A Spherically Symmetric Charge Distribution Is Characterised By A Charge Density Having The Following Variation : P(R) = P0 (1 - R/R) For R Lesser Than R P(R) = 0 For R Lesser And Equal To R. Where R Is The Distance From The Centre Of The Charge Distribution And Pois A Constant. What Is The Electric Field At An Internal Point?

The electric field at the internal point r is the sum of fields due to outer and inner spheres. However, the contribution due to the outer part is zero.

For the inner part, consider a shell of thickness dr’ at a distance of r’ from the centre of the sphere.

Therefore, the charge contained in it is

$$dq = \( 4\Pi {r}’^{2}d{r}’p_{0} (1 – \frac{{r}’}{R}) \)

Electric field dE is dE =\( d \frac{1}{4 \Pi \epsilon_{o}r^{2}} \\4\Pi {r}’^{2}{dr}’p_{0}(1-d \frac{{r}’}{R})$$ \)

Integrating above, we get

\( E =\frac{P_{0}}{\epsilon _{0}r^{2}}\int_{0}^{r} ({r}’^{2} – \frac{{r}’^{3}}{R})dr \\E = \frac{P_{0}}{\epsilon _{0}}(\frac{r}{3} – \frac{r^{3}}{4R}) \)

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