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Let the electrostatic field $E$ at distance $r$ from a point charge $q$ not be an inverse square but instead an inverse cubic, e.g. $E =k \cdot \frac{q}{r^{3}} \hat{ r }$, here $k$ is a constant.
Consider the following two statements:
$(I)$ Flux through a spherical surface enclosing the charge is $\phi=q_{\text {enclosed }} / \varepsilon_{0}$.
$(II)$ A charge placed inside uniformly charged shell will experience a force.
Which of the above statements are valid?
Only statement $I$ is valid
Only statement $II$ is valid
Both statements $I$ and $II$ are invalid
Both statements $I$ and $II$ are valid
Solution
$(b)$ If $E=\frac{k q}{r^{3}} \cdot \hat{ r }$, then flux through a shell enclosing a charge $q$ is
$\phi=\int E \cdot d A =\int \frac{k q}{r^{3}} \cdot d A$
$=\frac{k q}{r^{3}} \int d A=\frac{k q}{r^{3}} \cdot 4 \pi r^{2}$
$\therefore \quad \phi=\frac{k q(4 \pi)}{r}=\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{4 \pi q}{r}$
$\Rightarrow \quad \phi=\frac{q}{\varepsilon_{0} r} \neq \frac{q}{\varepsilon_{0}}$
Also, when $E=\frac{k q}{r^{3}}$, Gauss' law is not valid and electric field at an interior point of a charged shell is non-zero.
$\therefore$ Charge placed inside shell experience a force.
