An inductor is a device placed in a circuit to oppose a change in current; that is, to maintain, and regulate, a steady current in that section of the circuit. Generally an inductor is thought of as a coil of wire wound around either an air or ferromagnetic core. Shown below is the symbol for an inductor.

 

 

The unit used to measure the inductance, or the size of an inductor, is a henry (H). We will now derive an expression for inductance.

 

The flux through an area, A, resulting for a perpendicular field, , is expressed as

 

 

If there are N coils, then the total flux would be expressed as

 

  where 

 

where

 

 

When current is running through the coils, a uniform magnetic field is produced down the center of the inductor. That is, a given amount of magnetic flux is present. If the current were to change, the amount of flux would change. This changing flux induces an opposing emf in the coil. This self-induced emf is sometimes called a back emf.

 

Using Ampere's Law we can derive an expression for the magnetic along the axis of a tightly wound inductor as

 

      

 

Substituting in this expression for the magnetic field we get

 

 

This expression of total flux per unit current is known as the inductance of the coil, L.

 

 

One henry of inductance occurs when 1 weber of flux is generated by 1 amp of current circulating through the coil(s). Notice that the inductance of a coil is a constant for its geometry or physical characteristics: permeability (µ_o), coils per unit length(), and cross-sectional area, A.

 

 

 

Faraday's Law states the emf induced is a coil is proportional to the rate of change of flux. Later, Lenz added that the induced emf will be established in such a way as to oppose these changes and return the coil to its original condition. (This is actually a statement of conservation of energy. If the flux generated by the coil did NOT oppose the change in external flux then the induced emf would continue without limit.)

 

 

Substituting in our previous results we can develop an expression for the emf induced in an inductor.

 

 

Notice that the faster the current tries to change, the greater the self-induced emf will become. When the current is steady, , there will be NO self-induced voltage in the solenoid, or inductor .

 

 

 

Let's begin by looking at an example of an inductor-resistance circuit, an RL circuit.

 

When the switch is initially closed, we can use Kirchoff's loop rule to write an equation for the loop ABCDA where I represents the current in the circuit.

 

 

Since the inductor opposes change, it will initially act to thwart any current flowing through the circuit. The expression L/R is called the LR time constant.

 

 

 

 

As the current builds, the battery is supplying power to run the circuit. The energy not dissipated across the resistor is stored in the inductor's magnetic field. To derive an expression for this energy, we begin by multiplying every term by I.

 

 

where each term now represents power, or the rate at which work/energy is done/stored. In particular, for our inductor,

 

 

The magnetic energy density is the energy per unit volume stored in a magnetic field.

 

 

Recall that the electric energy density stored in an electric field (presented when we studied the energy stored in a charged capacitor) is

 

 

Once a maximum current is reached, the inductor can no longer resist change and it effectively disappears - having no further impact on the circuit. Notice that if the battery were to be removed from the circuit, the current should ordinarily immediately fall to zero. However, when the switch is closed the inductor would once again want to resist the change in current. Consequently the current would fall off in agreement with an exponential decay as dictated by the time constant, L/R.