id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
101 | Consider an uncharged ellipsoid subjected to a uniform external electric field. When the external electric field is oriented in any direction relative to the ellipsoid's $x, y, z$ axes, find the charge distribution $\sigma$ on its surface. | [] | Expression | Magnetism | $\sigma$: Surface charge distribution on the ellipsoid.
$\nu_i$: Components of a vector representing the normal direction or related geometric factors.
$n^{-1}_{ik}$: Inverse of a tensor related to depolarization factors.
$\mathfrak{C}_k$: Components of the external electric field vector, used in the final answer.
$\nu... | |
102 | For a prolate rotational ellipsoid conductor, find the external potential $\varphi$ when its symmetry axis is perpendicular to the external field (specifically referring to the scenario described in the solution where the field is in the $z$ axis direction). | [] | Expression | Magnetism | $\varphi$: External potential around the conductor.
$\mathfrak{C}$: A constant related to the strength of the uniform external electric field.
$z$: The Cartesian coordinate along the axis of the external electric field.
$\xi$: A coordinate in the prolate spheroidal coordinate system.
$a$: Semi-major axis of the prolate... | |
103 | For an oblate rotating ellipsoidal conductor, when its symmetry axis is perpendicular to the external field (specifically referring to the case described in the solution where the field is in the $x$ axis direction), find the potential $\varphi$ outside the conductor. | [] | Expression | Magnetism | $\varphi$: Potential outside the conductor.
$\mathfrak{c}$: A constant related to the magnitude of the external electric field.
$x$: Cartesian coordinate, specifically the direction of the external field.
$a$: Equatorial semi-axis length of the oblate ellipsoid.
$c$: Polar semi-axis length of the oblate ellipsoid.
$\xi... | |
104 | Consider a uniform electric field of magnitude \( \mathfrak{C} \) existing along the positive z-axis in the half-space \( z<0 \) (i.e., at \( z \to -\infty \), the electric field is \( \vec{E} = \mathfrak{C}\hat{k} \), corresponding to the potential \( \varphi = -\mathfrak{C}z \)). This electric field is constrained by... | [] | Expression | Magnetism | $\varphi$: Electric potential.
$\mathfrak{C}$: Magnitude of the uniform electric field.
$z$: Cartesian coordinate along the z-axis.
$a$: Radius of the circular hole.
$\xi$: Oblate spheroidal coordinate.
$\eta$: Oblate spheroidal coordinate. | |
105 | In the same physical scenario as the previous sub-question (that is, in the half-space $z<0$, there exists a uniform electric field $\mathfrak{C}$ along the positive $z$-axis, constrained by a grounded conductive plane at $z=0$ with a circular hole of radius $a$), the expression for the electric potential $\varphi$ is ... | [] | Expression | Magnetism | $\sigma$: Surface charge density.
$\mathfrak{C}$: Magnitude of the uniform electric field along the positive z-axis.
$a$: Radius of the circular hole.
$\rho$: Cylindrical radial distance. | |
106 | Assume a charged spherical conductor is cut in half, try to determine the mutual repulsive force between the two hemispheres. | [] | Expression | Magnetism | $F$: Mutual repulsive force between the two hemispheres
$a$: Radius of the sphere
$C$: Constant appearing in the final answer
$\varphi$: Electric potential appearing in the final answer
$r$: Distance or radius appearing in the final answer | |
107 | A spherical conductor is cut into two halves, determine the mutual repulsive force between the two hemispheres. The conductor sphere is uncharged and is in a uniform external electric field $\mathfrak{C}$ perpendicular to the interface. | [] | Expression | Magnetism | $F$: Mutual repulsive force between the two hemispheres
$a$: Radius of the spherical conductor
$\mathfrak{C}$: Uniform external electric field | |
108 | For waves propagating on the charged surface of a liquid conductor in a gravitational field, determine the stability conditions of this surface. | [] | Expression | Magnetism | $\sigma_0$: Surface charge density of the liquid conductor
$g$: Gravitational acceleration
$\rho$: Density of the liquid conductor
$\alpha$: Surface tension coefficient of the liquid
$\pi$: Mathematical constant pi | |
109 | Find the stability condition for a charged spherical droplet with respect to small deformations. | [] | Expression | Magnetism | $e$: Total charge of the spherical droplet.
$a$: Radius of the spherical droplet.
$\alpha$: Surface tension coefficient. | |
110 | Find the stability condition (Rayleigh, 1882) of a charged spherical droplet relative to splitting into two identical smaller droplets (large deformation). Assume the original droplet has a charge of $e$ and a radius of $a$, while each smaller droplet after splitting has a charge of $e/2$ and a radius of $a/2^{1/3}$. | [] | Expression | Magnetism | $e$: Charge of the original spherical droplet.
$a$: Radius of the original spherical droplet.
$\alpha$: Surface tension coefficient. | |
111 | An infinitely long straight charged wire (with charge per unit length of $e$) is parallel to the interface between two media with different dielectric constants ($\varepsilon_1$ and $\varepsilon_2$ respectively), and the distance from the interface is $h$. Determine the potential $\varphi_1$ in medium 1 ($\varepsilon_1... | [] | Expression | Magnetism | $\varphi_1$: Potential in medium 1.
$e$: Charge per unit length of the infinitely long straight charged wire.
$\varepsilon_1$: Dielectric constant of medium 1.
$r$: Distance from the observation point to the original wire.
$\varepsilon_2$: Dielectric constant of medium 2.
$r^{\prime}$: Distance from the observation poi... | |
112 | An infinitely long straight conductor (with a linear charge of $e$) is parallel to the interface between two media with different dielectric constants ($\varepsilon_1$ and $\varepsilon_2$ respectively) and at a distance $h$ from the interface. Determine the potential $\varphi_2$ within medium 2 ($\varepsilon_2$). | [] | Expression | Magnetism | $\varphi_2$: Potential within medium 2
$e$: Linear charge density of the infinitely long straight conductor
$\varepsilon_1$: Dielectric constant of the first medium
$\varepsilon_2$: Dielectric constant of the second medium
$r$: Distance from the observation point to the original wire | |
113 | Find the torque $K$ acting on a rotational ellipsoid in a uniform electric field $\mathfrak{C}$, with a dielectric constant of $\varepsilon$. The volume of the ellipsoid is $V$, $\alpha$ is the angle between the direction of $\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization coefficien... | [] | Expression | Magnetism | $K$: Torque acting on a rotational ellipsoid.
