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The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
The magnetic field strength at the center of a circular loop is given byB=μ0I2R(at center of loop),B=μ0I2R(at center of loop),whereRis the radius of the loop. RHR-2 gives the direction of the field about the loop. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampèreâs law. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Ampèreâs law can be used to determine the magnetic field from a thin wire or thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
The magnetic field strength inside a solenoid isB=μ0nI(inside a solenoid)B=μ0nI(inside a solenoid)wherenis the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
The magnetic field strength inside a toroid isB=μoNI2Ïr(within the toroid)B=μoNI2Ïr(within the toroid)whereNis the number of windings. The field inside a toroid is not uniform and varies with the distance as 1/r. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Materials are classified as paramagnetic, diamagnetic, or ferromagnetic, depending on how they behave in an applied magnetic field. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field. This is a negative magnetic susceptibility. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed, the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis. | https://openstax.org/books/university-physics-volume-2/pages/12-summary |
Φ m = â« S B â · n ^ d A Φ m = â« S B â · n ^ d A | https://openstax.org/books/university-physics-volume-2/pages/13-key-equations |
ε = â N d Φ m d t ε = â N d Φ m d t | https://openstax.org/books/university-physics-volume-2/pages/13-key-equations |
ε = B l v ε = B l v | https://openstax.org/books/university-physics-volume-2/pages/13-key-equations |
ε = â® E â · d l â = â d Φ m d t ε = â® E â · d l â = â d Φ m d t | https://openstax.org/books/university-physics-volume-2/pages/13-key-equations |
ε = N B A Ï sin ( Ï t ) ε = N B A Ï sin ( Ï t ) | https://openstax.org/books/university-physics-volume-2/pages/13-key-equations |
back emf : emf generated by a running motor, because it consists of a coil turning in a magnetic field; it opposes the voltage powering the motor | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
eddy current : current loop in a conductor caused by motional emf | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
electric generator : device for converting mechanical work into electric energy; it induces an emf by rotating a coil in a magnetic field | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
Faradayâs law : induced emf is created in a closed loop due to a change in magnetic flux through the loop | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
induced electric field : created based on the changing magnetic flux with time | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
induced emf : short-lived voltage generated by a conductor or coil moving in a magnetic field | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
Lenzâs law : direction of an induced emf opposes the change in magnetic flux that produced it; this is the negative sign in Faradayâs law | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
magnetic damping : drag produced by eddy currents | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
magnetic flux : measurement of the amount of magnetic field lines through a given area | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
motionally induced emf : voltage produced by the movement of a conducting wire in a magnetic field | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
peak emf : maximum emf produced by a generator | https://openstax.org/books/university-physics-volume-2/pages/13-key-terms |
The magnetic flux through an enclosed area is defined as the amount of field lines cutting through a surface areaAdefined by the unit area vector. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
The units for magnetic flux are webers, where1Wb=1T·m2.1Wb=1T·m2. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
The induced emf in a closed loop due to a change in magnetic flux through the loop is known as Faradayâs law. If there is no change in magnetic flux, no induced emf is created. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
We can use Lenzâs law to determine the directions of induced magnetic fields, currents, and emfs. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
The direction of an induced emf always opposes the change in magnetic flux that causes the emf, a result known as Lenzâs law. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
The relationship between an induced emfεεin a wire moving at a constant speedvthrough a magnetic fieldBis given byε=Blv.ε=Blv. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
An induced emf from Faradayâs law is created from a motional emf that opposes the change in flux. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
