Paragraph: The nuclear charge $(\mathrm{Ze})$ is non-uniformly distributed within a nucleus of radius $R$.…
Paragraph:
The nuclear charge $(\mathrm{Ze})$ is non-uniformly distributed within a nucleus of radius $R$. The charge density $\rho(r)$ [charge per unit volume] is dependent only on the radial distance $r$ from the centre of the nucleus as shown in figure. The electric field is only along the radial direction. Question:
The electric field at $r=R$ is
independent of $a$
directly proportional to $a$
directly proportional to $a^2$
inversely proportional to $a$
Solution
At $r=R$, from Gauss's law
or
$
\begin{aligned}
E\left(4 \pi R^2\right) & =\frac{q_{\text {net }}}{\varepsilon_0}=\frac{Z e}{\varepsilon_0} \\
E & =\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Z e}{R^2}
\end{aligned}
$
$E$ is independent of ' $a$ '
$\therefore$ correct option is (a).
!