$\varepsilon$: Dielectric constant of the ellipsoid.
$n$: Depolarization coefficient along the symmetry axis of the ellipsoid.
$V$: Volume of the ellipsoid.
$\alpha$: Angle between the direction of the electric field $\mathfrak{C}$ and the symmetry axis of the ellipsoid.
$\... | |
114 | For a conductive rotating ellipsoid ($\varepsilon \rightarrow \infty$), in a uniform electric field $\mathfrak{C}$, find the torque $K$ acting on it. The volume of the ellipsoid is $V$, $\alpha$ is the angle between the direction of $\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization fa... | [] | Expression | Magnetism | $K$: Torque acting on the ellipsoid.
$n$: Depolarization factor along the symmetry axis of the ellipsoid.
$V$: Volume of the ellipsoid.
$\mathfrak{C}$: Uniform external electric field.
$\alpha$: Angle between the direction of the electric field $\mathfrak{C}$ and the symmetry axis of the ellipsoid. | |
115 | A hollow dielectric sphere (dielectric constant $\varepsilon$, with inner and outer radii $b$ and $a$, respectively) is placed in a uniform external electric field $\mathfrak{E}$. Determine the field inside the cavity of the sphere. | [] | Expression | Magnetism | $\boldsymbol{E}_3$: Electric field inside the cavity of the sphere
$\mathfrak{C}$: Symbol used in the final answer, representing the external electric field (likely a typo for $\mathfrak{E}$)
$\varepsilon$: Dielectric constant of the hollow dielectric sphere
$b$: Inner radius of the hollow dielectric sphere
$a$: Outer ... | |
116 | Determine the height $h$ by which the liquid surface inside a vertical parallel-plate capacitor rises. | [] | Expression | Magnetism | $h$: Height by which the liquid surface inside a vertical parallel-plate capacitor rises
$\varepsilon$: Dielectric constant (or relative permittivity) of the liquid
$\rho$: Density of the liquid
$g$: Acceleration due to gravity
$E$: Electric field strength between the capacitor plates
$\pi$: Mathematical constant pi | |
117 | If the object is not in a vacuum, but in a medium with a dielectric constant of $\varepsilon^{(e)}$, find the formula of $\mathscr{F}-\mathscr{F}_{0}$.
If the answer exists in an integral, then find the integrand | [] | Expression | Magnetism | $\mathfrak{C}$: A vector field, used in the integral expression for the free energy difference.
$\boldsymbol{D}$: Electric displacement field.
$\varepsilon^{(e)}$: Dielectric constant of the medium.
$\boldsymbol{E}$: Electric field. | |
118 | Consider a capacitor composed of two conducting surfaces separated by a distance $h$, with $h$ being smaller than the dimensions of the capacitor plates. The space between the capacitor plates is filled with a material of dielectric constant $\varepsilon_{1}$. A small sphere with radius $a \ll h$ and dielectric constan... | [] | Expression | Magnetism | $C$: Total capacitance of the capacitor with the sphere.
$C_{0}$: Original capacitance of the capacitor in the absence of the sphere.
$a$: Radius of the small sphere.
$h$: Distance between the capacitor plates.
$\varepsilon_{1}$: Dielectric constant of the material filling the space between the capacitor plates.
$\vare... | |
119 | Try to determine the potential $\varphi$ produced by a point charge $e$ inside an anisotropic homogeneous medium (with the point charge located at the origin and the principal axes of the dielectric tensor $\varepsilon_{ik}$ aligned along the $x, y, z$ axes), and express it using the principal dielectric constants $\va... | [] | Expression | Magnetism | $\varphi$: Electric potential produced by the point charge.
$e$: Magnitude of the point charge.
$\varepsilon^{(x)}$: Principal dielectric constant along the x-axis.
$\varepsilon^{(y)}$: Principal dielectric constant along the y-axis.
$\varepsilon^{(z)}$: Principal dielectric constant along the z-axis.
$x$: Spatial coor... | |
120 | Determine the potential $\varphi$ generated by a point charge $e$ in an anisotropic homogeneous medium, using tensor notation that does not depend on the choice of coordinate system. | [] | Expression | Magnetism | $\varphi$: Potential generated by a point charge.
$e$: Point charge.
$|\varepsilon|$: Determinant of the dielectric tensor $\varepsilon_{i k}$, given by $|\varepsilon| = \varepsilon^{(x)}\varepsilon^{(y)}\varepsilon^{(z)}$.
$\varepsilon_{i k}^{-1}$: Inverse of the dielectric tensor $\varepsilon_{i k}$.
$x_i$: i-th comp... | |
121 | An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes, respectively) is in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_x, \mathfrak{C}_y... | [] | Expression | Magnetism | $K_x$: x component of the torque acting on the sphere
$a$: Radius of the anisotropic dielectric sphere
$\mathfrak{C}_y$: y-component of the uniform external electric field
$\mathfrak{C}_z$: z-component of the uniform external electric field
$\varepsilon^{(y)}$: Principal value of the dielectric tensor along the y-axis
... | |
122 | An anisotropic dielectric sphere with a radius of $a$ (the principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ directions, respectively) is placed in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_... | [] | Expression | Magnetism | $K_y$: $y$-component of the torque acting on the sphere
$a$: Radius of the anisotropic dielectric sphere
$\mathfrak{C}_z$: $z$-component of the uniform external electric field
$\mathfrak{C}_x$: $x$-component of the uniform external electric field
$\varepsilon^{(z)}$: Principal value of the dielectric tensor along the $... | |
123 | An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\varepsilon^{(x)}, \varepsilon^{(y)}, \varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes respectively) is placed in a uniform external electric field $\boldsymbol{\mathfrak{C}} = (\mathfrak{C}_x, \mathfra... | [] | Expression | Magnetism | $K_z$: z component of the torque acting on the sphere
$a$: Radius of the anisotropic dielectric sphere
$\mathfrak{C}_x$: x-component of the uniform external electric field
$\mathfrak{C}_y$: y-component of the uniform external electric field
$\varepsilon^{(x)}$: Principal value of the dielectric tensor along the x-axis
... | |
124 | Consider a dielectric sphere (with radius $a$) placed in a uniform external electric field $\mathfrak{C}$, sliced into two halves by a plane perpendicular to the field direction. Find the attraction force between the two hemispheres. | [] | Expression | Magnetism | $F$: Attraction force between the two hemispheres
$\varepsilon$: Dielectric constant of the sphere
$a$: Radius of the dielectric sphere
$\mathfrak{C}$: Uniform external electric field | |
125 | Try to determine the shape change of a dielectric sphere in a uniform external electric field. | [] | Expression | Magnetism | $a$: Semi-axis length of the deformed sphere along the x-axis
$b$: Semi-axis length of the deformed sphere along the y-axis (or perpendicular to x-axis)
$R$: Original radius of the sphere
$\mathfrak{C}$: Uniform external electric field
$\pi$: Mathematical constant pi
$\mu$: Shear modulus of the material
$\varepsilon_{0... | |
126 | Determine the Young's modulus of a non-pyroelectric piezoelectric material parallel plate thin slab under the following conditions (the ratio of tensile stress to relative tensile strain): The slab is under tensile stress between the plates of a short-circuited capacitor. | [] | Expression | Magnetism | $E$: Young's modulus, defined as the ratio of tensile stress to relative tensile strain.
