_0} \left( B_x^2 - B_y^2 - B_z^2 \right),$$ $$T_{xy} = \epsilon_0 (E_x E_y) + \frac{1}{\mu_0} (B_x B_y),$$ and so on. Because it carries two indices, where a vector has only one, $T_{ij}$ is sometimes written with a d...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 1655, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{8.23}$$ Meanwhile, for the equatorial disk, $$d\mathbf{a} = -r \, dr \, d\phi \, \hat{\mathbf{z}},\tag{8.24}$$ and (since we are now inside the sphere) $$\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^3} \mathbf{r} = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^3} r(\cos\phi \,\hat{\mathbf{x}} + \sin\phi \,\hat...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 2043, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Sands, The Feynman Lectures on Physics (Reading, Mass.: Addison-Wesley, 1964), Vol. II, Section 27-6. Similarly, $\overrightarrow{T}$ plays a dual role: $\overrightarrow{T}$ itself is the electromagnetic stress (force per unit area) acting on a surface, and $-\overrightarrow{T}$ describes the flow of momentum...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 1728, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
By now, the electromagnetic fields (which started out as mediators of forces between charges) have taken on a life of their own. They carry energy (Eq. 8.5) $$u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right), \tag{8.31}$$ and momentum (Eq. 8.29) $$\mathbf{g} = \epsilon_0 (\mathbf{E} \times \m...	{ "Header 1": "8.2.4 ■ Angular Momentum", "token_count": 2046, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
This is perhaps a good place to revisit the old paradox that magnetic forces do no work (Eq. 5.11). What about that magnetic crane lifting the carcass of a junked car? Somebody is doing work on the car, and if it's not the magnetic field, who <sup>10</sup>In Ex. 8.4 we turned the current off slowly, to keep things ...	{ "Header 1": "8.3 MAGNETIC FORCES DO NO WORK<sup>12</sup>", "token_count": 1927, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Meanwhile, however, a Faraday emf is induced in the upper loop, due to the changing flux from the lower loop: $$\Phi_b = M I_a \implies \mathcal{E}_b = -I_a \frac{dM}{dt},$$ and the work done by the power supply in ring b (to sustain the current Ib) is $$dW_b = -\mathcal{E}_b I_b dt = \frac{3\pi}{2} \mu...	{ "Header 1": "8.3 MAGNETIC FORCES DO NO WORK<sup>12</sup>", "token_count": 1276, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 8.13<sup>16</sup> A very long solenoid of radius a, with n turns per unit length, carries a current Is. Coaxial with the solenoid, at radius b a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is induced in the ring. - (a)...	{ "Header 1": "More Problems on Chapter 8", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
8.45 is valid only for ideal dipoles. But g is linear in B, and therefore, if E is held fixed, obeys the superposition principle: For a collection of magnetic dipoles, the total momentum is the (vector) sum of the momenta for each one separately. In particular, if E is uniform over a localized ste...	{ "Header 1": "More Problems on Chapter 8", "token_count": 373, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using J <sup>f</sup> in place of J. Show that the Poynting vector becomes $$\mathbf{S} = \mathbf{E} \times \mathbf{H},\tag{8.46}$$ and the rate of change of the energy density in the fields is $$\frac{\partial u}{\partial t} = \ma...	{ "Header 1": "Problem 8.23", "token_count": 874, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
What is a "wave"? I don't think I can give you an entirely satisfactory answer—the concept is intrinsically somewhat vague—but here's a start: A wave is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Immediately I must add qualifiers: In the presence of absorption, the w...	{ "Header 1": "9.1.1 The Wave Equation", "token_count": 2030, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
A is the amplitude of the wave (it is positive, and represents the maximum displacement from equilibrium). The argument of the cosine is called the phase, and $\delta$ is the phase constant (obviously, you can add any integer multiple of $2\pi$ to $\delta$ without changing f(z,t); ordinarily, one uses...	{ "Header 1": "9.1.1 The Wave Equation", "token_count": 774, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$e^{i\theta} = \cos\theta + i\sin\theta,\tag{9.15}$$ the sinusoidal wave (Eq. 9.12) can be written $$f(z,t) = \operatorname{Re}\left[Ae^{i(kz-\omega t + \delta)}\right],\tag{9.16}$$ where Re(ξ ) denotes the real part of the complex number ξ . This invites us to introduce the complex wave function $$\tilde{...	{ "Header 1": "(ii) Complex notation. In view of Euler's formula,", "token_count": 269, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$f_3 = f_1 + f_2 = \text{Re}(\tilde{f}_1) + \text{Re}(\tilde{f}_2) = \text{Re}(\tilde{f}_1 + \tilde{f}_2) = \text{Re}(\tilde{f}_3),$$ with ˜f<sup>3</sup> = ˜f<sup>1</sup> + ˜f2. You simply add the corresponding complex wave functions, and then take the real part. In particular, if they have the same frequenc...	{ "Header 1": "Example 9.1. Suppose you want to combine two sinusoidal waves:", "token_count": 765, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
So far, I have assumed the string is infinitely long—or at any rate long enough that we don't need to worry about what happens to a wave when it reaches the end. As a matter of fact, what happens depends a lot on how the string is attached—that is, on the specific boundary conditions to which the wave is subject. Sup...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "token_count": 1877, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{9.31}$$ If the second string is lighter than the first $(\mu_2 < \mu_1$ , so that $v_2 > v_1)$ , all three waves have the same phase angle $(\delta_R = \delta_T = \delta_I)$ , and the outgoing amplitudes are $$A_R = \left(\frac{v_2 - v_1}{v_2 + v_1}\right) A_I, \quad A_T = \left(\frac{2v_2}{v_2 + v_1}\rig...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "token_count": 1343, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
<sup>2</sup>Notice that you can always switch the sign of <sup>n</sup>ˆ, provided you simultaneously advance the phase constant by 180◦, since both operations change the sign of the wave. In terms of the polarization angle θ, $$\hat{\mathbf{n}} = \cos\theta \,\hat{\mathbf{x}} + \sin\theta \,\hat{\mathbf{y...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "Header 3": "FIGURE 9.8", "token_count": 496, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In regions of space where there is no charge or current, Maxwell's equations read (i) $$\nabla \cdot \mathbf{E} = 0$$ , (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ , (ii) $\nabla \cdot \mathbf{B} = 0$ , (iv) $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\par...	{ "Header 1": "9.2.1 The Wave Equation for E and B", "token_count": 1047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
For reasons discussed in Sect. 9.1.2, we may confine our attention to sinusoidal waves of frequency ω. Since different frequencies in the visible range correspond to different colors, such waves are called monochromatic (Table 9.1). Suppose, <sup>4</sup>As Maxwell himself put it, "We can scarcely avoid the infe...	{ "Header 1": "9.2.2 Monochromatic Plane Waves", "token_count": 2032, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The wave as a whole is said to be polarized in the x direction (by convention, we use the direction of E to specify the polarization of an electromagnetic wave). There is nothing special about the z direction, of course—we can easily generalize to monochromatic plane waves traveling in an arbitrary direction....	{ "Header 1": "9.2.2 Monochromatic Plane Waves", "token_count": 677, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
According to Eq. 8.5, the energy per unit volume in electromagnetic fields is $$u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right). \tag{9.53}$$ In the case of a monochromatic plane wave (Eq. 9.48) $$B^2 = \frac{1}{c^2} E^2 = \mu_0 \epsilon_0 E^2, \tag{9.54}$$ so the *electric and magnetic cont...	{ "Header 1": "9.2.3 Energy and Momentum in Electromagnetic Waves", "token_count": 1823, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
