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Consider a uniform charge distribution with charge density $2 C / m^{3}$ throughout in space. If a Gaussian sphere has a variable radius which changes at the rate of 2 m/s, then value of rate of change of flux is proportional to $r^{k} (r = r a d i u s o f s p h e r e) .$ Then, find the value of k.

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Detailed Solution

Flux linked with a Gaussian sphere of radius r

= change enclosed $\times \frac{1}{ε_{0}}$

= volume enclosed $\times c h \arg e d e n s i t y \times \frac{1}{ε_{0}}$

$= \frac{4}{3} π r^{3} \times ρ \times \frac{1}{ε_{0}}$

So, rate of change of flux $= \frac{d ϕ_{E}}{d t} = \frac{d}{d t} \frac{4}{3} π \frac{ρ}{ε_{0}} \times r^{3}$

$= \frac{4}{3} π \frac{ρ}{ε_{0}} \times 3 r^{2} \frac{d r}{d t} = 4 π r^{2} \frac{ρ}{ε_{0}} (\frac{d r}{d t}) = 8 π r^{2} \frac{ρ}{ε_{0}}$

$∴ \frac{d ϕ_{E}}{d t} \propto r^{2}, k = 2$

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