$\mu_{zzzz}$: Component of the elastic compliance tensor, defined by $u_{zz}=\mu_{zzzz} \sigma_{zz}$. | |
127 | Try to derive the expression or equation for the velocity of sound within a piezoelectric medium. | [] | Expression | Magnetism | $\rho$: Density of the medium.
$\omega$: Angular frequency.
$\delta_{i k}$: Kronecker delta, defined as $\delta_{i k} = 1$ if $i=k$ and $0$ if $i \neq k$.
$\lambda_{i l k m}$: Elastic stiffness tensor components.
$k_{l}$: Wave vector component.
$k_{m}$: Wave vector component.
$\pi$: Mathematical constant pi, approximat... | |
128 | The piezoelectric crystals belonging to the $C_{6 v}$ crystal class are constrained by the surface plane ( $xz$ plane) of the symmetry axis ($z$ axis). Try to determine the velocity of surface waves propagating perpendicular to the symmetry axis (along the $x$ axis) which undergo displacement $u_{z}$ and potential $\va... | [] | Expression | Magnetism | $\omega$: Angular frequency.
$k$: Wave number.
$\bar{\lambda}$: Effective elastic constant, $\bar{\lambda}=\lambda+\frac{4 \pi \beta^{2}}{\varepsilon}$.
$\rho$: Mass density of the medium.
$\Lambda$: Dimensionless parameter, $\Lambda = \frac{4 \pi \beta^{2}}{\bar{\lambda} \varepsilon(1+\varepsilon)}$. | |
129 | Given the second-order tensor $\sigma_{ik}$, with its symmetric part $s_{ik}$ and its antisymmetric part formed by the axial vector $\boldsymbol{a}$ (specific definitions are found in the symbols table), express the determinant $|\sigma|$ of the tensor $\sigma_{ik}$ in terms of the components of $s_{ik}$ and $\boldsymb... | [] | Expression | Magnetism | $|\sigma|$: Determinant of the second-order tensor $\sigma_{ik}$.
$|s|$: Determinant of the symmetric part $s_{ik}$.
$s_{ik}$: Symmetric part of the second-order tensor $\sigma_{ik}$.
$a_i$: Component of the axial vector $\boldsymbol{a}$ along the $i$-th axis. | |
130 | Given a second-order tensor $\sigma_{ik}$, its symmetric part is $s_{ik}$, and the antisymmetric part is formed by the axial vector $\boldsymbol{a}$. Given its determinant as $|\sigma|=|s|+s_{i k} a_{i} a_{k}$. Try to express the axial vector $b_i$ of the antisymmetric part of its inverse tensor $\sigma_{ik}^{-1}$ usin... | [] | Expression | Magnetism | $b_i$: Components of the axial vector $\boldsymbol{b}$, which forms the antisymmetric part of the inverse tensor $\sigma_{ik}^{-1}$.
$|\sigma|$: Determinant of the tensor $\sigma_{ik}$.
$s_{ik}$: Symmetric part of the second-order tensor $\sigma_{ik}$.
$a_k$: Components of the axial vector $\boldsymbol{a}$. | |
131 | Assuming two parallel planar plates (made of the same metal $A$) are immersed in an electrolyte solution $AX$. In the case of a very small current $j$, derive the expression for the effective resistivity of the solution $\frac{\mathscr{E}}{l j}$. | [] | Expression | Magnetism | $\mathscr{E}$: Potential difference between the two plates.
$l$: Distance between the parallel plates.
$j$: Current density.
$\sigma$: Conductivity of the solution.
$\rho$: A parameter representing a property of the electrolyte solution, appearing in the integral expressions.
$D$: Diffusion coefficient.
$\zeta$: A pote... | |
132 | Consider a circular line current with a radius $a$. Try to find the radial component $B_r$ of the magnetic field in cylindrical coordinates. | [] | Expression | Magnetism | $B_r$: Radial component of the magnetic induction.
$A_{\varphi}$: Azimuthal component of the magnetic vector potential.
$z$: Axial coordinate in the cylindrical coordinate system.
$J$: Magnitude of the current in the circular line.
$c$: Speed of light.
$r$: Radial coordinate in the cylindrical coordinate system.
$a$: R... | |
133 | Consider a circular line current with a radius of $a$. Try to find the axial component $B_z$ of the magnetic field it produces in cylindrical coordinates. | [] | Expression | Magnetism | $B_z$: Axial component of the magnetic field.
$r$: Radial cylindrical coordinate.
$A_{\varphi}$: Azimuthal component of the magnetic vector potential.
$J$: Current flowing in the circular line.
$c$: Speed of light.
$a$: Radius of the circular line current.
$z$: Axial cylindrical coordinate.
$K$: Complete elliptic integ... | |
134 | Try to find the 'internal' part of the self-inductance $L_i$ of a closed thin wire with a circular cross-section.
