A capacitor $4\,\mu F$ charged to $50\, V$ is connected to another capacitor of $2\,\mu F$ charged to $100 \,V$ with plates of like charges connected together. The total energy before and after connection in multiples of $({10^{ - 2}}\,J)$ is
Diffcult
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(a) The total energy before connection
$ = \frac{1}{2} \times 4 \times {10^{ - 6}} \times {(50)^2} + \frac{1}{2} \times 2 \times {10^{ - 6}} \times {(100)^2}$
$ = 1.5 \times {10^{ - 2}}\,J$
When connected in parallel
$4 \times 50 + 2 \times 100 = 6 \times V$ $==>$ $V = \frac{{200}}{3}$
Total energy after connection
$ = \frac{1}{2} \times 6 \times {10^{ - 6}} \times {\left( {\frac{{200}}{3}} \right)^2} = 1.33 \times {10^{ - 2}}\,J$
$ = \frac{1}{2} \times 4 \times {10^{ - 6}} \times {(50)^2} + \frac{1}{2} \times 2 \times {10^{ - 6}} \times {(100)^2}$
$ = 1.5 \times {10^{ - 2}}\,J$
When connected in parallel
$4 \times 50 + 2 \times 100 = 6 \times V$ $==>$ $V = \frac{{200}}{3}$
Total energy after connection
$ = \frac{1}{2} \times 6 \times {10^{ - 6}} \times {\left( {\frac{{200}}{3}} \right)^2} = 1.33 \times {10^{ - 2}}\,J$
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