An RL Circuit with a Battery

Let's put an inductor (i.e., a coil with an inductance L) in series with a battery of emf e and a resistor of resistance R. This is known as an RL circuit. There are some similarities between the RL circuit and the RC circuit, and some important differences.

Consider what happens with the resistor and the battery. When the switch is closed we have a current; when the switch is open we have no current. Now add an inductor to the circuit. When we close the switch now the current tries to jump up to the same value we had with the resistor but the inductor opposes this because a change in current means a change in flux for the coil. If the inductor adds negligible resistance to the circuit the current eventually reaches the same value it had with the resistor but the current follows an exponential curve to get there.

To see why, apply Kirchoff's loop rule. With the switch closed we have:
e – IR – L
dI
dt
= 0

Solving this for current gives I(t) = Io [1 - e-t/t ]
where Io =
e
R
is the maximum current
and the time constant t =
L
R

Look at the voltage vs. time graphs. One graph represents the resistor voltage as a function of time. The other graph represents the inductor voltage as a function of time. Which graph is which?

  1. Red is the resistor voltage; Blue is the inductor voltage. (32/65) (49%)
  2. Blue is the resistor voltage; Red is the inductor voltage. (33/65) (51%)















The first choice is correct. Red is the resistor voltage, and blue is the inductor voltage. The resistor voltage is proportional to the current, so the graph of resistor voltage looks like the current graph. The inductor has an impact right after you open or close the switch, so its voltage has a large magnitude then...and then the inductor voltage decreases to zero.

The resistor voltage has the same form as the current:

DVR = e [1 - e-t/t ]

The potential difference across the inductor is:

DVL = e e-t/t

The graph of current as a function of time in the RL circuit has the same form as the graph of the capacitor voltage as a function of time in the RC circuit, while the graph of the inductor voltage as a function of time in the RL circuit has the same form as the graph of current vs. time in the RC circuit.