[Note that this only works if the two waves have the same $\mathbf{k}$ and $\omega$ , but they need not have the same amplitude or phase.] For example, $$\langle u \rangle = \frac{1}{4} \operatorname{Re} \left( \epsilon_0 \tilde{\mathbf{E}} \cdot \tilde{\mathbf{E}}^* + \frac{1}{\mu_0} \tilde{\mathbf{B}} \cdot \til...	{ "Header 1": "9.2.3 Energy and Momentum in Electromagnetic Waves", "token_count": 1964, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose the x y plane forms the boundary between two linear media. A plane wave of frequency ω, traveling in the z direction and polarized in the x direction, approaches the interface from the left (Fig. 9.13): $$\tilde{\mathbf{E}}_{I}(z,t) = \tilde{E}_{0_{I}} e^{i(k_{1}z - \omega t)} \,\hat{\mathbf{x}}, \tilde...	{ "Header 1": "9.3.2 Reflection and Transmission at Normal Incidence", "token_count": 1938, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section, I treated reflection and transmission at normal incidence—that is, when the incoming wave hits the interface head-on. We now turn to the more general case of oblique incidence, in which the incoming wave meets the boundary at an arbitrary angle θ<sup>I</sup> (Fig. 9.14). Of course, normal inc...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 1786, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.74) become: (i) $$\epsilon_{1} \left( \tilde{\mathbf{E}}_{0_{I}} + \tilde{\mathbf{E}}_{0_{R}} \right)_{z} = \epsilon_{2} \left( \tilde{\mathbf{E}}_{0_{T}} \right)_{z}$$ (ii) $\left( \tilde{\mathbf{B}}_{0_{I}} + \tilde{\mathbf{B}}_{0_{R}} \right)_{z} = \left( \tilde{\mathbf{B}}_{0_{T}} \right)_{z}$ (iii) $\left(...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 2028, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
![](_page_429_Figure_2.jpeg) FIGURE 9.17 (The cosines are there because I am talking about the average power per unit area of interface, and the interface is at an angle to the wave front.) The reflection and transmission coefficients for waves polarized parallel to the plane of incidence are $$R \equiv \...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 854, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In Sect. 9.3 I stipulated that the free charge density $\rho_f$ and the free current density $\mathbf{J}_f$ are zero, and everything that followed was predicated on that assumption. Such a restriction is perfectly reasonable when you're talking about wave propagation through a vacuum or through insulating materials...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "token_count": 1969, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
We may as well orient our axes so that $\mathbf{E}$ is polarized along the x direction: $$\tilde{\mathbf{E}}(z,t) = \tilde{E}_0 e^{-\kappa z} e^{i(kz - \omega t)} \,\hat{\mathbf{x}}.\tag{9.130}$$ Then (iii) gives $$\tilde{\mathbf{B}}(z,t) = \frac{\tilde{k}}{\omega} \tilde{E}_0 e^{-\kappa z} e^{i(kz - \omega t)}...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "token_count": 715, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
- (a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface? - (b) Silver is an excellent conductor, but it's expensive. Suppose you were designing a microwave experiment to operate at a frequency of 10<sup>10</sup> Hz. How thick would you make the...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "Header 2": "Problem 9.19", "token_count": 476, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The boundary conditions we used to analyze reflection and refraction at an interface between two dielectrics do not hold in the presence of free charges and currents. Instead, we have the more general relations (Eq. 7.64): (i) $$\epsilon_1 E_1^{\perp} - \epsilon_2 E_2^{\perp} = \sigma_f$$ , (iii) $\mathbf{E}_1^{\par...	{ "Header 1": "9.4.2 ■ Reflection at a Conducting Surface", "token_count": 1581, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the preceding sections, we have seen that the propagation of electromagnetic waves through matter is governed by three properties of the material: the permittivity $\epsilon$ , the permeability $\mu$ , and the conductivity $\sigma$ . Actually, each of these parameters depends to some extent on the frequency of th...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 2039, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{9.157}$$ The resulting dipole moment is the real part of $$\tilde{p}(t) = q\tilde{x}(t) = \frac{q^2/m}{\omega_0^2 - \omega^2 - i\gamma\omega} E_0 e^{-i\omega t}.$$ (9.158) The imaginary term in the denominator means that p is out of phase with E—lagging behind by an angle $\tan^{-1}[\gamma\omega/(\omega_0^2...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 2019, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
If you agree to stay away from the resonances, the damping can be ignored, and the formula for the index of refraction simplifies: $$n = 1 + \frac{Nq^2}{2m\epsilon_0} \sum_{j} \frac{f_j}{\omega_j^2 - \omega^2}.$$ (9.172) For most substances the natural frequencies $\omega_j$ are scattered all over the spectrum ...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 1941, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
As we shall soon see, confined waves are not (in general) transverse; in order to fit the boundary conditions we shall have to include longitudinal components $(E_z \text{ and } B_z)^{23}$ $$\tilde{\mathbf{E}}_0 = E_x \,\hat{\mathbf{x}} + E_y \,\hat{\mathbf{y}} + E_z \,\hat{\mathbf{z}}, \qquad \tilde{\mathbf{B...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 1535, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose we have a wave guide of rectangular shape (Fig. 9.24), with height a and width b, and we are interested in the propagation of TE waves. The problem is to solve Eq. 9.181ii, subject to the boundary condition 9.175ii. We'll do it by separation of variables. Let $$B_{\tau}(x, y) = X(x)Y(y),$$ so that $$Y\fra...	{ "Header 1": "9.5.2 ■ TE Waves in a Rectangular Wave Guide", "token_count": 2052, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.179) yield $$k = \omega/c \tag{9.193}$$ (so the waves travel at speed c, and are nondispersive), $$cB_{y} = E_{x} \quad \text{and} \quad cB_{x} = -E_{y} \tag{9.194}$$ (so E and B are mutually perpendicular), and (together with $\nabla \cdot \mathbf{E} = 0$ , $\nabla \cdot \mathbf{B} = 0$ ): $$\fra...	{ "Header 1": "9.5.2 ■ TE Waves in a Rectangular Wave Guide", "token_count": 519, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
! Problem 9.33 The "inversion theorem" for Fourier transforms states that $$\tilde{\phi}(z) = \int_{-\infty}^{\infty} \tilde{\Phi}(k)e^{ikz} dk \quad \Longleftrightarrow \quad \tilde{\Phi}(k) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{\phi}(z)e^{-ikz} dz. \quad (9.198)$$ Use this to determine A˜(k), in...	{ "Header 1": "More Problems on Chapter 9", "token_count": 2039, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.22).] - (c) Do the same for polarization perpendicular to the plane of incidence (use the results of Prob. 9.17). - (d) In the case of polarization perpendicular to the plane of incidence, show that the (real) evanescent fields are $$\mathbf{E}(\mathbf{r},t) = E_0 e^{-\kappa z} \cos(kx - \omega t) \,\hat{\mathbf{y}...	{ "Header 1": "More Problems on Chapter 9", "token_count": 396, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 10.1 ■ THE POTENTIAL FORMULATION #### 10.1.1 ■ Scalar and Vector Potentials In this chapter we seek the general solution to Maxwell's equations, (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ , (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ , (ii) $\nabla \cdot \mat...	{ "Header 1": "Potentials and Fields", "token_count": 1965, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
10.4 and 10.5) can be written in the more symmetrical form $$\Box^{2}V + \frac{\partial L}{\partial t} = -\frac{1}{\epsilon_{0}}\rho,$$ $$\Box^{2}\mathbf{A} - \nabla L = -\mu_{0}\mathbf{J},$$ (10.6) where $$\Box^2 \equiv \nabla^2 - \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} \quad \text{and} \quad L \equiv...	{ "Header 1": "Potentials and Fields", "token_count": 400, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Equations 10.4 and 10.5 are ugly, and you might be inclined to abandon the potential formulation altogether. However, we have succeeded in reducing six problems—finding E and B (three components each)—down to four: V (one component) and A (three more). Moreover, Eqs. 10.2 and 10.3 do not uniquely defi...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2032, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(called the d'Alembertian) occurs in both equations: (i) $$\Box^2 V = -\frac{1}{\epsilon_0} \rho$$ , (ii) $\Box^2 \mathbf{A} = -\mu_0 \mathbf{J}$ . This democratic treatment of V and A is especially nice in the context of special relativity, where the d'Alembertian is the natural generalization of the Laplac...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2051, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 10.9 Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.] #### 10.2 ■ CONTINUOUS DISTRIBUTIONS #### 10.2.1 ■ Retarded Potentials In the static case, Eq. 10.16 reduces to (four copies of) Poisson's equation, $$\nabla^2 V = -\frac{1}{\epsilon_0} \rho, \quad \nabla^2 \mathbf{A} = -\mu...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2038, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