\footnotetext{
(1) The assertion in the main text that the self-inductance does not depend on the current distribution actually applies not only to the approximation (34.1), but also to subsequent approxim... | [] | Expression | Magnetism | $L_i$: Internal part of the self-inductance of a closed thin wire
$l$: Length of the closed wire
$\mu_i$: Permeability of the internal medium of the wire | |
135 | Try to determine the self-inductance of a thin circular ring (radius $b$) made from a wire with a circular cross-section (radius $a$). | [] | Expression | Magnetism | $L$: Total self-inductance of the circular ring.
$\pi$: Mathematical constant pi.
$b$: Radius of the thin circular ring.
$a$: Radius of the wire's circular cross-section.
$\mu_i$: Internal permeability of the wire material. | |
136 | A current flows through a wire loop $(\mu_{i}=1)$; try to determine the elongation of the loop under the magnetic field generated by this current. | [] | Expression | Magnetism | $\Delta b$: Elongation or change in the radius of the wire loop.
$b$: Radius of the wire loop.
$J$: Current flowing through the wire loop.
$a$: Radius of the wire's cross-section.
$c$: Speed of light in vacuum.
$E$: Young's modulus of the wire material.
$\sigma$: Poisson's ratio of the wire material. | |
137 | Seek the first-order correction value of the cylindrical helical tube's self-inductance, due to the field distortion near both ends of the cylindrical helical tube when $l / h$ (with $\mu_{e}=1$). | [] | Expression | Magnetism | $L$: Self-inductance of the helical tube
$b$: Radius of the helical tube
$n$: Turns per unit length
$h$: Length of the helical tube | |
138 | If a planar circuit is placed on the surface of a semi-infinite medium with a permeability of $\mu_{e}$, find the factor by which the self-inductance of the planar circuit changes. We neglect the internal part of the conductor's self-inductance. | [] | Expression | Magnetism | $\mu_{e}$: Permeability of the semi-infinite medium | |
139 | Given a straight wire carrying a current $J$ parallel to an infinitely long cylindrical conductor with a radius $a$ (permeability $\mu$) at a distance $l$ from the axis of the cylinder, determine the force on the straight wire. | [] | Expression | Magnetism | $F$: Force per unit length on the conductor
$J$: Current carried by the straight wire
$a$: Radius of the infinitely long cylindrical conductor
$\mu$: Permeability of the cylindrical conductor
$b$: Distance from the axis of the cylinder to the straight wire
$c$: Speed of light | |
140 | Try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \gg 4 \pi M)$, where the microcrystals have uniaxial symmetry. | [] | Expression | Magnetism | $\bar{M}$: Average magnetization intensity of the polycrystal
$M$: Magnitude of the magnetization intensity of a single microcrystal
$\beta$: Material parameter related to anisotropy
$H$: Magnitude of the strong magnetic field | |
141 | For cubic symmetric microcrystals, try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \gg 4 \pi M)$. | [] | Expression | Magnetism | $\bar{M}$: Average magnetization intensity of a polycrystal
$M$: Magnitude of the magnetization vector
$\beta$: Anisotropy constant parameter for cubic symmetric microcrystals
$H$: Magnitude of the magnetic field vector | |
142 | Try to find out the relative elongation of a ferromagnetic cubic crystal depending on the magnetization direction $\boldsymbol{m}$ and the measurement direction $\boldsymbol{n}$. | [] | Expression | Magnetism | $\frac{\delta l}{l}$: Relative elongation of the crystal.
$a_1$: Material constant related to magnetostriction, specifically for terms involving squares of magnetization and measurement direction components along the same axis.
$m_x$: $x$-component of the magnetization direction $\boldsymbol{m}$.
$n_x$: $x$-component o... | |
143 | The easy magnetization axes of a cubic ferromagnet align along the three edges of the cube (specifically the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$ axis, and the domain walls are distributed parallel to the (100) plane. Determine the surface tension of the domain wall i... | [] | Expression | Magnetism | $\Delta_{(100)}$: Surface tension of the domain wall for the (100) plane
$\alpha$: Exchange stiffness constant
$\beta$: Anisotropy constant | |
144 | The easy magnetization axes of a cubic ferromagnet are along the three edges of the cube (namely the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$-axis, and the domain walls are distributed parallel to the (110) plane. Determine the surface tension of the domain walls in this ... | [] | Expression | Magnetism | $\Delta_{(110)}$: Surface tension of the domain walls for the (110) plane.
$\alpha$: Exchange stiffness constant.
$\beta$: Anisotropy constant. | |
145 | If the transition between magnetic domains is not achieved through the rotation of $M$ but by changing the magnitude of $M$ (i.e., when $M$ changes sign after passing through zero), determine the surface tension of the domain wall in a uniaxial crystal. The free energy's dependence on $M$ (at $\boldsymbol{H}=0$) takes ... | [] | Expression | Magnetism | $\Delta$: Surface tension of the domain wall
$\alpha_1$: Coefficient related to inhomogeneity in free energy density
$|A|$: Absolute value of coefficient A | |
146 | The parallel plane magnetic domains extend perpendicularly to the surface of the ferromagnetic material without changing the direction of magnetization . Try to derive and present the exact mathematical expression for the magnetic field energy per unit surface area near the surface of the ferromagnet (expressed in term... | [] | Expression | Magnetism | $a$: Domain width
$M$: Magnetization (magnitude of surface charge density)
$\pi$: Mathematical constant pi
$\zeta(3)$: Riemann zeta function evaluated at 3 | |
147 | Find the magnetic moment ${ }^{(1)}$ of a superconducting disk perpendicular to the external magnetic field. | [] | Expression | Superconductivity | $\mathscr{M}$: Magnetic moment of the superconducting disk.
$a$: Semi-axis of the spheroid, representing the radius of the disk in the limit.
$\pi$: Mathematical constant pi.
$\mathfrak{H}$: External magnetic field. | |
148 | Seek the heat capacity of a superconducting ellipsoid in the intermediate state. | [] | Expression | Superconductivity | $\mathscr{C}_{t}$: Heat capacity of the object in the intermediate state.
$\mathscr{C}_{s}$: Heat capacity of the object in the superconducting state.
$V$: Volume of the object (superconducting ellipsoid).
$T$: Temperature.
$\pi$: Mathematical constant pi.
$n$: Demagnetization factor.
