Electrostatics 5 Question 26
28. Two identical capacitors, have the same capacitance $C$. One of them is charged to potential $V _1$ and the other to $V _2$. Likely charged plates are then connected. Then, the decrease in energy of the combined system is
(a) $\frac{1}{4} C\left(V _1^{2}-V _2^{2}\right)$
(b) $\frac{1}{4} C\left(V _1^{2}+V _2^{2}\right)$
(c) $\frac{1}{4} C\left(V _1-V _2\right)^{2}$
(d) $\frac{1}{4} C\left(V _1+V _2\right)^{2}$
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Solution:
- $\Delta U=$ decrease in potential energy
$$ \begin{aligned} & =U _i-U _f \\ & =\frac{1}{2} C\left(V _1^{2}+V _2^{2}\right)-\frac{1}{2}(2 C) \frac{V _1+V _2}{2} \\ & =\frac{1}{4} C\left(V _1-V _2\right)^{2} \end{aligned} $$
