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  • A solid ball of radius R has a charge density

A solid ball of radius R has a charge density \rho given by \rho=\rho _{0} (1-r/R) for 0 ≤ r ≤ R. The electric field outside the ball is :

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  Total charge on sphere:

q=\int_{0}^{R}\rho dv=\int_{0}^{R}\rho_{0}\left ( 1- \frac{r}{R}\right ).4\Pi r^{2}dr

q=\rho_{0}.4\Pi \left [ \int_{0}^{R}r^{2}dr-\int_{0}^{R}{\frac{r^{3}}{R}}dr \right ]

q=\rho_{0}.4\Pi\left ( \frac{R^{3}}{3}-\frac{R^{3}}{4} \right )=\rho_{0}\left ( 4\Pi \right )\frac{R^{3}}{12}

q=\frac{\rho_{0}.\Pi R^{3}}{3}

E=\frac{1}{4\pi \epsilon _{0}}\frac{q}{r^{2}}=\frac{1}{4\pi \epsilon _{0}}.\frac{\rho _{0}\pi r^{3}}{3.r^{2}}

E=\frac{\rho _{0}R^{3}}{12\epsilon _{0}r^{2}}

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Safeer PP

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