Current enters a junction through branch A

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Electric Circuits · Electric Circuits

Problem

Current

I_A(t)=I_0 e^{-t/\tau}

enters a junction through branch A, while current

I_B(t)=I_0\left(1-e^{-t/\tau}\right)

enters the same junction through branch B. The only branch leaving the junction is connected to an initially uncharged capacitor of capacitance

C

; the current in that branch is

I_C(t)

(directed away from the junction). The switch that starts the process is closed at

t=0

. (a) Express

I_C(t)

in terms of

I_0

t

, and

\tau

so that charge is conserved instantaneously. (b) Derive an expression for the charge

q(t)

on the capacitor as a function of time. (c) Using your result from part (b), determine the potential difference

V(t)

across the capacitor plates. (d) Evaluate

q(t)

and

V(t)

in the limits

t\to0

and

t\to\infty

, and interpret the physical meaning of each limit in the context of the circuit.