First slide

Gauss's law and its applications

Question

Consider a uniform charge distribution with charge density 2 C/m³ 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 charge of flux is proportional to r^k, (r= radius of sphere). Then, the value of k is _____________.

Moderate

Solution

Flux linked with a Gaussian sphere of radius r
$\begin{array}{l} = change enclosed \times \frac{1}{ε_{0}} \\ = volume enclosed \times charge density \times \frac{1}{ε_{0}} = \frac{4}{3} π r^{3} \times ρ \times \frac{1}{ε_{0}} \\ So, rate of change of flux = \frac{d ϕ_{E}}{d t} = \frac{d}{d t} \frac{4}{3} π \frac{ρ}{ε_{0}} \times r^{3} \\ = \frac{4}{3} π \frac{ρ}{ε_{0}} \times 3 r^{2} \frac{d r}{d t} = 4 π r^{2} \frac{ρ}{ε_{0}} (\frac{d r}{d t}) = 8 π r^{2} \frac{ρ}{ε_{0}} \\ ∴ \frac{d ϕ_{E}}{d t} \propto r^{2}, k = 2 \end{array}$

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