• Placing an inductance (L) and a resistance (R) in a simple circuit gives us the "LR" circuit, at right.  When the switch is closed to include the power supply in the circuit we can apply Kirchoff's loop theorem to obtain the following equation,

Note that in applying the loop theorem we make use of the fact that the induced emf across the inductance opposes the changing emf that causes it.

• The solution of this differential equation is given by,

shown at right, where I_max is ε/R  .  Also shown is the time dependence of the voltage across the inductance.

• The time constant of the inductance, given by  is a measure of the time needed for the current in the circuit to reach 63% of its maximum value.

• Taking the power supply out of the circuit leads to an even simpler differential equation
  

the solution of which is

shown at right.  In this figure I₀ = ε/R.