Kirchhoff’s Law states that the sum of voltage drops around a circuit (loop) is zero.?
For a passive RL circuit, in which an inductance of L Henries, a resistor of R Ohms,
a switch and a battery of V0 Volts are all connected in series, we obtain the following
first-order differential equation, in which I(t) Amperes is the current at time t seconds
after the switch is closed.
L*(dI(t)/dt) + RI(t) V0 = 0
(a) Use the integrating factor method to find the general solution I(t) of this equation.
(b) The switch is closed at t = 0, at which time the current is zero Amperes. Find
the particular solution satisfying this initial condition.
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Verified answer
(a)
dI/dt + (R/L) I = (Vo/L)
φ = e^[ ∫ (R/L) dt]
φ = e^[(R/L)t]
{I e^[(R/L)t]}' = (Vo/L) e^[(R/L)t]
I e^[(R/L)t] = (Vo/R) e^[(R/L)t] + C
I = (Vo/R) + C e^[-(R/L)t]
(b)
I(0) = (Vo/R) + C = 0
C = -Vo/R
I = (Vo/R) - (Vo/R)e^[-(R/L)t]
I = (Vo/R)(1 - e^[-(R/L)t])