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  • The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has volume charge density 

The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has 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

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Qenc - charge enclosed by closed surface.

 

 \int \vec{E} \cdot \vec{ds} = \frac{Q+q}{\epsilon_o} \Rightarrow E \times 4 \pi r^2 = \frac{Q+q}{\epsilon_o} \:\:\:\:\:\: -(i)

q = \int_{a}^{r}\frac{A}{x} 4 \pi x^2 dx = 4 \pi A \int_{a}^{r}xdx

    = 4 \pi A \left [ \frac{x^2}{2} \right ]_{a}^{r} = 2 \pi A(r^2-a^2)

Now putting the value of q in equation (i)

E \times 4 \pi r^2 = \frac{1}{\epsilon_o}\left [ Q + 2 \pi A(r^2-a^2) \right ]

E = \frac{1}{4 \pi \epsilon_o}\left [\frac{Q}{r^2} + 2 \pi A-\frac{2 \pi A a^2}{r^2} \right ]

E will be constant if it is independent of r 

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

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Safeer PP

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