Question

A circuit contains a resistor and capacitor wired in series with a battery as shown above. The resistor $R_1$ has a resistance $R$ and the Capacitor $C_1$ has a capacitance $C$. The capacitor is initially uncharged.
Which of the following expresses the charge on the capacitor as a function of time?

• A. The charge is constant but not zero.
• B. Zero.
• C. $Q\left(t\right)=Ct$
• D. $Q\left(t\right)=VC{e}^{\frac{t}{RC}}$
• E. $Q\left(t\right)=VC(1-e^{\frac{t}{RC}})$

Answer 1

The correct option is A The charge is constant but not zero.

Applying Kirchhoff's law for the given circuit ,

$V=IR+Q/C$ ....................eq1,

where $Q=$ charge on capacitor ,

when capacitor is fully charged $Q=Q_0$ , and current $I=0$ ,

by eq1 $V=0+Q_0/C$ ,

putting this value in eq1 , we get

$Q_0/C=IR+Q/C$

or $Q_0-Q/C=IR$ ,

but $i=dQ/dt$ ,

so $Q_0-Q/C=\left(dQ/dt\right)R$ ,

or $dQ/Q_0-Q=dt/CR$ ,

or $\int dQ/(Q_0+Q)=\left(1/CR\right)\int dt$ ,

or $-{\log }_e(Q_0-Q)=t/CR+k$ (k=integration constant) ,

when $t=0,Q=0$ ,

we have , $-{\log }_e(Q_0-Q)=t/CR+-{\log }_e{Q}_0$ ,

or ${\log }_e(Q_0-Q)/Q_0=-t/CR$ ,

or $(Q_0-Q)/Q_0=e^{-t/CR}$ ,

or $Q=Q_0(1-e^{-t/CR})$ ,

or $Q=VC(1-e^{-t/CR})$

this is the charge on capacitor at any time .