14.8: Inductance (Summary)
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Key Terms
henry (H) | unit of inductance, 1H=1Ω⋅s; it is also expressed as a volt second per ampere |
inductance | property of a device that tells how effectively it induces an emf in another device |
inductive time constant | denoted by τ, the characteristic time given by quantity L/R of a particular series RL circuit |
inductor | part of an electrical circuit to provide self-inductance, which is symbolized by a coil of wire |
LC circuit | circuit composed of an ac source, inductor, and capacitor |
magnetic energy density | energy stored per volume in a magnetic field |
mutual inductance | geometric quantity that expresses how effective two devices are at inducing emfs in one another |
RLC circuit | circuit with an ac source, resistor, inductor, and capacitor all in series. |
self-inductance | effect of the device inducing emf in itself |
Key Equations
Mutual inductance by flux | M=N2Φ21I=N1Φ12I2 |
Mutual inductance in circuits | ε1=−MdI2dt |
Self-inductance in terms of magnetic flux | NΦm=LI |
Self-inductance in terms of emf | ε=−LdIdt |
Self-inductance of a solenoid | Lsolenoid=μ0N2Al |
Self-inductance of a toroid | Ltoroid=μ0N2h2πlnR2R1. |
Energy stored in an inductor | U=12LI2 |
Current as a function of time for a RL circuit | I(t)=εR(1−e−t/τL) |
Time constant for a RL circuitτ | τL=L/R |
Charge oscillation in LC circuits | q(t)=q0cos(ωt+ϕ) |
Angular frequency in LC circuits | ω=√1LC |
Current oscillations in LC circuits | i(t)=−ωq0sin(ωt+ϕ) |
Charge as a function of time in RLC circuit | q(t)=q0e−Rt/2Lcos(ω′t+ϕ) |
Angular frequency in RLC circuit | ω′=√1LC−(R2L)2 |
Summary
14.2 Mutual Inductance
- Inductance is the property of a device that expresses how effectively it induces an emf in another device.
- Mutual inductance is the effect of two devices inducing emfs in each other.
- A change in current dI1/dt in one circuit induces an emf (ε2) in the second:
ε2=−MdI1dt,
where M is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.
- Symmetrically, a change in current dI2/dt through the second circuit induces an emf (ε1) in the first:
ε1=−MdI2dt,
where M is the same mutual inductance as in the reverse process.
14.3 Self-Inductance and Inductors
- Current changes in a device induce an emf in the device itself, called self-inductance,
ε=−LdIdt,
where L is the self-inductance of the inductor and dI/dt is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law. The unit of self-inductance and inductance is the henry (H), where 1H=1Ω⋅s.
- The self-inductance of a solenoid is
L=μ0N2Al,
where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and μ0=4π×10−7T⋅m/A is the permeability of free space.
- The self-inductance of a toroid is
L=μ0N2h2πlnR2R1,
where N is its number of turns in the toroid, R1 and R2 are the inner and outer radii of the toroid, h is the height of the toroid, and μ0=4π×10−7T⋅m/A is the permeability of free space.
14.4 Energy in a Magnetic Field
- The energy stored in an inductor U is
U=12LI2.
- The self-inductance per unit length of coaxial cable is
Ll=μ02πlnR2R1.
14.5 RL Circuits
- When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
I(t)=εR(1−e−Rt/L)=εR(1−e−t/τL) (turning on),
where the initial current is I0=ε/R..
- The characteristic time constant τ is τL=L/R, where L is the inductance and R is the resistance.
- In the first time constant τ, the current rises from zero to 0.632I0, and to 0.632 of the remainder in every subsequent time interval τ.
- When the inductor is shorted through a resistor, current decreases as
I(t)=εRe−t/τL (turning off).
Current falls to 0.368I0 in the first time interval τ, and to 0.368 of the remainder toward zero in each subsequent time τ.
14.6 Oscillations in an LC Circuit
- The energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency ω=√1LC.
- The charge and current in the circuit are given by
q(t)=q0cos(ωt+ϕ),
i(t)=−ωq0sin(ωt+ϕ).
14.7 RLC Series Circuits
- The underdamped solution for the capacitor charge in an RLC circuit is
q(t)=q0e−Rt/2Lcos(ω′t+ϕ).
- The angular frequency given in the underdamped solution for the RLC circuit is
ω′=√1LC−(R2L)2.