A changing magnetic flux induces an electric field. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Both the changing magnetic flux and the induced electric field are related to the induced emf from Faradayâs law. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Current loops induced in moving conductors are called eddy currents. They can create significant drag, called magnetic damping. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Manipulation of eddy currents has resulted in applications such as metal detectors, braking in trains or roller coasters, and induction cooktops. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
An electric generator rotates a coil in a magnetic field, inducing an emf given as a function of time byε=NBAÏsin(Ït)ε=NBAÏsin(Ït)whereAis the area of anN-turn coil rotated at a constant angular velocityÏÏin a uniform magnetic fieldBâ.Bâ. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
The peak emf of a generator isε0=NBAÏε0=NBAÏ. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Any rotating coil produces an induced emf. In motors, this is called back emf because it opposes the emf input to the motor. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Hard drives utilize magnetic induction to read/write information. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
Other applications of magnetic induction can be found in graphics tablets, electric and hybrid vehicles, and in transcranial magnetic stimulation. | https://openstax.org/books/university-physics-volume-2/pages/13-summary |
M = N 2 Φ 21 I 1 = N 1 Φ 12 I 2 M = N 2 Φ 21 I 1 = N 1 Φ 12 I 2 | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
ε 1 = â M d I 2 d t ε 1 = â M d I 2 d t | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
N Φ m = L I N Φ m = L I | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
ε = â L d I d t ε = â L d I d t | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
L solenoid = μ 0 N 2 A l L solenoid = μ 0 N 2 A l | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
L toroid = μ 0 N 2 h 2 Ï ln R 2 R 1 . L toroid = μ 0 N 2 h 2 Ï ln R 2 R 1 . | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
U = 1 2 L I 2 U = 1 2 L I 2 | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
I ( t ) = ε R ( 1 â e â t / Ï L ) I ( t ) = ε R ( 1 â e â t / Ï L ) | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
Ï L = L / R Ï L = L / R | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
q ( t ) = q 0 cos ( Ï t + Ï ) q ( t ) = q 0 cos ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
Ï = 1 L C Ï = 1 L C | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
i ( t ) = â Ï q 0 sin ( Ï t + Ï ) i ( t ) = â Ï q 0 sin ( Ï t + Ï ) | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
q ( t ) = q 0 e â R t / 2 L cos ( Ï â² t + Ï ) q ( t ) = q 0 e â R t / 2 L cos ( Ï â² t + Ï ) | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
Ï â² = 1 L C â ( R 2 L ) 2 Ï â² = 1 L C â ( R 2 L ) 2 | https://openstax.org/books/university-physics-volume-2/pages/14-key-equations |
henry (H) : unit of inductance,1H=1Ω·s1H=1Ω·s; it is also expressed as a volt second per ampere | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
inductance : property of a device that tells how effectively it induces an emf in another device | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
inductive time constant : denoted byÏÏ, the characteristic time given by quantityL/Rof a particular seriesRLcircuit | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
inductor : part of an electrical circuit to provide self-inductance, which is symbolized by a coil of wire | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
LCcircuit : circuit composed of an ac source, inductor, and capacitor | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
magnetic energy density : energy stored per volume in a magnetic field | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
mutual inductance : geometric quantity that expresses how effective two devices are at inducing emfs in one another | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
RLCcircuit : circuit with an ac source, resistor, inductor, and capacitor all in series. | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
self-inductance : effect of the device inducing emf in itself | https://openstax.org/books/university-physics-volume-2/pages/14-key-terms |
Inductance is the property of a device that expresses how effectively it induces an emf in another device. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
Mutual inductance is the effect of two devices inducing emfs in each other. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
A change in currentdI1/dtdI1/dtin one circuit induces an emf(ε2)(ε2)in the second:ε2=âMdI1dt,ε2=âMdI1dt,whereMis defined to be the mutual inductance between the two circuits and the minus sign is due to Lenzâs law. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
