The circuit above is composed of a constant voltage source E, an inductor L, a resistor R and two switches S

_{1} and S

_{2}. Prior to time t

_{0} , both switches S

_{1} and S

_{2} are open.

a) Switch S

_{1} is closed at time t

_{0} = 0s. What is the current in the inductor L immediately after S

_{1} was closed?

b) What is the current in the inductor L at a later time t

_{1}, after the voltage on resistor R

_{1} has stabilized?

c) At time t

_{2}, after the current in R has stabilized, S

_{2} is closed and S

_{1} is opened at the same time. Write a differential equation and solve it to find the current in the inductor as a function of time passed from the moment t

_{0}, E, R, L and t

_{2}.

d) Sketch the shape of the current in the inductor L, from time t

_{0} until the current stabilizes after the switch S

_{2} is closed.

**Solution:**
a) The current through the inductor L is zero immediately after S

_{1} is closed. The current in an inductor is proportional with the integral of the voltage on the inductor.

b) If the current through the inductor has stabilized, di/dt = 0, and the voltage on the inductor is 0V:

u

_{L} = L(di/dt) = 0V.

According to the Kirchoff's Second law, E = 0 + RI and I = E/R.

c) Kirchoff's Second law:
Ldi/dt + Ri = 0

di/i = (-R/L)dt

ln(i) = (-R/L)t + C

At t = t

_{2}, the current is E/R, so ln(E/R) = (-R/L)t

_{2} + C.

ln(i) = (-R/L)t + ln(E/R)+ + (R/L)t

_{2}
ln(Ri/E) = (-R/L)(t - t

_{2})

Ri/E = e

^{(-R/L)(t - t2)}
i = (E/R)e

^{-(t - t2)R/L} for any t > t

_{2}
d)