$H_{\mathrm{cr}}^{\prime}$: First ... | |
149 | An isotropic conducting sphere with radius $a$ is in a uniform periodic external magnetic field. Determine the expression for its magnetic polarizability $\alpha$. | [] | Expression | Magnetism | $\alpha$: Magnetic polarizability of the sphere.
$\pi$: Mathematical constant pi.
$a$: Radius of the isotropic conducting sphere.
$k$: Wave number or propagation constant inside the sphere, defined as $k=\frac{1+\mathrm{i}}{\delta}$. | |
150 | An isotropic conducting sphere with a radius of $a$ is in a uniform periodic external magnetic field, with its magnetic susceptibility given by $\alpha = \alpha^{\prime} + \mathrm{i} \alpha^{\prime \prime}$. Determine the expression for the real part of its magnetic susceptibility $\alpha^{\prime}$. | [] | Expression | Magnetism | $\alpha^{\prime}$: Real part of the magnetic susceptibility
$\pi$: Mathematical constant pi
$\delta$: Skin depth
$a$: Radius of the isotropic conducting sphere | |
151 | An isotropic conductive sphere with a radius of $a$ is in a uniform periodic external magnetic field. Its magnetic susceptibility is $\alpha = \alpha^{\prime} + \mathrm{i} \alpha^{\prime \prime}$. Determine the expression for the imaginary part of its magnetic susceptibility $\alpha^{\prime \prime}$. | [] | Expression | Magnetism | $\alpha^{\prime \prime}$: Imaginary part of the magnetic susceptibility $\alpha$.
$\delta$: Skin depth of the conductive sphere.
$a$: Radius of the isotropic conductive sphere.
$\pi$: Mathematical constant pi.
$\phi$: A symbol appearing in the final answer, likely a typo for the radius $a$. | |
152 | Find the smallest value of the attenuation coefficient for the magnetic field inside a conducting sphere. | [] | Expression | Magnetism | $\gamma_1$: Smallest value of the attenuation coefficient for the magnetic field
$\pi$: Pi, a mathematical constant
$c$: Speed of light
$\sigma$: Electrical conductivity of the conducting sphere
$a$: Radius of the conducting sphere | |
153 | Two inductively coupled circuits respectively contain self-inductances $L_{1}$ and $L_{2}$ and capacitances $C_{1}$ and $C_{2}$. Determine the intrinsic frequencies of electric oscillations within these coupled circuits (we neglect resistances $R_{1}$ and $R_{2}$). | [] | Expression | Magnetism | $\omega_{1,2}^{2}$: Squared intrinsic frequencies of electric oscillations for the coupled circuits.
$c$: Speed of light.
$L_1$: Self-inductance of the first circuit.
$C_1$: Capacitance of the first circuit.
$L_2$: Self-inductance of the second circuit.
$C_2$: Capacitance of the second circuit.
$L_{12}$: Mutual inducta... | |
154 | A uniformly magnetized sphere rotates uniformly around an axis parallel to the magnetization direction. Determine the unipolar induced electromotive force between one pole of the uniformly magnetized sphere and the equator. | [] | Expression | Magnetism | $\mathscr{E}$: Unipolar induced electromotive force (EMF) between one pole and the equator of the sphere.
$\Omega$: Angular velocity of the sphere.
$\mathscr{M}$: Total magnetic moment of the sphere.
$a$: Radius of the sphere.
$c$: Speed of light. | |
155 | Determine the current generated inside the superconducting ring when its uniform rotation stops. | [] | Expression | Superconductivity | $J$: Current generated inside the superconducting ring
$m$: Mass (likely electron mass)
$c$: Speed of light
$b$: Radius of the superconducting ring
$\Omega$: Angular velocity of the uniform rotation of the ring
$e$: Elementary charge (likely electron charge)
$a$: Radius of the wire's circular cross-section | |
156 | Try to determine the absorption coefficient of Alfven waves in an incompressible fluid (assuming this coefficient is very small). | [] | Expression | Magnetism | $\gamma$: Absorption coefficient of the wave.
$\omega$: Angular frequency of the wave.
$u_{\mathrm{A}}$: Alfven speed, $u_{\mathrm{A}} = \frac{|H_x|}{\sqrt{4\pi\rho}}$.
$\eta$: Dynamic viscosity.
$\rho$: Mass density of the fluid.
$c$: Speed of light in vacuum.
$\sigma$: Electrical conductivity. | |
157 | Try to find the law of rotational discontinuity expanding with time. | [] | Expression | Magnetism | $c$: Speed of light.
$\pi$: Mathematical constant pi.
$\sigma$: Electrical conductivity of the medium.
$\nu$: Kinematic viscosity of the fluid.
$t$: Time. | |
158 | A dielectric sphere in vacuum rotates in a constant magnetic field $\mathfrak{H}$, determine the electric field produced around the sphere. | [] | Expression | Magnetism | $D_{i k}$: Electric quadrupole moment tensor of the sphere.
$a$: Radius of the sphere.
$c$: Speed of light in vacuum.
$\varepsilon$: Permittivity of the dielectric sphere.
$\mu$: Permeability of the dielectric sphere.
$\mathfrak{H}_{i}$: i-th component of the constant magnetic field $\mathfrak{H}$.
$\Omega_{k}$: k-th c... | |
159 | A magnetized dielectric sphere (with dielectric constant $\varepsilon$) rotates uniformly in a vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with angular velocity $\Omega$. This rotation generates an electric field around the sphere. To describe this electric field, it is necessary t... | [] | Expression | Magnetism | $D_{zz}$: $zz$-component of the electric quadrupole tensor.
$\varepsilon$: Dielectric constant of the sphere.
$c$: Speed of light in vacuum.
$a$: Radius of the sphere.
$\Omega$: Angular velocity of the sphere's rotation.
$\mathscr{M}$: Total magnetic moment of the sphere. | |
160 | A magnetized metallic sphere (considered as the case of dielectric constant $\varepsilon \rightarrow \infty$) rotates uniformly in vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with an angular velocity $\Omega$. This rotation will generate an electric field around the sphere. Determi... | [] | Expression | Magnetism | $D_{zz}$: $zz$ component of the electric quadrupole moment tensor.
$c$: Speed of light in vacuum.
$\Omega$: Angular velocity of the sphere's rotation.
