University Physics Volume 2

Summary

14.1Mutual 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 $dI1/dtdI1/dt$ in one circuit induces an emf $(ε2)(ε2)$ in the second:
$ε2=−MdI1dt,ε2=−MdI1dt,$
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 $dI2/dtdI2/dt$ through the second circuit induces an emf $(ε1)(ε1)$ in the first:
$ε1=−MdI2dt,ε1=−MdI2dt,$
where M is the same mutual inductance as in the reverse process.

14.2Self-Inductance and Inductors

• Current changes in a device induce an emf in the device itself, called self-inductance,
$ε=−LdIdt,ε=−LdIdt,$
where L is the self-inductance of the inductor and $dI/dtdI/dt$ 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 $1H=1Ω·s1H=1Ω·s$.
• The self-inductance of a solenoid is
$L=μ0N2Al,L=μ0N2Al,$
where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and $μ0=4π×10−7T·m/Aμ0=4π×10−7T·m/A$ is the permeability of free space.
• The self-inductance of a toroid is
$L=μ0N2h2πlnR2R1,L=μ0N2h2πlnR2R1,$
where N is its number of turns in the toroid, $R1andR2R1andR2$ are the inner and outer radii of the toroid, h is the height of the toroid, and $μ0=4π×10−7T·m/Aμ0=4π×10−7T·m/A$ is the permeability of free space.

14.3Energy in a Magnetic Field

• The energy stored in an inductor U is
$U=12LI2.U=12LI2.$
• The self-inductance per unit length of coaxial cable is
$Ll=μ02πlnR2R1.Ll=μ02πlnR2R1.$

14.4RL 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
$I(t)=εR(1−e−Rt/L)=εR(1−e−t/τL)I(t)=εR(1−e−Rt/L)=εR(1−e−t/τL)$ (turning on),
where the initial current is $I0=ε/R.I0=ε/R.$
• The characteristic time constant $ττ$ is $τL=L/R,τL=L/R,$ where L is the inductance and R is the resistance.
• In the first time constant $τ,τ,$ the current rises from zero to $0.632I0,0.632I0,$ and to 0.632 of the remainder in every subsequent time interval $τ.τ.$
• When the inductor is shorted through a resistor, current decreases as
$I(t)=εRe−t/τLI(t)=εRe−t/τL$ (turning off).
Current falls to $0.368I00.368I0$ in the first time interval $ττ$, and to 0.368 of the remainder toward zero in each subsequent time $τ.τ.$

14.5Oscillations 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 $ω=1LCω=1LC$.
• The charge and current in the circuit are given by
$q(t)=q0cos(ωt+ϕ), i(t)=−ωq0sin(ωt+ϕ).q(t)=q0cos(ωt+ϕ), i(t)=−ωq0sin(ωt+ϕ).$

14.6RLC Series Circuits

• The underdamped solution for the capacitor charge in an RLC circuit is
$q(t)=q0e−Rt/2Lcos(ω′t+ϕ).q(t)=q0e−Rt/2Lcos(ω′t+ϕ).$
• The angular frequency given in the underdamped solution for the RLC circuit is
$ω′=1LC−(R2L)2.ω′=1LC−(R2L)2.$
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