Symmetrically, a change in currentdI2/dtdI2/dtthrough the second circuit induces an emf(ε1)(ε1)in the first:ε1=âMdI2dt,ε1=âMdI2dt,whereMis the same mutual inductance as in the reverse process. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
Current changes in a device induce an emf in the device itself, called self-inductance,ε=âLdIdt,ε=âLdIdt,whereLis the self-inductance of the inductor anddI/dtdI/dtis the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenzâs law. The unit of ... | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The self-inductance of a solenoid isL=μ0N2Al,L=μ0N2Al,whereNis its number of turns in the solenoid,Ais its cross-sectional area,lis its length, andμ0=4ÏÃ10â7T·m/Aμ0=4ÏÃ10â7T·m/Ais the permeability of free space. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The self-inductance of a toroid isL=μ0N2h2ÏlnR2R1,L=μ0N2h2ÏlnR2R1,whereNis its number of turns in the toroid,R1andR2R1andR2are the inner and outer radii of the toroid,his the height of the toroid, andμ0=4ÏÃ10â7T·m/Aμ0=4ÏÃ10â7T·m/Ais the permeability of free space. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The energy stored in an inductorUisU=12LI2.U=12LI2. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The self-inductance per unit length of coaxial cable isLl=μ02ÏlnR2R1.Ll=μ02ÏlnR2R1. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
When a series connection of a resistor and an inductorâanRLcircuitâis connected to a voltage source, the time variation of the current isI(t)=εR(1âeâRt/L)=εR(1âeât/ÏL)I(t)=εR(1âeâRt/L)=εR(1âeât/ÏL)(turning on),where the initial current isI0=ε/R.I0=ε/R. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The characteristic time constantÏÏisÏL=L/R,ÏL=L/R,whereLis the inductance andRis the resistance. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
In the first time constantÏ,Ï,the current rises from zero to0.632I0,0.632I0,and to 0.632 of the remainder in every subsequent time intervalÏ.Ï. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
When the inductor is shorted through a resistor, current decreases asI(t)=εReât/ÏLI(t)=εReât/ÏL(turning off).Current falls to0.368I00.368I0in the first time intervalÏÏ, and to 0.368 of the remainder toward zero in each subsequent timeÏ.Ï. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The energy transferred in an oscillatory manner between the capacitor and inductor in anLCcircuit occurs at an angular frequencyÏ=1LCÏ=1LC. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The charge and current in the circuit are given byq(t)=q0cos(Ït+Ï),i(t)=âÏq0sin(Ït+Ï).q(t)=q0cos(Ït+Ï),i(t)=âÏq0sin(Ït+Ï). | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The underdamped solution for the capacitor charge in anRLCcircuit isq(t)=q0eâRt/2Lcos(Ïâ²t+Ï).q(t)=q0eâRt/2Lcos(Ïâ²t+Ï). | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
The angular frequency given in the underdamped solution for theRLCcircuit isÏâ²=1LCâ(R2L)2.Ïâ²=1LCâ(R2L)2. | https://openstax.org/books/university-physics-volume-2/pages/14-summary |
v = V 0 sin Ï t v = V 0 sin Ï t | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
i = I 0 sin Ï t i = I 0 sin Ï t | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
V 0 I 0 = 1 Ï C = X C V 0 I 0 = 1 Ï C = X C | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
V rms = V 0 2 V rms = V 0 2 | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
I rms = I 0 2 I rms = I 0 2 | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
V 0 I 0 = Ï L = X L V 0 I 0 = Ï L = X L | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
Ï = tan â1 X L â X C R Ï = tan â1 X L â X C R | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
I 0 = V 0 Z I 0 = V 0 Z | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
Z = R 2 + ( X L â X C ) 2 Z = R 2 + ( X L â X C ) 2 | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
P ave = 1 2 I 0 V 0 cos Ï P ave = 1 2 I 0 V 0 cos Ï | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
P ave = 1 2 I 0 V 0 = I rms V rms = I rms 2 R P ave = 1 2 I 0 V 0 = I rms V rms = I rms 2 R | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
Ï 0 = 1 L C Ï 0 = 1 L C | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
Q = Ï 0 Î Ï Q = Ï 0 Î Ï | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
Q = Ï 0 L R Q = Ï 0 L R | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
V S V P = N S N P V S V P = N S N P | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
I S = N P N S I P I S = N P N S I P | https://openstax.org/books/university-physics-volume-2/pages/15-key-equations |
ac current : current that fluctuates sinusoidally with time at a fixed frequency | https://openstax.org/books/university-physics-volume-2/pages/15-key-terms |
ac voltage : voltage that fluctuates sinusoidally with time at a fixed frequency | https://openstax.org/books/university-physics-volume-2/pages/15-key-terms |
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