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 in one circuit induces an emf 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 through the second circuit induces an emf 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 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 .
- 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 is the permeability of free space.
- The self-inductance of a toroid is
where N is its number of turns in the toroid, are the inner and outer radii of the toroid, h is the height of the toroid, and 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
where the initial current is
- The characteristic time constant is where L is the inductance and R is the resistance.
- In the first time constant the current rises from zero to and to 0.632 of the remainder in every subsequent time interval
- When the inductor is shorted through a resistor, current decreases as
Current falls to in the first time interval , and to 0.368 of the remainder toward zero in each subsequent time
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 .
- The charge and current in the circuit are given by
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