1.63), whereas $$\nabla \cdot \left(\frac{\hat{\mathbf{\lambda}}}{r^2}\right) = 4\pi \,\delta^3(\mathbf{\lambda})$$ (Eq. 1.100). So $$\nabla^2 V = \frac{1}{4\pi\epsilon_0} \int \left[ \frac{1}{c^2} \frac{\ddot{\rho}}{\imath} - 4\pi\rho \delta^3(\mathbf{z}) \right] d\tau' = \frac{1}{c^2} \frac{\partial^2 V}{\parti...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 545, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$I(t) = \begin{cases} 0, & \text{for } t \le 0, \\ I_0, & \text{for } t > 0. \end{cases}$$ That is, a constant current $I_0$ is turned on abruptly at t = 0. Find the resulting electric and magnetic fields. ![](_page_465_Picture_6.jpeg) FIGURE 10.4 #### Solution The wire is presumably electrically n...	{ "Header 1": "Example 10.2. An infinite straight wire carries the current", "token_count": 1375, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Given the retarded potentials $$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r}',t_r)}{\imath} d\tau', \quad \mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}',t_r)}{\imath} d\tau', \quad (10.33)$$ it is, in principle, a straightforward matter to determine the fiel...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 1943, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 10.3 ■ POINT CHARGES #### 10.3.1 ■ Liénard-Wiechert Potentials My next project is to calculate the (retarded) potentials, $V(\mathbf{r}, t)$ and $\mathbf{A}(\mathbf{r}, t)$ , of a point charge q that is moving on a specified trajectory $$\mathbf{w}(t) \equiv \text{position of } q \text{ at time } t.$$ (...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 2051, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Moreover, since the current density is $\rho \mathbf{v}$ (Eq. 5.26), the vector potential is $$\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \frac{\rho(\mathbf{r}',t_r)\mathbf{v}(t_r)}{\imath} d\tau' = \frac{\mu_0}{4\pi} \frac{\mathbf{v}}{\imath} \int \rho(\mathbf{r}',t_r) d\tau',$$ or $$\mathbf{A}(\mathbf...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 2025, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
We are now in a position to calculate the electric and magnetic fields of a point charge in arbitrary motion, using the Liénard-Wiechert potentials:17 $$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \frac{qc}{(ic - \mathbf{v} \cdot \mathbf{v})}, \quad \mathbf{A}(\mathbf{r},t) = \frac{\mathbf{v}}{c^2} V(\mathbf{r},t), \q...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 2046, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{10.64}$$ Putting all this back into Eq. 10.58, and using the "BAC-CAB" rule to reduce the triple cross products, $$\nabla(\mathbf{z} \cdot \mathbf{v}) = \mathbf{a}(\mathbf{z} \cdot \nabla t_r) + \mathbf{v} - \mathbf{v}(\mathbf{v} \cdot \nabla t_r) - \mathbf{z} \times (\mathbf{a} \times \nabla t_r) + \mathbf{v}...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
As we shall see in Chapter 11, it is this term that is responsible for electromagnetic radiation; accordingly, it is called the radiation field—or, since it is proportional to a, the acceleration field. The same terminology applies to the magnetic field. Back in Chapter 2, I commented that if we could write d...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 1908, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 10.24 Suppose you take a plastic ring of radius a and glue charge on it, so that the line charge density is λ0\|sin(θ/2)\|. Then you spin the loop about its axis at an angular velocity ω. Find the (exact) scalar and vector potentials at the center of the ring. Answer: A = (μ0λ0ωa/3π ) sin[ω(t − *a...	{ "Header 1": "More Problems on Chapter 10", "token_count": 1770, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \frac{\hat{\mathbf{r}}}{r^2} \cdot [\mathbf{p} + (r/c)\dot{\mathbf{p}}]$$ $$\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \left[ \frac{\dot{\mathbf{p}}}{r} \right]$$ $$\mathbf{E}(\mathbf{r},t) = -\frac{\mu_0}{4\pi} \left\{ \frac{\ddot{\mathbf{p}} - \hat{\mathbf{r}}(\ha...	{ "Header 1": "More Problems on Chapter 10", "token_count": 372, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 11.1 ■ DIPOLE RADIATION #### 11.1.1 ■ What is Radiation? When charges accelerate, their fields can transport energy irreversibly out to infinity—a process we call radiation.<sup>1</sup> Let us assume the source is localized<sup>2</sup> near the origin; we would like to calculate the energy it is radiatin...	{ "Header 1": "Radiation", "token_count": 945, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Picture two tiny metal spheres separated by a distance d and connected by a fine wire (Fig. 11.2); at time t the charge on the upper sphere is q(t), and the charge on the lower sphere is −q(t). Suppose that we drive the charge back and forth through the wire, from one end to the other, at an angular frequen...	{ "Header 1": "11.1.2 Electric Dipole Radiation", "token_count": 1888, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\end{split}$$ (I dropped the first and last terms, in accordance with approximation 3.) Likewise, $$\frac{\partial \mathbf{A}}{\partial t} = -\frac{\mu_0 p_0 \omega^2}{4\pi r} \cos[\omega (t - r/c)](\cos \theta \,\hat{\mathbf{r}} - \sin \theta \,\hat{\boldsymbol{\theta}}),$$ and therefore $$\mathbf{E} = -\nabla...	{ "Header 1": "11.1.2 Electric Dipole Radiation", "token_count": 2005, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose now that we have a wire loop of radius b (Fig. 11.8), around which we drive an alternating current: $$I(t) = I_0 \cos(\omega t). \tag{11.23}$$ This is a model for an oscillating magnetic dipole, $$\mathbf{m}(t) = \pi b^2 I(t) \,\hat{\mathbf{z}} = m_0 \cos(\omega t) \,\hat{\mathbf{z}},\tag{11.24}$$ !...	{ "Header 1": "11.1.3 Magnetic Dipole Radiation", "token_count": 1959, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The energy flux for magnetic dipole radiation is $$\mathbf{S}(\mathbf{r},t) = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B}) = \frac{\mu_0}{c} \left\{ \frac{m_0 \omega^2}{4\pi c} \left( \frac{\sin \theta}{r} \right) \cos[\omega(t - r/c)] \right\}^2 \hat{\mathbf{r}}, \quad (11.38)$$ the intensity is $$\langle \m...	{ "Header 1": "11.1.3 Magnetic Dipole Radiation", "token_count": 887, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the previous sections, we studied the radiation produced by two specific systems: oscillating electric dipoles and oscillating magnetic dipoles. Now I want to apply the same procedures to a configuration of charge and current that is entirely arbitrary, except that it is localized within some finite volume near the ...	{ "Header 1": "11.1.4 Radiation from an Arbitrary Source", "token_count": 1769, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
11.48 it follows that $$\nabla t_0 = -\frac{1}{c} \nabla r = -\frac{1}{c} \hat{\mathbf{r}},$$ and hence $$\nabla V \cong \nabla \left[ \frac{1}{4\pi\epsilon_0} \frac{\hat{\mathbf{r}} \cdot \dot{\mathbf{p}}(t_0)}{rc} \right] \cong \frac{1}{4\pi\epsilon_0} \left[ \frac{\hat{\mathbf{r}} \cdot \ddot{\mathbf{p}}(t_0)}...	{ "Header 1": "11.1.4 Radiation from an Arbitrary Source", "token_count": 1105, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) In the case of an oscillating electric dipole, $$p(t) = p_0 \cos(\omega t), \quad \ddot{p}(t) = -\omega^2 p_0 \cos(\omega t),$$ and we recover the results of Sect. 11.1.2. (b) For a single point charge q, the dipole moment is $$\mathbf{p}(t) = q\mathbf{d}(t),$$ where d is the position of q with re...	{ "Header 1": "Example 11.2.", "token_count": 2045, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(It's like the flies that stay with the garbage truck.) Now $\mathbf{E}_{rad}$ is perpendicular to $\hat{\lambda}$ , so the second term in Eq. 11.64 vanishes: $$\mathbf{S}_{\text{rad}} = \frac{1}{\mu_0 c} E_{\text{rad}}^2 \, \hat{\mathbf{z}}. \tag{11.67}$$ If the charge is instantaneously at rest (at time $t_...	{ "Header 1": "Example 11.2.", "token_count": 1847, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### Solution In this case $(\mathbf{u} \times \mathbf{a}) = c(\hat{\mathbf{a}} \times \mathbf{a})$ , so $$\frac{dP}{d\Omega} = \frac{q^2c^2}{16\pi^2\epsilon_0} \frac{\|\hat{\mathbf{i}} \times (\hat{\mathbf{i}} \times \mathbf{a})\|^2}{(c - \hat{\mathbf{i}} \cdot \mathbf{v})^5}.$$ Now $$\hat{\mathbf{a}} \ti...	{ "Header 1": "Example 11.2.", "token_count": 2009, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }

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Since B is due in part to bound currents (which we don't yet know), we cannot compute it directly. However, this is one of those symmetrical cases in which we can get H from the free current alone, using Ampère's law in the form of Eq. 6.20: $$\mathbf{H} = nI \,\hat{\mathbf{z}}$$ (Fig. 6.22). According to E...	{ "Header 1": "Solution", "token_count": 1061, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In a linear medium, the alignment of atomic dipoles is maintained by a magnetic field imposed from the outside. Ferromagnets—which are emphatically not linear11—require no external fields to sustain the magnetization; the alignment is "frozen in." Like paramagnetism, ferromagnetism involves the magnetic dipoles assoc...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 2047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
![](_page_310_Picture_2.jpeg) FIGURE 6.30 (b) Show that the interaction energy of two magnetic dipoles separated by a displacement $\mathbf{r}$ is given by $$U = \frac{\mu_0}{4\pi} \frac{1}{r^3} [\mathbf{m}_1 \cdot \mathbf{m}_2 - 3(\mathbf{m}_1 \cdot \hat{\mathbf{r}})(\mathbf{m}_2 \cdot \hat{\mathbf{r}})]...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 1986, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Show that the magnetic field inside the sphere (0 < r < R) is $$\frac{\mu}{4\pi} \left\{ \frac{1}{r^3} [3(\mathbf{m} \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{m}] + \frac{2(\mu_0 - \mu)\mathbf{m}}{(2\mu_0 + \mu)R^3} \right\}.$$ What is the field outside the sphere? ! Problem 6.29 You are asked to...	{ "Header 1": "6.4.2 Ferromagnetism", "token_count": 460, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 7.1 ■ ELECTROMOTIVE FORCE #### 7.1.1 ■ Ohm's Law To make a current flow, you have to push on the charges. How fast they move, in response to a given push, depends on the nature of the material. For most substances, the current density J is proportional to the force per unit charge, f: $$\mathbf{J} = \s...	{ "Header 1": "Electrodynamics", "token_count": 1312, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The field between the cylinders is $$\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 s} \hat{\mathbf{s}},$$ where λ is the charge per unit length on the inner cylinder. The current is therefore $$I = \int \mathbf{J} \cdot d\mathbf{a} = \sigma \int \mathbf{E} \cdot d\mathbf{a} = \frac{\sigma}{\epsilon_0} \lambda L.$$ ...	{ "Header 1": "Solution", "token_count": 2043, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
How do you account for that? Exploit this observation to determine the current flowing between two metal spheres, each of radius a, immersed deep in the sea and held quite far apart (Fig. 7.4b), if the potential difference between them is V. (This arrangement can be used to measure the conductivity of sea water.) ...	{ "Header 1": "Solution", "token_count": 753, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
If you think about a typical electric circuit—a battery hooked up to a light bulb, say (Fig. 7.7)—a perplexing question arises: In practice, the current is the same all the way around the loop; why is this the case, when the only obvious driving force is inside the battery? Off hand, you might expect a large current ...	{ "Header 1": "7.1.2 Electromotive Force", "token_count": 1761, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section, I listed several possible sources of electromotive force, batteries being the most familiar. But I did not mention the commonest one of all: the generator. Generators exploit motional emfs, which arise when you move a wire through a magnetic field. Figure 7.10 suggests a primitive model f...	{ "Header 1": "7.1.3 Motional emf", "token_count": 2031, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The flux rule assumes you have a single wire loop—it can move, rotate, stretch, or distort (continuously), but beware of switches, sliding contacts, or extended conductors allowing a variety of current paths. A standard "flux rule paradox" involves the circuit in Figure 7.14. When the switch is thrown (from a to b)...	{ "Header 1": "7.1.3 Motional emf", "token_count": 221, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The speed of a point on the disk at a distance s from the axis is v = ωs, so the force per unit charge is fmag = v × B = ωs Bsˆ. The emf is therefore $$\mathcal{E} = \int_0^a f_{\text{mag}} \, ds = \omega B \int_0^a s \, ds = \frac{\omega B a^2}{2},$$ and the current is $$I = \frac{\mathcal{...	{ "Header 1": "Solution", "token_count": 1251, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In 1831 Michael Faraday reported on a series of experiments, including three that (with some violence to history) can be characterized as follows: Experiment 1. He pulled a loop of wire to the right through a magnetic field (Fig. 7.21a). A current flowed in the loop. Experiment 2. He moved the magnet to t...	{ "Header 1": "7.2.1 Faraday's Law", "token_count": 426, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
It is this induced<sup>8</sup> electric field that accounts for the emf in Experiment 2.<sup>9</sup> Indeed, if (as Faraday found empirically) the emf is again equal to the rate of change of the flux, $$\mathcal{E} = \oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi}{dt},\tag{7.14}$$ then E is related to the change...	{ "Header 1": "A changing magnetic field induces an electric field.", "token_count": 674, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$\mathcal{E} = -\frac{d\Phi}{dt} \tag{7.17}$$ #### will appear in the loop. Many people call this "Faraday's law." Maybe I'm overly fastidious, but I find this confusing. There are really two totally different mechanisms underlying Eq. 7.17, and to identify them both as "Faraday's law" is a little like saying ...	{ "Header 1": "Whenever (and for whatever reason) the magnetic flux through a loop changes, an emf", "token_count": 729, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The induced current will flow in such a direction that the flux it produces tends to cancel the change. (As the front end of the magnet in Ex. 7.5 enters the ring, the flux increases, so the current in the ring must generate a field to the right—it therefore flows clockwise.) Notice that it is the change in flu...	{ "Header 1": "Nature abhors a change in flux.", "token_count": 369, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Before you turned on the current, the flux through the ring was zero. Afterward a flux appeared (upward, in the diagram), and the emf generated in the ring led to a current (in the ring) which, according to Lenz's law, was in such a direction that its field tended to cancel this new flux. This means that the cu...	{ "Header 1": "Solution", "token_count": 2025, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
To convince you of this, I deliberately set things up so that the magnetic field is zero at the location of the charge. The experimenter may tell you she never put in any electric field—all she did was switch off the magnetic field. But when she did that, an electric field automatically appeared, and it's this el...	{ "Header 1": "Solution", "token_count": 2038, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Some of the field lines pass <sup>&</sup>lt;sup>16</sup>This paradox was suggested by Tom Colbert. Refer to Problem 2.55. ![](_page_340_Picture_2.jpeg) FIGURE 7.30 FIGURE 7.31 through loop 2; let <sup>2</sup> be the flux of B<sup>1</sup> through 2. You might have a tough time actually *calculating...	{ "Header 1": "Solution", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Example 7.12. Suppose a current I is flowing around a loop, when someone suddenly cuts the wire. The current drops "instantaneously" to zero. This generates a whopping back emf, for although I may be small, d I/dt is enormous. (That's why you sometimes draw a spark when you unplug an iron or toaster elect...	{ "Header 1": "Solution", "token_count": 1401, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
It takes a certain amount of energy to start a current flowing in a circuit. I'm not talking about the energy delivered to the resistors and converted into heat—that is irretrievably lost, as far as the circuit is concerned, and can be large or small, depending on how long you let the current run. What I am concerned w...	{ "Header 1": "7.2.4 Energy in Magnetic Fields", "token_count": 1777, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