$\mathscr{M}$: Total magnetic moment of the sphere.
$a$: Radius of the sphere.
$\varepsilon$: Dielectric constant of the sphere, which approaches infinity for a metallic... | |
161 | Try to find the dispersion relation of magnetostatic oscillations in an unbounded medium. | [] | Expression | Magnetism | $\omega$: Angular frequency of magnetostatic oscillations.
$\gamma$: Gyromagnetic ratio.
$M$: Magnetization of the medium.
$\beta$: A parameter in the permeability tensor, related to the medium's properties.
$\theta$: Angle between the wave vector $\boldsymbol{k}$ and the easy magnetization axis (z-axis). | |
162 | The surface of an infinite parallel plate is perpendicular to the easy magnetization axis, and an external magnetic field $\mathfrak{H}$ is applied along this axis direction. Determine the non-uniform resonance frequency within this plate. | [] | Expression | Magnetism | $\omega$: Non-uniform resonance frequency (vibration frequency).
$\gamma$: Gyromagnetic ratio.
$M$: Magnetization.
$\beta$: A parameter in the expression for $\mu_{xx}$.
$\mathfrak{H}$: External magnetic field applied along the easy magnetization axis.
$\pi$: Mathematical constant pi.
$\theta$: Angle between the wave v... | |
163 | Calculate the reflection coefficient when light is almost grazing the surface of a material with $\varepsilon$ close to 1 from a vacuum. | [] | Expression | Magnetism | $R_{\perp}$: Reflection coefficient for perpendicular polarization
$R_{\|}$: Reflection coefficient for parallel polarization
$\varphi_{0}$: Grazing angle, defined as $\pi / 2 - \theta_0$
$\varepsilon$: Permittivity of the material | |
164 | Find the reflection coefficient $R_{\perp}$ when a wave is incident from vacuum onto the surface of a medium where both $\varepsilon$ and $\mu$ are different from 1, with the electric field vector perpendicular to the plane of incidence. | [] | Expression | Magnetism | $R_{\perp}$: Reflection coefficient when the electric field vector is perpendicular to the plane of incidence.
$\mu$: Relative permeability of the medium.
$\theta_0$: Angle of incidence from vacuum.
$\varepsilon$: Relative permittivity of the medium. | |
165 | Find the reflection coefficient $R_{\|}$ when a wave is incident on the surface of a medium with both $\varepsilon$ and $\mu$ different from 1, and when the electric field vector is parallel to the plane of incidence. | [] | Expression | Magnetism | $R_{\|}$: Reflection coefficient when the electric field vector is parallel to the plane of incidence.
$\varepsilon$: Permittivity of the medium.
$\mu$: Permeability of the medium.
$\theta_{0}$: Angle of incidence. | |
166 | For metals with impedance determined by formula \begin{align*}
\zeta = (1 - i)\sqrt{\frac{\omega\mu}{8\pi\sigma}}
\end{align*} (a special case with a flat surface having low impedance), and assuming its permeability $\mu=1$, try to determine the ratio of its thermal radiation intensity to the absolute blackbody surface... | [] | Expression | Magnetism | $I$: Thermal radiation intensity from the metal surface
$I_0$: Absolute blackbody surface radiation intensity
$\omega$: Angular frequency of radiation
$\pi$: Mathematical constant pi
$\sigma$: Conductivity of the metal | |
167 | Determine the dependence of the radiation intensity of a dipole emitter immersed in a homogeneous isotropic medium on the medium's permittivity $\varepsilon$ and magnetic permeability $\mu$. | [] | Expression | Magnetism | $I$: Radiation intensity of the dipole emitter in the medium.
$I_{0}$: Radiation intensity of the dipole emitter in vacuum.
$\mu$: Magnetic permeability of the homogeneous isotropic medium.
$\varepsilon$: Permittivity of the homogeneous isotropic medium. | |
168 | For the E wave in a circular waveguide with a radius of $a$, provide the expression for its attenuation coefficient $\alpha$. | [] | Expression | Magnetism | $\alpha$: Attenuation coefficient of the E wave in the circular waveguide
$\omega$: Angular frequency of the E wave
$\zeta^{\prime}$: Surface impedance of the waveguide material
$c$: Speed of light in vacuum
$a$: Radius of the circular waveguide
$k_z$: Longitudinal wave number (propagation constant) of the E wave | |
169 | Provide the expression for the attenuation coefficient $\alpha$ of H modes in a circular (radius $a$) cross-section waveguide. | [] | Expression | Magnetism | $\alpha$: Attenuation coefficient of H modes in the waveguide.
$c$: Speed of light in vacuum.
$\varkappa$: Transverse wave number (used interchangeably with $\kappa$).
$\zeta'$: Surface impedance, representing losses in the waveguide walls.
$\omega$: Angular frequency of the electromagnetic wave.
$k_z$: Longitudinal wa... | |
170 | Linearly polarized light is scattered by small particles with random orientations, where the particles' polarizability tensor has three distinct principal values. It is known that the scalar constants describing the average of the electric dipole moment tensor are $A = \frac{1}{15}(2 \alpha_{i i} \alpha_{k k}^{*}-\alph... | [] | Expression | Magnetism | $I_y$: Intensity of scattered light in the y-direction.
$I_x$: Intensity of scattered light in the x-direction.
$B$: Scalar constant describing the average of the electric dipole moment tensor, $B = \frac{1}{30}(3 \alpha_{i k} \alpha_{i k}^{*}-\alpha_{i i} \alpha_{k k}^{*})$
$A$: Scalar constant describing the average ... | |
171 | Try to express the components of the ray vector $s$ in terms of the components of $\boldsymbol{n}$ within the principal dielectric axes. | [] | Expression | Magnetism | $s_x$: x-component of the ray vector $\boldsymbol{s}$
$n_x$: x-component of the wave vector $\boldsymbol{n}$
$\varepsilon^{(x)}$: Principal dielectric constant (permittivity) along the x-axis
$\varepsilon^{(y)}$: Principal dielectric constant (permittivity) along the y-axis
$\varepsilon^{(z)}$: Principal dielectric con... | |
172 | Find the polarization of the reflected light when linearly polarized light is perpendicularly incident from vacuum onto the surface of an anisotropic object induced by a magnetic field. | [] | Expression | Magnetism | $g$: A parameter related to the magnetic field induced anisotropy, appearing in the approximation for $E_{1y}$ and the final ratio.
