Question

The nuclear charge (Ze) is non-uniformly distribute within a nucleus of radius R. The charge density $\rho \left(r\right)$ [charge per unit volume] is dependent on the radial distance r from the centre of the nucleus as shown in figure. Select correct alternative/s.

• A. Electric field at r=R is independent of b
• B. Electric potential at r=R is proportional to b
• C. Electric field at r=R is proportional to a
• D. Electric potential at r=R is proportional to a

Answer 1

Answer: (A)

Electric field at r=R is independent of b

$\begin{array}{c}\text{According to Gauss I aw}\\ \oint E\cdot dS=\frac{q_{in}}{{ϵ}_0}⟶\text{(i)}\end{array}$

$\text{also,}$

$q_{\text{in}}=\int \rho dV⟶\left(ii\right)$

$\begin{array}{c}\rho =\text{Charge density}\\ \text{dV}\to \text{small wlume}\\ \text{From graph,}\\ \rho =\frac{d}{a}r(0⩽r⩽a)\end{array}$

$\rho =d\begin{array}{cc}(a⩽r& ⩽R\text{and also}a<b<R)\end{array}$

$\begin{array}{c}V=\frac{4}{3}\pi r^3\\ \text{differentiating}\\ dV=4\pi r^{2}dr\\ q_{\text{in}}=\int \rho dV\\ ={\int }_0^a\frac{dr}{a}4\pi r^{2}dr+{\int }_a^{R}dy\pi r^{2}dr\end{array}$

$\begin{array}{cc}& =\frac{4\pi d\times (a^4-0)}{a\times 4}+4\pi d\frac{R^3-a^3}{3}\\ =& 4xd(\frac{a^3}{4}+\frac{R^3}{3}-\frac{a^3}{3})\\ q_{in}=& 4\pi d{(\frac{R^3}{3}-\frac{a^3}{12})}^3\end{array}$

$\begin{array}{c}\text{from eq(i)}\\ E\times \frac{4}{3}\pi R^3=q_{in}/{\epsilon }_0\\ E=d(1-\frac{a^3}{4R^3})⟶\left(\text{independent of}b\right)\end{array}$