According to Ampère's law, the field between the cylinders is $$\mathbf{B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol{\phi}}.$$ Elsewhere, the field is zero. Thus, the energy per unit volume is $$\frac{1}{2\mu_0} \left( \frac{\mu_0 I}{2\pi s} \right)^2 = \frac{\mu_0 I^2}{8\pi^2 s^2}.$$ The energy in a cylindrica...	{ "Header 1": "Solution", "token_count": 1319, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
So far, we have encountered the following laws, specifying the divergence and curl of electric and magnetic fields: (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ (Gauss's law), (ii) $$\nabla \cdot \mathbf{B} = 0$$ (no name), (iii) $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$...	{ "Header 1": "7.3.1 Electrodynamics Before Maxwell", "token_count": 896, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The problem is on the right side of Eq. 7.36, which should be zero, but isn't. Applying the continuity equation (5.29) and Gauss's law, the offending term can be rewritten: $$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial t} (\epsilon_0 \nabla \cdot \mathbf{E}) = -\nabla \...	{ "Header 1": "7.3.2 ■ How Maxwell Fixed Ampère's Law", "token_count": 493, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
<sup>&</sup>lt;sup>20</sup>For the history of this subject, see A. M. Bork, Am. J. Phys. 31, 854 (1963). <sup>&</sup>lt;sup>21</sup>See footnote 8 (page 313) for commentary on the word "induce." The same issue arises here: Should a changing electric field be regarded as an independent source of magnetic field (al...	{ "Header 1": "A changing electric field induces a magnetic field.", "token_count": 737, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The inner one (radius a) carries a charge Q(t), and the outer one (radius b) an opposite charge -Q(t). The space between them is filled with Ohmic material of conductivity $\sigma$ , so a radial current flows: $$\mathbf{J} = \sigma \mathbf{E} = \sigma \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \,\hat{\mathbf{r}}; \quad ...	{ "Header 1": "A changing electric field induces a magnetic field.", "Header 2": "Example 7.14. Imagine two concentric metal spherical shells (Fig. 7.44).", "token_count": 1148, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section we put the finishing touches on Maxwell's equations: (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ (Gauss's law), (ii) $\nabla \cdot \mathbf{B} = 0$ (no name), (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ (Faraday's law), (iv) $\nabla \times \mathbf...	{ "Header 1": "7.3.3 ■ Maxwell's Equations", "token_count": 932, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
There is a pleasing symmetry to Maxwell's equations; it is particularly striking in free space, where ρ and J vanish: $$\nabla \cdot \mathbf{E} = 0, \qquad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},$$ $$\nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \fra...	{ "Header 1": "7.3.4 Magnetic Charge", "token_count": 1354, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Maxwell's equations in the form 7.40 are complete and correct as they stand. However, when you are working with materials that are subject to electric and magnetic polarization there is a more convenient way to write them. For inside polarized matter there will be accumulations of "bound" charge and current, over whi...	{ "Header 1": "7.3.5 ■ Maxwell's Equations in Matter", "token_count": 1904, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In general, the fields E, B, D, and H will be discontinuous at a boundary between two different media, or at a surface that carries a charge density $\sigma$ or a current density K. The explicit form of these discontinuities can be deduced from Maxwell's equations (7.56), in their integral form ...	{ "Header 1": "7.3.6 ■ Boundary Conditions", "token_count": 1942, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
[Answer: (0V0/πa)tan(φ/2)] Problem 7.43 The magnetic field outside a long straight wire carrying a steady current I is $$\mathbf{B} = \frac{\mu_0}{2\pi} \frac{I}{s} \,\hat{\boldsymbol{\phi}}.$$ The electric field inside the wire is uniform: $$\mathbf{E} = \frac{I\rho}{\pi a^2} \,\hat{\mathbf{z}},$...	{ "Header 1": "7.3.6 ■ Boundary Conditions", "token_count": 1826, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) Referring to Prob. 5.52(a) and Eq. 7.18, show that $$\mathbf{E} = -\frac{\partial \mathbf{A}}{\partial t},\tag{7.66}$$ for Faraday-induced electric fields. Check this result by taking the divergence and curl of both sides. (b) A spherical shell of radius R carries a uniform surface charge σ. It spins about ...	{ "Header 1": "Problem 7.49", "token_count": 1364, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
- (a) Use the Neumann formula (Eq. 7.23) to calculate the mutual inductance of the configuration in Fig. 7.37, assuming a is very small (a b, a z). Compare your answer to Prob. 7.22. - (b) For the general case (not assuming a is small), show that $$M = \frac{\mu_0 \pi \beta}{2} \sqrt{ab\beta} \left( 1 + \fr...	{ "Header 1": "Problem 7.56", "token_count": 2047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The current travels down one strip and back along the other. In each case, it spreads out uniformly over the surface of the ribbon. - (a) Find the capacitance per unit length, C. - (b) Find the inductance per unit length, $\mathcal{L}$ . - (c) What is the product $\mathcal{LC}$ , numerically? [ $\mathcal{L}$ and ...	{ "Header 1": "Problem 7.56", "token_count": 1364, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
All of our cards are now on the table, and in a sense my job is done. In the first seven chapters we assembled electrodynamics piece by piece, and now, with Maxwell's equations in their final form, the theory is complete. There are no more laws to be learned, no further generalizations to be considered, and (with perha...	{ "Header 1": "Intermission", "token_count": 250, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In this chapter we study conservation of energy, momentum, and angular momentum, in electrodynamics. But I want to begin by reviewing the conservation of charge, because it is the paradigm for all conservation laws. What precisely does conservation of charge tell us? That the total charge in the universe is constant?...	{ "Header 1": "8.1.1 The Continuity Equation", "token_count": 568, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In Chapter 2, we found that the work necessary to assemble a static charge distribution (against the Coulomb repulsion of like charges) is (Eq. 2.45) $$W_{\rm e} = \frac{\epsilon_0}{2} \int E^2 d\tau,$$ where E is the resulting electric field. Likewise, the work required to get currents going (against the back ...	{ "Header 1": "8.1.2 Poynting's Theorem", "token_count": 1885, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{8.11}$$ What if no work is done on the charges in V—what if, for example, we are in a region of empty space, where there is no charge? In that case dW/dt = 0, so $$\int \frac{\partial u}{\partial t} d\tau = -\oint \mathbf{S} \cdot d\mathbf{a} = -\int (\mathbf{\nabla} \cdot \mathbf{S}) d\tau,$$ and hence ...	{ "Header 1": "8.1.2 Poynting's Theorem", "token_count": 898, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Imagine a point charge q traveling in along the x axis at a constant speed v. Because it is moving, its electric field is not given by Coulomb's law; nevertheless, $\mathbf{E}$ still points radially outward from the instantaneous position of the charge (Fig. 8.2a), as we'll see in Chapter 10. Since, moreover, a mov...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 2015, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Thus $$T_{xx} = \frac{1}{2} \epsilon_0 \left( E_x^2 - E_y^2 - E_z^2 \right) + \frac{1}{2\mu_0} \left( B_x^2 - B_y^2 - B_z^2 \right),$$ $$T_{xy} = \epsilon_0 (E_x E_y) + \frac{1}{\mu_0} (B_x B_y),$$ and so on. Because it carries two indices, where a vector has only one, $T_{ij}$ is sometimes written with a d...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 1655, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{8.23}$$ Meanwhile, for the equatorial disk, $$d\mathbf{a} = -r \, dr \, d\phi \, \hat{\mathbf{z}},\tag{8.24}$$ and (since we are now inside the sphere) $$\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^3} \mathbf{r} = \frac{1}{4\pi\epsilon_0} \frac{Q}{R^3} r(\cos\phi \,\hat{\mathbf{x}} + \sin\phi \,\hat...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 2043, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Sands, The Feynman Lectures on Physics (Reading, Mass.: Addison-Wesley, 1964), Vol. II, Section 27-6. Similarly, $\overrightarrow{T}$ plays a dual role: $\overrightarrow{T}$ itself is the electromagnetic stress (force per unit area) acting on a surface, and $-\overrightarrow{T}$ describes the flow of momentum...	{ "Header 1": "8.1.2 Poynting's Theorem", "Header 2": "8.2.1 ■ Newton's Third Law in Electrodynamics", "token_count": 1728, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
