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14.1 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/dtdI1/dt in one circuit induces an emf (ε2)(ε2) in the second:
    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:
    where M is the same mutual inductance as in the reverse process.

14.2 Self-Inductance and Inductors

  • Current changes in a device induce an emf in the device itself, called self-inductance,
    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
    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
    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.

14.3 Energy in a Magnetic Field

  • The energy stored in an inductor U is
  • The self-inductance per unit length of coaxial cable is

14.4 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(1eRt/L)=εR(1et/τL)I(t)=εR(1eRt/L)=εR(1et/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)=εRet/τLI(t)=εRet/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.

14.5 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=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).

14.6 RLC Series Circuits

  • The underdamped solution for the capacitor charge in an RLC circuit is
  • The angular frequency given in the underdamped solution for the RLC circuit is
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