Question

A sphere of radius $R$ has charge density per unit volume, $\rho \left(r\right)=A(R-r),$ for $0<r<R$ ( Here $A$ is constant). If total electric charge inside the sphere is $Q$, then the value of $A$ in terms of $Q$ and $R$ is

• A. $\frac{2Q}{\pi R^4}$
• B. $\frac{3Q}{\pi R^4}$
• C. $\frac{3Q}{2\pi R^4}$
• D. $\frac{3Q}{\pi R}$

Answer 1

The correct option is B $\frac{3Q}{\pi R^4}$


Volume of elemental shell of radius $x$ , thickness $dx$ is $4\pi x^{2}dx$
Let $dQ$ is the charge of the shell element, then
$dQ=\rho \left(4\pi x^{2}dx\right)$
$dQ=A(R-x)4\pi x^{2}dx$
$dQ=4\pi A(R{x}^2-x^3)dx$
Intergrating
${\int }_0^{Q}dQ=4\pi A[R{\int }_0^R{x}^{2}dx-{\int }_0^R{x}^{3}dx]$
$Q=4\pi A[R\times \frac{R^3}{3}-\frac{R^4}{4}]$
$Q=4\pi A[\frac{R^4}{3}-\frac{R^4}{4}]$
$Q=4\pi A\times \frac{R^4}{12}$
$\Rightarrow A=\frac{3Q}{\pi R^4}$