By now, the electromagnetic fields (which started out as mediators of forces between charges) have taken on a life of their own. They carry energy (Eq. 8.5) $$u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right), \tag{8.31}$$ and momentum (Eq. 8.29) $$\mathbf{g} = \epsilon_0 (\mathbf{E} \times \m...	{ "Header 1": "8.2.4 ■ Angular Momentum", "token_count": 2046, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
This is perhaps a good place to revisit the old paradox that magnetic forces do no work (Eq. 5.11). What about that magnetic crane lifting the carcass of a junked car? Somebody is doing work on the car, and if it's not the magnetic field, who <sup>10</sup>In Ex. 8.4 we turned the current off slowly, to keep things ...	{ "Header 1": "8.3 MAGNETIC FORCES DO NO WORK<sup>12</sup>", "token_count": 1927, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Meanwhile, however, a Faraday emf is induced in the upper loop, due to the changing flux from the lower loop: $$\Phi_b = M I_a \implies \mathcal{E}_b = -I_a \frac{dM}{dt},$$ and the work done by the power supply in ring b (to sustain the current Ib) is $$dW_b = -\mathcal{E}_b I_b dt = \frac{3\pi}{2} \mu...	{ "Header 1": "8.3 MAGNETIC FORCES DO NO WORK<sup>12</sup>", "token_count": 1276, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 8.13<sup>16</sup> A very long solenoid of radius a, with n turns per unit length, carries a current Is. Coaxial with the solenoid, at radius b a, is a circular ring of wire, with resistance R. When the current in the solenoid is (gradually) decreased, a current Ir is induced in the ring. - (a)...	{ "Header 1": "More Problems on Chapter 8", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
8.45 is valid only for ideal dipoles. But g is linear in B, and therefore, if E is held fixed, obeys the superposition principle: For a collection of magnetic dipoles, the total momentum is the (vector) sum of the momenta for each one separately. In particular, if E is uniform over a localized ste...	{ "Header 1": "More Problems on Chapter 8", "token_count": 373, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) Carry through the argument in Sect. 8.1.2, starting with Eq. 8.6, but using J <sup>f</sup> in place of J. Show that the Poynting vector becomes $$\mathbf{S} = \mathbf{E} \times \mathbf{H},\tag{8.46}$$ and the rate of change of the energy density in the fields is $$\frac{\partial u}{\partial t} = \ma...	{ "Header 1": "Problem 8.23", "token_count": 874, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
What is a "wave"? I don't think I can give you an entirely satisfactory answer—the concept is intrinsically somewhat vague—but here's a start: A wave is a disturbance of a continuous medium that propagates with a fixed shape at constant velocity. Immediately I must add qualifiers: In the presence of absorption, the w...	{ "Header 1": "9.1.1 The Wave Equation", "token_count": 2030, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
A is the amplitude of the wave (it is positive, and represents the maximum displacement from equilibrium). The argument of the cosine is called the phase, and $\delta$ is the phase constant (obviously, you can add any integer multiple of $2\pi$ to $\delta$ without changing f(z,t); ordinarily, one uses...	{ "Header 1": "9.1.1 The Wave Equation", "token_count": 774, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$e^{i\theta} = \cos\theta + i\sin\theta,\tag{9.15}$$ the sinusoidal wave (Eq. 9.12) can be written $$f(z,t) = \operatorname{Re}\left[Ae^{i(kz-\omega t + \delta)}\right],\tag{9.16}$$ where Re(ξ ) denotes the real part of the complex number ξ . This invites us to introduce the complex wave function $$\tilde{...	{ "Header 1": "(ii) Complex notation. In view of Euler's formula,", "token_count": 269, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$f_3 = f_1 + f_2 = \text{Re}(\tilde{f}_1) + \text{Re}(\tilde{f}_2) = \text{Re}(\tilde{f}_1 + \tilde{f}_2) = \text{Re}(\tilde{f}_3),$$ with ˜f<sup>3</sup> = ˜f<sup>1</sup> + ˜f2. You simply add the corresponding complex wave functions, and then take the real part. In particular, if they have the same frequenc...	{ "Header 1": "Example 9.1. Suppose you want to combine two sinusoidal waves:", "token_count": 765, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
So far, I have assumed the string is infinitely long—or at any rate long enough that we don't need to worry about what happens to a wave when it reaches the end. As a matter of fact, what happens depends a lot on how the string is attached—that is, on the specific boundary conditions to which the wave is subject. Sup...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "token_count": 1877, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{9.31}$$ If the second string is lighter than the first $(\mu_2 < \mu_1$ , so that $v_2 > v_1)$ , all three waves have the same phase angle $(\delta_R = \delta_T = \delta_I)$ , and the outgoing amplitudes are $$A_R = \left(\frac{v_2 - v_1}{v_2 + v_1}\right) A_I, \quad A_T = \left(\frac{2v_2}{v_2 + v_1}\rig...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "token_count": 1343, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
<sup>2</sup>Notice that you can always switch the sign of <sup>n</sup>ˆ, provided you simultaneously advance the phase constant by 180◦, since both operations change the sign of the wave. In terms of the polarization angle θ, $$\hat{\mathbf{n}} = \cos\theta \,\hat{\mathbf{x}} + \sin\theta \,\hat{\mathbf{y...	{ "Header 1": "9.1.3 ■ Boundary Conditions: Reflection and Transmission", "Header 3": "FIGURE 9.8", "token_count": 496, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In regions of space where there is no charge or current, Maxwell's equations read (i) $$\nabla \cdot \mathbf{E} = 0$$ , (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ , (ii) $\nabla \cdot \mathbf{B} = 0$ , (iv) $\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\par...	{ "Header 1": "9.2.1 The Wave Equation for E and B", "token_count": 1047, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
For reasons discussed in Sect. 9.1.2, we may confine our attention to sinusoidal waves of frequency ω. Since different frequencies in the visible range correspond to different colors, such waves are called monochromatic (Table 9.1). Suppose, <sup>4</sup>As Maxwell himself put it, "We can scarcely avoid the infe...	{ "Header 1": "9.2.2 Monochromatic Plane Waves", "token_count": 2032, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The wave as a whole is said to be polarized in the x direction (by convention, we use the direction of E to specify the polarization of an electromagnetic wave). There is nothing special about the z direction, of course—we can easily generalize to monochromatic plane waves traveling in an arbitrary direction....	{ "Header 1": "9.2.2 Monochromatic Plane Waves", "token_count": 677, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
According to Eq. 8.5, the energy per unit volume in electromagnetic fields is $$u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right). \tag{9.53}$$ In the case of a monochromatic plane wave (Eq. 9.48) $$B^2 = \frac{1}{c^2} E^2 = \mu_0 \epsilon_0 E^2, \tag{9.54}$$ so the *electric and magnetic cont...	{ "Header 1": "9.2.3 Energy and Momentum in Electromagnetic Waves", "token_count": 1823, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