$\theta$: Angle between the incident direction and the vector $\boldsymbol{g}$.
$n_{0}$: Base refractive index of the anisotropic medium. | |
173 | Attempt to find the limiting law of the dependence of the surface tension coefficient $\alpha$ of liquid nitrogen near absolute zero on temperature | [] | Expression | Superconductivity | $\alpha$: Surface tension coefficient of liquid nitrogen; free energy per unit surface area of a liquid.
$\alpha_0$: Surface tension coefficient at absolute zero temperature ($T=0$).
$T$: Absolute temperature. (Note: The symbol $\tau$ is used inconsistently in some intermediate integral expressions, but it represents t... | |
174 | Try to find the dispersion relation of impurity particles in a moving superfluid $\varepsilon_{\text {imp}}(p)$, given that in a stationary fluid the dispersion relation is $\varepsilon_{\text {imp}}^{(0)}(p)$. | [] | Expression | Superconductivity | $\varepsilon_{\text {imp}}$: Dispersion relation of impurity particles in a moving superfluid.
$\varepsilon_{\text {imp}}^{(0)}$: Dispersion relation of impurity particles in a stationary fluid.
$p$: Momentum of the impurity atom in the moving superfluid's reference frame, defined as $p=p_{0}+m v$.
$m$: Mass of the imp... | |
175 | Try to find the dispersion relation of small oscillations of a rectilinear vortex line (W. Thomson, 1880). | [] | Expression | Superconductivity | $\omega$: Angular frequency of the oscillation.
$\kappa$: Vortex strength (circulation).
$k$: Wave number of the oscillation.
$a$: Vortex core radius. | |
176 | A neutron with an initial velocity $v$ scatters within a liquid. Determine the conditions under which an excitation with momentum $p$ and energy $\varepsilon(p)$ can be produced. | [] | Expression | Superconductivity | $V$: Magnitude of the initial velocity of the neutron
$\varepsilon(p)$: Energy of the excitation, as a function of its momentum $p$
$p$: Magnitude of the momentum of the excitation
$m$: Mass of the neutron | |
177 | Try to find the magnetic moment of a superconducting sphere of radius $R \ll \delta$ in a magnetic field under the London situation. | [] | Expression | Superconductivity | $\boldsymbol{M}$: Magnetic moment
$R$: Radius of the superconducting sphere
$\delta$: London penetration depth
$\mathfrak{G}$: External magnetic field | |
178 | For superconductors with parameter $\kappa \ll 1$, find the first-order correction to the penetration depth in a weak magnetic field. | [] | Expression | Superconductivity | $\delta_{\text{eff}}$: Effective penetration depth of the magnetic field into the superconductor.
$\delta$: London penetration depth, the uncorrected penetration depth.
$\kappa$: Ginzburg-Landau parameter, a dimensionless parameter characterizing superconductors.
$\mathfrak{S}$: Magnetic field strength at the surface o... | |
179 | Attempt to find the critical field of a superconducting small sphere with radius $R \ll \delta$. | [] | Expression | Superconductivity | $H_{\mathrm{c}}^{\text {(sphere)}}$: Critical magnetic field of the small superconducting sphere
$H_{\mathrm{c}}$: Critical magnetic field
$\delta$: Superconducting penetration depth
$R$: Radius of the small superconducting sphere | |
180 | Calculate the interaction energy of two vortices separated by $d \gg \xi$. | [] | Expression | Superconductivity | $\varepsilon_{12}$: Interaction energy per unit length between the two vortices.
$\phi_0$: Magnetic flux quantum.
$\delta$: Penetration depth.
$d$: Separation distance between two vortices. | |
181 | A thin film (thickness $d \ll \xi(T)$) is placed in a weak magnetic field perpendicular to its plane. Find the magnetic moment of the film when the temperature $T$ $>T_{\mathrm{c}}, T-T_{\mathrm{c}} \ll T_{\mathrm{c}}$. | [] | Expression | Superconductivity | $M$: Magnetic moment of the film.
$S$: Area of the film.
$e$: Elementary charge.
$T_{\mathrm{c}}$: Critical temperature.
$\mathfrak{S}$: Magnetic field, appearing in the calculation of the number of eigenfunctions, the derived free energy formula, and the provided final answer.
$\pi$: Mathematical constant pi.
$m$: Mas... | |
182 | Under the conditions of the previous question, determine the magnetic moment of a small sphere with radius $R \ll \xi(T)$ | [] | Expression | Superconductivity | $e$: Elementary charge
$T_{\mathrm{c}}$: Critical temperature
$R$: Radius of the small sphere
$\mathscr{S}_{\mathrm{g}}$: Magnetic field strength (or flux density)
$m$: Mass of the charge carrier (e.g., electron)
$c$: Speed of light in vacuum
$\alpha$: A constant related to the temperature dependence
$T$: Temperature | |
183 | Find the energy spectrum of spin wave quanta in an uniaxial ferromagnet of the 'easy magnetization plane' type $(K<0)$. | [] | Expression | Magnetism | $\varepsilon$: Energy of spin wave quanta
$k$: Magnitude of the wave vector
$\beta$: A constant related to the equations of motion for magnetization
$M$: Magnetization, representing the equilibrium magnetization $M_0$
$\alpha$: Exchange stiffness constant
$|K|$: Magnitude of the anisotropy constant
$\pi$: Mathematical ... | |
184 | Calculate the spin wave quantum part of thermodynamic quantities (energy) at temperature $T \ll \varepsilon(0)$. | [] | Expression | Magnetism | $E_{\operatorname{mag}}$: Spin wave quantum part of the thermodynamic energy.
$V$: Volume.
$K$: Anisotropy constant.
$T$: Temperature.
$\pi$: Mathematical constant pi.
$A$: Coefficient related to exchange stiffness, where for cubic crystals $A=2 \beta M \alpha$, and for 'easy magnetization axis'-type uniaxial crystals ... | |
185 | In the exchange approximation, determine the spatial correlation function of magnetization fluctuations at distances $r \gg a$. | [] | Expression | Magnetism | $\varphi_{i k}(\boldsymbol{r})$: Spatial correlation function of magnetization fluctuations.
