Question

The nuclear charge $\left(Ze\right)$ is non-uniformly distributed within a nucleus of radius $R$. The charge density $\rho \left(r\right)$( 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. The electric field within the nucleus is generally observed to be linearly dependent on $r$. This implies

• A. $a=0$
• B. $a=\frac{R}{2}$
• C. $a=R$
• D. $a=\frac{2R}{3}$

Answer 1

The correct option is C $a=R$

The nucleus can be considered as a solid non-conducting sphere with a total charge $q=Ze$.
It is given that electric field inside nucleus is linearly dependent on $r$.
$\Rightarrow E\propto r$
The electric field at $r$ due to a uniformly charged solid non-conducting sphere of radius $R$ is
$\Rightarrow E=\frac{\rho r}{3{ϵ}_0}$ where $(\rho =\frac{q}{\frac{4}{3}\pi R^3})$
$\therefore$
For $E$ to be directly proportional to $r$, volume charge density should be constant.
$\Rightarrow \rho$ = constant for region $r=0$ to $r=R$
$\therefore a=R$