[Note that this only works if the two waves have the same $\mathbf{k}$ and $\omega$ , but they need not have the same amplitude or phase.] For example, $$\langle u \rangle = \frac{1}{4} \operatorname{Re} \left( \epsilon_0 \tilde{\mathbf{E}} \cdot \tilde{\mathbf{E}}^* + \frac{1}{\mu_0} \tilde{\mathbf{B}} \cdot \til...	{ "Header 1": "9.2.3 Energy and Momentum in Electromagnetic Waves", "token_count": 1964, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose the x y plane forms the boundary between two linear media. A plane wave of frequency ω, traveling in the z direction and polarized in the x direction, approaches the interface from the left (Fig. 9.13): $$\tilde{\mathbf{E}}_{I}(z,t) = \tilde{E}_{0_{I}} e^{i(k_{1}z - \omega t)} \,\hat{\mathbf{x}}, \tilde...	{ "Header 1": "9.3.2 Reflection and Transmission at Normal Incidence", "token_count": 1938, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the last section, I treated reflection and transmission at normal incidence—that is, when the incoming wave hits the interface head-on. We now turn to the more general case of oblique incidence, in which the incoming wave meets the boundary at an arbitrary angle θ<sup>I</sup> (Fig. 9.14). Of course, normal inc...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 1786, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.74) become: (i) $$\epsilon_{1} \left( \tilde{\mathbf{E}}_{0_{I}} + \tilde{\mathbf{E}}_{0_{R}} \right)_{z} = \epsilon_{2} \left( \tilde{\mathbf{E}}_{0_{T}} \right)_{z}$$ (ii) $\left( \tilde{\mathbf{B}}_{0_{I}} + \tilde{\mathbf{B}}_{0_{R}} \right)_{z} = \left( \tilde{\mathbf{B}}_{0_{T}} \right)_{z}$ (iii) $\left(...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 2028, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
![](_page_429_Figure_2.jpeg) FIGURE 9.17 (The cosines are there because I am talking about the average power per unit area of interface, and the interface is at an angle to the wave front.) The reflection and transmission coefficients for waves polarized parallel to the plane of incidence are $$R \equiv \...	{ "Header 1": "9.3.3 Reflection and Transmission at Oblique Incidence", "token_count": 854, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In Sect. 9.3 I stipulated that the free charge density $\rho_f$ and the free current density $\mathbf{J}_f$ are zero, and everything that followed was predicated on that assumption. Such a restriction is perfectly reasonable when you're talking about wave propagation through a vacuum or through insulating materials...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "token_count": 1969, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
We may as well orient our axes so that $\mathbf{E}$ is polarized along the x direction: $$\tilde{\mathbf{E}}(z,t) = \tilde{E}_0 e^{-\kappa z} e^{i(kz - \omega t)} \,\hat{\mathbf{x}}.\tag{9.130}$$ Then (iii) gives $$\tilde{\mathbf{B}}(z,t) = \frac{\tilde{k}}{\omega} \tilde{E}_0 e^{-\kappa z} e^{i(kz - \omega t)}...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "token_count": 715, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
- (a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface? - (b) Silver is an excellent conductor, but it's expensive. Suppose you were designing a microwave experiment to operate at a frequency of 10<sup>10</sup> Hz. How thick would you make the...	{ "Header 1": "9.4.1 ■ Electromagnetic Waves in Conductors", "Header 2": "Problem 9.19", "token_count": 476, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The boundary conditions we used to analyze reflection and refraction at an interface between two dielectrics do not hold in the presence of free charges and currents. Instead, we have the more general relations (Eq. 7.64): (i) $$\epsilon_1 E_1^{\perp} - \epsilon_2 E_2^{\perp} = \sigma_f$$ , (iii) $\mathbf{E}_1^{\par...	{ "Header 1": "9.4.2 ■ Reflection at a Conducting Surface", "token_count": 1581, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the preceding sections, we have seen that the propagation of electromagnetic waves through matter is governed by three properties of the material: the permittivity $\epsilon$ , the permeability $\mu$ , and the conductivity $\sigma$ . Actually, each of these parameters depends to some extent on the frequency of th...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 2039, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{9.157}$$ The resulting dipole moment is the real part of $$\tilde{p}(t) = q\tilde{x}(t) = \frac{q^2/m}{\omega_0^2 - \omega^2 - i\gamma\omega} E_0 e^{-i\omega t}.$$ (9.158) The imaginary term in the denominator means that p is out of phase with E—lagging behind by an angle $\tan^{-1}[\gamma\omega/(\omega_0^2...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 2019, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
If you agree to stay away from the resonances, the damping can be ignored, and the formula for the index of refraction simplifies: $$n = 1 + \frac{Nq^2}{2m\epsilon_0} \sum_{j} \frac{f_j}{\omega_j^2 - \omega^2}.$$ (9.172) For most substances the natural frequencies $\omega_j$ are scattered all over the spectrum ...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 1941, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
As we shall soon see, confined waves are not (in general) transverse; in order to fit the boundary conditions we shall have to include longitudinal components $(E_z \text{ and } B_z)^{23}$ $$\tilde{\mathbf{E}}_0 = E_x \,\hat{\mathbf{x}} + E_y \,\hat{\mathbf{y}} + E_z \,\hat{\mathbf{z}}, \qquad \tilde{\mathbf{B...	{ "Header 1": "9.4.3 ■ The Frequency Dependence of Permittivity", "token_count": 1535, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose we have a wave guide of rectangular shape (Fig. 9.24), with height a and width b, and we are interested in the propagation of TE waves. The problem is to solve Eq. 9.181ii, subject to the boundary condition 9.175ii. We'll do it by separation of variables. Let $$B_{\tau}(x, y) = X(x)Y(y),$$ so that $$Y\fra...	{ "Header 1": "9.5.2 ■ TE Waves in a Rectangular Wave Guide", "token_count": 2052, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.179) yield $$k = \omega/c \tag{9.193}$$ (so the waves travel at speed c, and are nondispersive), $$cB_{y} = E_{x} \quad \text{and} \quad cB_{x} = -E_{y} \tag{9.194}$$ (so E and B are mutually perpendicular), and (together with $\nabla \cdot \mathbf{E} = 0$ , $\nabla \cdot \mathbf{B} = 0$ ): $$\fra...	{ "Header 1": "9.5.2 ■ TE Waves in a Rectangular Wave Guide", "token_count": 519, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
! Problem 9.33 The "inversion theorem" for Fourier transforms states that $$\tilde{\phi}(z) = \int_{-\infty}^{\infty} \tilde{\Phi}(k)e^{ikz} dk \quad \Longleftrightarrow \quad \tilde{\Phi}(k) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde{\phi}(z)e^{-ikz} dz. \quad (9.198)$$ Use this to determine A˜(k), in...	{ "Header 1": "More Problems on Chapter 9", "token_count": 2039, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
9.22).] - (c) Do the same for polarization perpendicular to the plane of incidence (use the results of Prob. 9.17). - (d) In the case of polarization perpendicular to the plane of incidence, show that the (real) evanescent fields are $$\mathbf{E}(\mathbf{r},t) = E_0 e^{-\kappa z} \cos(kx - \omega t) \,\hat{\mathbf{y}...	{ "Header 1": "More Problems on Chapter 9", "token_count": 396, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 10.1 ■ THE POTENTIAL FORMULATION #### 10.1.1 ■ Scalar and Vector Potentials In this chapter we seek the general solution to Maxwell's equations, (i) $$\nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho$$ , (iii) $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ , (ii) $\nabla \cdot \mat...	{ "Header 1": "Potentials and Fields", "token_count": 1965, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
10.4 and 10.5) can be written in the more symmetrical form $$\Box^{2}V + \frac{\partial L}{\partial t} = -\frac{1}{\epsilon_{0}}\rho,$$ $$\Box^{2}\mathbf{A} - \nabla L = -\mu_{0}\mathbf{J},$$ (10.6) where $$\Box^2 \equiv \nabla^2 - \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} \quad \text{and} \quad L \equiv...	{ "Header 1": "Potentials and Fields", "token_count": 400, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Equations 10.4 and 10.5 are ugly, and you might be inclined to abandon the potential formulation altogether. However, we have succeeded in reducing six problems—finding E and B (three components each)—down to four: V (one component) and A (three more). Moreover, Eqs. 10.2 and 10.3 do not uniquely defi...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2032, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(called the d'Alembertian) occurs in both equations: (i) $$\Box^2 V = -\frac{1}{\epsilon_0} \rho$$ , (ii) $\Box^2 \mathbf{A} = -\mu_0 \mathbf{J}$ . This democratic treatment of V and A is especially nice in the context of special relativity, where the d'Alembertian is the natural generalization of the Laplac...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2051, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 10.9 Derive Eq. 10.23. [Hint: Start by dotting v into Eq. 10.17.] #### 10.2 ■ CONTINUOUS DISTRIBUTIONS #### 10.2.1 ■ Retarded Potentials In the static case, Eq. 10.16 reduces to (four copies of) Poisson's equation, $$\nabla^2 V = -\frac{1}{\epsilon_0} \rho, \quad \nabla^2 \mathbf{A} = -\mu...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 2038, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