$\beta$: Constant related to spin or magnetic moment.
$M$: Magnetization.
$\mathrm{e}$: Base of the natural logarithm.
$\varepsilon(k)$: Energy of a spin-wave quantum state, dependent on the magnitude of the wave vector $k$.
$k... | |
186 | Given $S \gg 1$, try to find the correction terms related to the interaction of heat capacity of a cubic lattice. In this lattice, only the exchange integral between a pair of neighboring atoms (along the cubic axes) is non-zero. | [] | Expression | Magnetism | $C_{\text{int}}$: Interaction heat capacity, $C_{\text{int}}=\frac{15 \pi \zeta^{2}(5 / 2) N}{S}(\frac{T}{4 \pi S J_{0}})^{4}$
$\pi$: Mathematical constant pi
$\zeta$: Riemann zeta function
$N$: Number of atoms in the lattice
$S$: Spin quantum number, given $S \gg 1$
$T$: Temperature
$J_0$: Exchange integral between a ... | |
187 | Try to determine the interaction law between an atom and a metal wall at 'large' distances. | [] | Expression | Magnetism | $U(L)$: Interaction energy between a single atom and the wall as a function of distance $L$.
$\alpha_{2}$: A coefficient related to the interaction energy between a single atom and the wall.
$\hbar$: Reduced Planck's constant.
$c$: Speed of light in vacuum.
$\pi$: Mathematical constant pi.
$L$: Distance between the ato... | |
188 | Determine the correlation function $\nu(r)$ of a Bose liquid at temperature $T \ll T_{\lambda}$, at distances $r \gtrsim \hbar u / T$. | [] | Expression | Strongly Correlated Systems | $\nu(r)$: Correlation function of a Bose liquid at distance $r$.
$T$: Temperature of the Bose liquid.
$m$: Mass of the particles in the Bose liquid.
$u$: Speed of sound (or phonon velocity) in the liquid.
$\hbar$: Reduced Planck's constant.
$r$: Distance. | |
189 | In a Bose superfluid, the condensate wave function exhibits fluctuations. Try to find the asymptotic form of this fluctuation correlation function at large distances. | [] | Expression | Strongly Correlated Systems | $G(r)$: Simultaneous correlation function.
$T$: Temperature.
$n_0$: Condensate density.
$m$: Mass of a boson.
$\pi$: Mathematical constant pi.
$\hbar$: Reduced Planck's constant.
$\rho_s$: Superfluid density.
$r$: Distance. | |
190 | Calculate the bandwidth of the body-centered cubic lattice; | [] | Expression | Semiconductors | $\Delta E$: Bandwidth of the body-centered cubic lattice
$J$: Overlap integral | |
191 | Calculate the effective mass of electrons at the bottom of the band; | [] | Expression | Semiconductors | $m_{\mathrm{b}}^{*}$: Effective mass of electrons at the bottom of the band, defined as $m_{\mathrm{b}}^{*}=\frac{h^{2}}{8 \pi^{2} a^{2} J}$.
$h$: Planck's constant.
$\pi$: Mathematical constant pi.
$a$: Lattice constant of the body-centered cubic lattice.
$J$: Overlap integral, a parameter in the tight-binding approxi... | |
192 | Calculate the effective mass of electrons at the top of the band; | [] | Expression | Semiconductors | $m_{\mathrm{t}}^{*}$: Effective mass of electrons at the top of the band, $m_{\mathrm{t}}^{*}=-\frac{h^{2}}{8 \pi^{2} a^{2} J}$.
$h$: Planck's constant.
$a$: Lattice constant.
$J$: Overlap integral. | |
193 | Find the bandwidth $(\Delta E)$ ; | [] | Expression | Semiconductors | $\Delta E$: Bandwidth of the energy band
$J$: Hopping integral (or overlap integral) between nearest neighbors | |
194 | The energy $E$ near the top of the valence band of a semiconductor crystal can be expressed as: $E(k)=E_{\text {max }}-10^{26} k^{2}(\mathrm{erg})$. Now, remove an electron with the wave vector $k=10^{7} \mathrm{i} / \mathrm{cm}$, and find the speed of the hole left by this electron. | [] | Expression | Semiconductors | i: Unit vector in the x-direction | |
195 | In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:
$\psi_{k}(x)=\sin \frac{\pi}{a} x$:
Try to use Bloch's theorem to point out the wave vector $\mathbf{k}$ values within the reduced Brillouin zone. | [] | Expression | Semiconductors | $k$: Wave vector.
$\pi$: Mathematical constant pi, approximately 3.14159.
$a$: Lattice constant or period of the one-dimensional periodic potential. | |
196 | In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:
$\psi_{k}(x)=i \cos \frac{3 \pi}{a} x$ ；
Try to use Bloch's theorem to point out the wave vector $\mathbf{k}$ values within the reduced Brillouin zone. | [] | Expression | Semiconductors | $k$: Wave number, representing the magnitude of the wave vector in one dimension.
$\pi$: Mathematical constant pi.
$a$: Periodicity length of the one-dimensional potential (lattice constant). | |
197 | The electron wave function moving in a one-dimensional periodic potential field has the following form: $\psi_{k}(x)=\sum_{l=-\infty}^{\infty}(-1)^{l} f(x-l a)$ . Here $a$ is the lattice constant of the one-dimensional lattice, $f(x)$ is a certain function, try using Bloch's theorem to indicate the values of the wave v... | [] | Expression | Semiconductors | $k$: Wave vector
$\pi$: Mathematical constant pi
$a$: Lattice constant of the one-dimensional lattice | |
198 | Try to find: the width of the energy band; | [] | Expression | Semiconductors | $\Delta E$: Width of the energy band
$h$: Planck's constant
$m_0$: Mass of the electron
$a$: Lattice constant of the one-dimensional crystal | |
199 | Try to find the velocity of the electron at wave vector $k$; | [] | Expression | Semiconductors | $v(\boldsymbol{k})$: Velocity of the electron at wave vector $\boldsymbol{k}$
$h$: Planck's constant
$m_0$: Mass of the electron
$a$: Lattice constant
$k$: Wave vector | |
200 | Find the effective mass of electrons at the bottom of the band. | [] | Expression | Semiconductors | $m_0$: Rest mass of the electron. |
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