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  • Let P(r)=\frac{Q}{\pi R^{4}}r be the charge density distribution for a solid sphere of radius R and total charge Q . For a point 'p' insid

Let P(r)=\frac{Q}{\pi R^{4}}r  be the charge density distribution for a solid sphere of radius R and total charge Q . For a point 'p' insid

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Let us consider a spherical shell of thickness dx and radius x. The volume of this spherical shell =4 \pi r^{2} d r 

The charge enclosed within shell  =\frac{Q r}{\pi R^{4}}\left[4 \pi r^{2} d r\right] 

The charge enclosed in a sphere of radius r1 is

=\frac{4 Q}{R^{4}} \int_{0}^{r_{1}} r^{3} d r=\frac{4 Q}{R^{4}}\left[\frac{r^{4}}{4}\right]_{0}^{r}-1=\frac{Q}{R^{4}} r_{1}^{4} 

 

∴ The electric field at point p inside the sphere at a distance r1 from the centre of the sphere is

E=\frac{1}{4 \pi \in_{0}} \frac{\left[\frac{Q}{R^{4}} r_{1}^{4}\right]}{r_{1}^{2}}=\frac{1}{4 \pi \in_{0}} \frac{Q}{R^{4}} r_{1}^{2}

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Satyajeet Kumar

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