1.63), whereas $$\nabla \cdot \left(\frac{\hat{\mathbf{\lambda}}}{r^2}\right) = 4\pi \,\delta^3(\mathbf{\lambda})$$ (Eq. 1.100). So $$\nabla^2 V = \frac{1}{4\pi\epsilon_0} \int \left[ \frac{1}{c^2} \frac{\ddot{\rho}}{\imath} - 4\pi\rho \delta^3(\mathbf{z}) \right] d\tau' = \frac{1}{c^2} \frac{\partial^2 V}{\parti...	{ "Header 1": "10.1.2 Gauge Transformations", "token_count": 545, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$I(t) = \begin{cases} 0, & \text{for } t \le 0, \\ I_0, & \text{for } t > 0. \end{cases}$$ That is, a constant current $I_0$ is turned on abruptly at t = 0. Find the resulting electric and magnetic fields. ![](_page_465_Picture_6.jpeg) FIGURE 10.4 #### Solution The wire is presumably electrically n...	{ "Header 1": "Example 10.2. An infinite straight wire carries the current", "token_count": 1375, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Given the retarded potentials $$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r}',t_r)}{\imath} d\tau', \quad \mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}',t_r)}{\imath} d\tau', \quad (10.33)$$ it is, in principle, a straightforward matter to determine the fiel...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 1943, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 10.3 ■ POINT CHARGES #### 10.3.1 ■ Liénard-Wiechert Potentials My next project is to calculate the (retarded) potentials, $V(\mathbf{r}, t)$ and $\mathbf{A}(\mathbf{r}, t)$ , of a point charge q that is moving on a specified trajectory $$\mathbf{w}(t) \equiv \text{position of } q \text{ at time } t.$$ (...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 2051, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Moreover, since the current density is $\rho \mathbf{v}$ (Eq. 5.26), the vector potential is $$\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \int \frac{\rho(\mathbf{r}',t_r)\mathbf{v}(t_r)}{\imath} d\tau' = \frac{\mu_0}{4\pi} \frac{\mathbf{v}}{\imath} \int \rho(\mathbf{r}',t_r) d\tau',$$ or $$\mathbf{A}(\mathbf...	{ "Header 1": "10.2.2 ■ Jefimenko's Equations", "token_count": 2025, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
We are now in a position to calculate the electric and magnetic fields of a point charge in arbitrary motion, using the Liénard-Wiechert potentials:17 $$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \frac{qc}{(ic - \mathbf{v} \cdot \mathbf{v})}, \quad \mathbf{A}(\mathbf{r},t) = \frac{\mathbf{v}}{c^2} V(\mathbf{r},t), \q...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 2046, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\tag{10.64}$$ Putting all this back into Eq. 10.58, and using the "BAC-CAB" rule to reduce the triple cross products, $$\nabla(\mathbf{z} \cdot \mathbf{v}) = \mathbf{a}(\mathbf{z} \cdot \nabla t_r) + \mathbf{v} - \mathbf{v}(\mathbf{v} \cdot \nabla t_r) - \mathbf{z} \times (\mathbf{a} \times \nabla t_r) + \mathbf{v}...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 2037, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
As we shall see in Chapter 11, it is this term that is responsible for electromagnetic radiation; accordingly, it is called the radiation field—or, since it is proportional to a, the acceleration field. The same terminology applies to the magnetic field. Back in Chapter 2, I commented that if we could write d...	{ "Header 1": "10.3.2 The Fields of a Moving Point Charge", "token_count": 1908, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Problem 10.24 Suppose you take a plastic ring of radius a and glue charge on it, so that the line charge density is λ0\|sin(θ/2)\|. Then you spin the loop about its axis at an angular velocity ω. Find the (exact) scalar and vector potentials at the center of the ring. Answer: A = (μ0λ0ωa/3π ) sin[ω(t − *a...	{ "Header 1": "More Problems on Chapter 10", "token_count": 1770, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
$$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \frac{\hat{\mathbf{r}}}{r^2} \cdot [\mathbf{p} + (r/c)\dot{\mathbf{p}}]$$ $$\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi} \left[ \frac{\dot{\mathbf{p}}}{r} \right]$$ $$\mathbf{E}(\mathbf{r},t) = -\frac{\mu_0}{4\pi} \left\{ \frac{\ddot{\mathbf{p}} - \hat{\mathbf{r}}(\ha...	{ "Header 1": "More Problems on Chapter 10", "token_count": 372, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### 11.1 ■ DIPOLE RADIATION #### 11.1.1 ■ What is Radiation? When charges accelerate, their fields can transport energy irreversibly out to infinity—a process we call radiation.<sup>1</sup> Let us assume the source is localized<sup>2</sup> near the origin; we would like to calculate the energy it is radiatin...	{ "Header 1": "Radiation", "token_count": 945, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Picture two tiny metal spheres separated by a distance d and connected by a fine wire (Fig. 11.2); at time t the charge on the upper sphere is q(t), and the charge on the lower sphere is −q(t). Suppose that we drive the charge back and forth through the wire, from one end to the other, at an angular frequen...	{ "Header 1": "11.1.2 Electric Dipole Radiation", "token_count": 1888, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
\end{split}$$ (I dropped the first and last terms, in accordance with approximation 3.) Likewise, $$\frac{\partial \mathbf{A}}{\partial t} = -\frac{\mu_0 p_0 \omega^2}{4\pi r} \cos[\omega (t - r/c)](\cos \theta \,\hat{\mathbf{r}} - \sin \theta \,\hat{\boldsymbol{\theta}}),$$ and therefore $$\mathbf{E} = -\nabla...	{ "Header 1": "11.1.2 Electric Dipole Radiation", "token_count": 2005, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
Suppose now that we have a wire loop of radius b (Fig. 11.8), around which we drive an alternating current: $$I(t) = I_0 \cos(\omega t). \tag{11.23}$$ This is a model for an oscillating magnetic dipole, $$\mathbf{m}(t) = \pi b^2 I(t) \,\hat{\mathbf{z}} = m_0 \cos(\omega t) \,\hat{\mathbf{z}},\tag{11.24}$$ !...	{ "Header 1": "11.1.3 Magnetic Dipole Radiation", "token_count": 1959, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
The energy flux for magnetic dipole radiation is $$\mathbf{S}(\mathbf{r},t) = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{B}) = \frac{\mu_0}{c} \left\{ \frac{m_0 \omega^2}{4\pi c} \left( \frac{\sin \theta}{r} \right) \cos[\omega(t - r/c)] \right\}^2 \hat{\mathbf{r}}, \quad (11.38)$$ the intensity is $$\langle \m...	{ "Header 1": "11.1.3 Magnetic Dipole Radiation", "token_count": 887, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
In the previous sections, we studied the radiation produced by two specific systems: oscillating electric dipoles and oscillating magnetic dipoles. Now I want to apply the same procedures to a configuration of charge and current that is entirely arbitrary, except that it is localized within some finite volume near the ...	{ "Header 1": "11.1.4 Radiation from an Arbitrary Source", "token_count": 1769, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
11.48 it follows that $$\nabla t_0 = -\frac{1}{c} \nabla r = -\frac{1}{c} \hat{\mathbf{r}},$$ and hence $$\nabla V \cong \nabla \left[ \frac{1}{4\pi\epsilon_0} \frac{\hat{\mathbf{r}} \cdot \dot{\mathbf{p}}(t_0)}{rc} \right] \cong \frac{1}{4\pi\epsilon_0} \left[ \frac{\hat{\mathbf{r}} \cdot \ddot{\mathbf{p}}(t_0)}...	{ "Header 1": "11.1.4 Radiation from an Arbitrary Source", "token_count": 1105, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(a) In the case of an oscillating electric dipole, $$p(t) = p_0 \cos(\omega t), \quad \ddot{p}(t) = -\omega^2 p_0 \cos(\omega t),$$ and we recover the results of Sect. 11.1.2. (b) For a single point charge q, the dipole moment is $$\mathbf{p}(t) = q\mathbf{d}(t),$$ where d is the position of q with re...	{ "Header 1": "Example 11.2.", "token_count": 2045, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
(It's like the flies that stay with the garbage truck.) Now $\mathbf{E}_{rad}$ is perpendicular to $\hat{\lambda}$ , so the second term in Eq. 11.64 vanishes: $$\mathbf{S}_{\text{rad}} = \frac{1}{\mu_0 c} E_{\text{rad}}^2 \, \hat{\mathbf{z}}. \tag{11.67}$$ If the charge is instantaneously at rest (at time $t_...	{ "Header 1": "Example 11.2.", "token_count": 1847, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }
#### Solution In this case $(\mathbf{u} \times \mathbf{a}) = c(\hat{\mathbf{a}} \times \mathbf{a})$ , so $$\frac{dP}{d\Omega} = \frac{q^2c^2}{16\pi^2\epsilon_0} \frac{\|\hat{\mathbf{i}} \times (\hat{\mathbf{i}} \times \mathbf{a})\|^2}{(c - \hat{\mathbf{i}} \cdot \mathbf{v})^5}.$$ Now $$\hat{\mathbf{a}} \ti...	{ "Header 1": "Example 11.2.", "token_count": 2009, "source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf" }