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 within the nucleus is generally observed to be linearly dependent on $r$. This implies

• A

  $a=0$
• B

  $a=\frac{R}{2}$
• Correct answer
  C

  $a=R$
• D

  $a=\frac{2 R}{3}$

Solution

In case of solid sphere of charge of uniform volume density. $$ E=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{q}{R^3} \cdot r \text { or } E \propto r $$ Thus, for $E$ to be linearly dependent on $r$, volume charge density should be constant. or $\quad a=R$ $\therefore$ correct option is (c).

Hence, the correct answer is option: 'C'.