An uncharged capacitor (Qi = 0) is connected to a resistor of resistance R and a

Question

An uncharged capacitor (Qi = 0) is connected to a resistor of resistance R and a battery of emf , all in series. A switch is closed, so the capacitor begins charging at time t = 0.

(R/(-Q/c))*dQ=dt

a)find a function Q(t) that describes the charge on the capacitor as a function of time. Express your result in terms of R,, t, and constants.

b) Using your result from Part (a), find a function I(t) that describes the current through the resistor as a function of time. Again express your result in terms of R,, t, and constants.

c) Using your result from Part (a), find the charge on the capacitor after a very long time (t ).

d) Using your result from Part (b), find the maximum current through the resistor, and at which time that occurs.

Explanation / Answer

RdQ/[E - Q/c] = dt

A. Integrating both sides
0|-Rcln[E - Q/c]|Q = t = -Rcln[E]+Rcln[E - Q/c] = t
ln[1 - Q/Ec] = -t/Rc
1 - Q/Ec = e^(-t/Rc)
Q= Ec[1 - e^(-t/RC)]

B. I(t) = dQ/dt = [Ee^(-t/RC)]/R
C. as t--> infinity, Q = Ec
D. I(0) = E/R
I(infinity) = 0
Max current = E/R at time t = 0

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