Chapter 32:  Inductance

 

Learning Goals

  1. Understand self inductance and how to compute it.
  2. Understand the time dependence of RL circuits.
  3. Understand energy concepts relating to magnetic fields.
  4. Understand mutual inductance.
  5. Understand oscillations in LC circuits.

 

Self Inductance

Since any circuit is linked by its own magnetic flux, Faraday’s Law requires that a self-induced emf be generated any time the current in a circuit changes in time:


where L is the self inductance of the circuit:


The self inductance depends only on the shape of the circuit - never on the current.

It is possible to shape a circuit so as to dramatically increase its self inductance. For example, wrapping wire into a coil of length d and area A with N turns produces an air-core soleniod with inductance


Actually, this formula is only correct for very long solenoids.

 

Mutual Inductance

The current in one circuit can induce currents in remote circuits whenever magnetic flux of one links the other.

 

RL Circuits

Circuits consisting of a DC voltage source V, inductance L , and resistance R connected in series exhibit exponential time dependencies. For example,

when the current is increasing, and


when the current is decreasing. In each case, t = L/R.

 

Energy Storage in Inductors and in Magnetic Fields

An inductor with current I stores magnetic energy equal to

A region of space containing a magnetic has an energy density given by


LC Oscillations

An inductor L and capacitor C connected in series will produce a circuit whose current oscillates harmonically in time with natural frequency

The energy in an oscillating LC circuit continuously transforms between energy stored in the inductor and energy stored in the capactor. If there is no resistance and if we may neglect radiation of electromagnetic energy (not a very good approximation at high frequencies), the total energy stored in the circuit remains constant: