Region between two concentric spheres ofradii ' a ' and ' b ', respectively (see fig

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1. Electric Charges and Fields

hard

The region between two concentric spheres ofradii '$a$' and '$b$', respectively (see figure), have volume charge density $\rho = \frac{A}{r}$ 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 :

$\frac{{2Q}}{{\pi \left( {{a^2} - {b^2}} \right)}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;$

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

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

$\;\frac{Q}{{2\pi \left( {{b^2} - {a^2}} \right)}}$

(JEE MAIN-2016)

Solution

Applying Gauss's law

$\oint_{\mathrm{S}} \overrightarrow{\mathrm{E}} \cdot \overrightarrow{\mathrm{ds}}=\frac{Q}{\epsilon_{0}}$

$\therefore \mathrm{E} \times 4 \pi \mathrm{r}^{2}=\frac{\mathrm{Q}+4 \pi \mathrm{ar}^{2}-4 \pi \mathrm{Aa}^{2}}{\epsilon_{0}}$

$\rho=\frac{\mathrm{d} \mathrm{r}}{\mathrm{d} \mathrm{v}}$

$Q=\rho 4 \pi r^{2}$

$\mathrm{Q}=\int_{\mathrm{a}}^{\mathrm{A}} \frac{\mathrm{A}}{\mathrm{r}} 4 \pi \mathrm{r}^{2} \mathrm{dr}=4 \pi \mathrm{A}\left[\mathrm{r}^{2}-\mathrm{a}^{2}\right]$

$\mathrm{E}=\frac{1}{4 \pi \epsilon_{0}}\left[\frac{\mathrm{Q}-4 \pi \mathrm{Aa}^{2}}{\mathrm{r}^{2}}+4 \pi \mathrm{A}\right]$

For $\mathrm{E}$ to be independent of $'{r}'$

$\mathrm{Q}-2 \pi \mathrm{Aa}^{2}=0$

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

Standard 12

Physics

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Solution

Similar Questions

A conducting sphere of radius $10 \;cm$ has an unknown charge. If the electric field $20\; cm$ from the centre of the sphere is $1.5 \times 10^{3} \;N / C$ and points radially inward, what is the net charge (in $n\;C$) on the sphere?

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