We know that the current through an inductor LL size 12{L} {} cannot be turned on or off instantaneously. The change in current changes flux, inducing an emf opposing the change (Lenz’s law). How long does the opposition last? Current will flow and can be turned off, but how long does it take? [link] shows a switching circuit that can be used to examine current through an inductor as a function of time.

When the switch is first moved to position 1 (at t=0t=0 size 12{t=0} {}), the current is zero and it eventually rises to I0=V/RI0=V/R size 12{I rSub { size 8{0} } = ital “V/R”} {}, where RR is the total resistance of the circuit. The opposition of the inductor LL size 12{L} {} is greatest at the beginning, because the amount of change is greatest. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. The opposing emf is proportional to the amount of change left. This is the hallmark of an exponential behavior, and it can be shown with calculus that

is the current in an RL circuit when switched on (Note the similarity to the exponential behavior of the voltage on a charging capacitor). The initial current is zero and approaches I0=V/RI0=V/R size 12{I rSub { size 8{0} } = ital “V/R”} {} with a characteristic time constant
ττ
for an RL circuit, given by

where ττ size 12{τ} {} has units of seconds, since
1H=1Ω·s1H=1Ω·s.
In the first period of time ττ size 12{τ} {}, the current rises from zero to 0.632I00.632I0 size 12{0 “.” “632”I rSub { size 8{0} } } {}, since I=I0(1−e−1)=I0(1−0.368)=0.632I0I=I0(1−e−1)=I0(1−0.368)=0.632I0 size 12{I=I rSub { size 8{0} } ( 1 – e rSup { size 8{ – 1} } ) =I rSub { size 8{0} } ( 1 – 0 “.” “368” ) =0 “.” “632”I rSub { size 8{0} } } {}. The current will go 0.632 of the remainder in the next time ττ size 12{τ} {}. A well-known property of the exponential is that the final value is never exactly reached, but 0.632 of the remainder to that value is achieved in every characteristic time ττ size 12{τ} {}. In just a few multiples of the time ττ size 12{τ} {}, the final value is very nearly achieved, as the graph in [link](b) illustrates.

The characteristic time ττ size 12{τ} {} depends on only two factors, the inductance LL size 12{L} {} and the resistance RR size 12{R} {}. The greater the inductance LL size 12{L} {}, the greater ττ size 12{τ} {} is, which makes sense since a large inductance is very effective in opposing change. The smaller the resistance RR size 12{R} {}, the greater ττ size 12{τ} {} is. Again this makes sense, since a small resistance means a large final current and a greater change to get there. In both cases—large LL size 12{L} {} and small RR size 12{R} {} —more energy is stored in the inductor and more time is required to get it in and out.

When the switch in [link](a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. But this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, (1/2)LI02(1/2)LI02 size 12{ ( “1/2” ) ital “LI” rSub { size 8{0} } rSup { size 8{2} } } {}, stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is I2RI2R size 12{ I rSup { size 8{2} } R} {}. Once again the behavior is exponential, and
II
is found to be

(See [link](c).) In the first period of time τ=L/Rτ=L/R size 12{τ=L/R} {} after the switch is closed, the current falls to 0.368 of its initial value, since I=I0e−1=0.368I0I=I0e−1=0.368I0 size 12{I=I rSub { size 8{0} } e rSup { size 8{ – 1} } =0 “.” “368”I rSub { size 8{0} } } {}. In each successive time ττ size 12{τ} {}, the current falls to 0.368 of the preceding value, and in a few multiples of ττ size 12{τ} {}, the current becomes very close to zero, as seen in the graph in [link](c).

In summary, when the voltage applied to an inductor is changed, the current also changes, but the change in current lags the change in voltage in an RL circuit. In Reactance, Inductive and Capacitive, we explore how an RL circuit behaves when a sinusoidal AC voltage is applied.