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  • A capacitor having initial charge   is connect

A capacitor having initial charge \mathrm{q_0=\frac{C E}{2}}  is connected to a cell ofe.m.f. E through a resistor R as shown. Find the total heat generated in the circuit after the switch S is closed.


 

Option: 1

\mathrm{\left (\frac{1}{12 }\right ) C E^2}


Option: 2

\mathrm{\left (\frac{1}{8 }\right ) C E^2}


Option: 3

\mathrm{ \left (\frac{1}{4 }\right ) C E^2}


Option: 4

None of these


Answers (1)

best_answer

Final charge on capacitor \mathrm{q_f=C E}
Charge flown = \mathrm{C E-\frac{C E}{2}=\frac{C E}{2}}
Work done by battery: \mathrm{W_b=E\left (\frac{C E }{2 } \right )=\frac{1}{2} C E^2}
Change in energy of capacitor is

\begin{aligned} &\mathrm{ \Delta U=\frac{1}{2 C}\left[q_f^2-q_0^2\right] }\\ &\mathrm{ =\frac{1}{2 C}\left[(C E)^2-(C E / 2)^2\right]=\frac{3 C E^2}{8} }\\ &\mathrm{ \text { Heat }=W_b-\Delta U=\frac{1}{8} C E^2}\\ \end{aligned}

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