Question 125 of 311

The region between two concentric spheres of radii 'a' and 'b', respectively (see figure), has volume charge density $\rho =\dfrac{A}{r}$ , 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$\dfrac{2Q}{\pi \left({a}^{2}-{b}^{2}\right)}$B$\dfrac{2Q}{\pi {a}^{2}}$C$\dfrac{Q}{2\pi {a}^{2}}$D$\dfrac{Q}{2\pi \left({b}^{2}-{a}^{2}\right)}$

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Answer: (C) $\dfrac{Q}{2\pi {a}^{2}}$

${\text{E=E}}_{1}+{\text{E}}_{2}$$\Rightarrow E=\dfrac{K\left[Q+\int _{a}^{r}4\pi {x}^{2}dx\dfrac{A}{x}\right]}{{r}^{2}}$

$E=K\left[\dfrac{Q}{{r}^{2}}+4\pi A\left\{\dfrac{{r}^{2}}{2{r}^{2}}-\dfrac{{a}^{2}}{2{r}^{2}}\right\}\right]$

$\dfrac{dE}{dr}=0$

$\Rightarrow -2\dfrac{Q}{{r}^{3}}+\dfrac{8\pi A{a}^{2}}{2{r}^{3}}=0$

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