The radii of a spherical capacitor are equal to a and b (b > a). The space between them is filled with a dielectric of dielectric constant K and resistivity $ρ$ . At t = 0, the inner electrode is given a charge q₀. Choose the correct option(s) :

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In a short time, the charge on the inner electrode will become zero

Charge q on the inner electrode as a function of time is given by $q = q_{0} e^{- \frac{t}{ρ K ε_{0}}}$

The total amount of heat generated during the spreading of charge will be given by $H = (\frac{1}{a} - \frac{1}{b}) \frac{q_{0}^{2}}{8 π ε_{0}}$

After a long time, the charge on the outer sphere will become q₀

answer is A, C.

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Detailed Solution

$\begin{array}{l} C = \frac{4 π ε_{0} abK}{b - a} \\ R = \int_{a}^{b} ρ \frac{d r}{4 π r^{2}} = \frac{ρ}{4 π} \frac{(b - a)}{a b} \\ τ = R C = ρ K ε_{0} \\ q = q_{0} e^{- t / τ} \\ q = q_{0} e^{- \frac{t}{ρ K ε_{0}}} \\ t = \infty, q = 0 \\ Amount of heat generated =H= \frac{{q^{2}}_{0}}{2 C} \\ H = (\frac{1}{a} - \frac{1}{b}) \frac{q_{0}^{2}}{8 π ε_{0} K} \end{array}$