A certain series RC circuit is formed using a resistance R, a capacitor without dielectric having a capacitance C=2F and a battery of emf E=3V. The circuit is completed and it is allowed to attain the steady state. After this, at t=0, half the thickness of the capacitor is filled with a dielectric of constant K=2 as shown in the figure. The system is again allowed to attain a steady state. What will be heat generated (in joule) in the circuit between t=0 and $t = \infty$ ?

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answer is 3.

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Detailed Solution

Initial charge on capacitor $= C E$
Initial potential energy of capacitor $= \frac{C E^{2}}{2}$
Now $C = \frac{ε_{0} A}{d}$
And new capacitane: $C' = \frac{ε_{0} A}{\frac{d}{2}},$ in series with $\frac{ε_{0} k A}{\frac{d}{2}}$
$\Rightarrow C' = \frac{ε_{0} A}{d} (\frac{2 \times 4}{2 + 4}) = \frac{4 ε_{0} A}{3 d} = \frac{4}{3} C$ $\Rightarrow$ New charge $= \frac{4}{3} C E$
New energy $= \frac{1}{2} \times \frac{4}{3} C E^{2} = \frac{2}{3} C E^{2}$
Now $W_{b} = Δ H + Δ U$
$\Rightarrow E (\frac{4}{3} C E - C E) = Δ H + \frac{2}{3} C E^{2} - \frac{1}{2} C E^{2} \Rightarrow Δ H = \frac{1}{6} C E^{2}$