Question

The region between two concentric sphere of radii 'a' and 'b' respectively (see figure), has volume charge density $p=\frac{A}{r}$, Where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is:

• A. $\frac{Q}{2\pi (b^2-a^2)}$
• B. $\frac{2Q}{\pi (a^2-b^2)}$
• C. $\frac{2Q}{\pi a^2}$
• D. $\frac{Q}{2\pi a^2}$

Answer 1

Answer: (D)

$\frac{Q}{2\pi a^2}$

$\begin{array}{c}\frac{Q+{\int }_a^r\frac{A}{r}A{\pi }^{2}dr}{E_o}=EA\pi r^2\\ EA{\in }_o{r}^2=Q+A\frac{A\pi }{r^2}\left(\frac{E^2-a^2}{2}\right)\\ E=\frac{1}{4\pi {\in }_o}\times A\times 2\pi \\ AsQ=2\pi A{a}^2\end{array}$

Since there is a charge at the center of the charge therefor, the electric field in the region between the two spheres will be given as:

$Q=2\pi A{a}^2$

$A=\frac{Q}{2\pi a^2}$