Question 651: Electrostats & Capacitance

Question

Let there be a spherically symmetric charge distribution with charge density varying as $$\rho \left( r \right) = {\rho _0}\left( {{5 \over 4} - {r \over R}} \right)$$ upto $$r=R,$$ and $$\rho \left( r \right) = 0$$ for $$r>R,$$ where $$r$$ is the distance from the erigin. The electric field at a distance $$r\left( {r < R} \right)$$ from the origin is given by

Accepted Answer

Let us consider a spherical shell of radius $$x$$ and thickness $$dx.$$
Charge on this shell
$$dq = ho .4{\pi ^2}dx = {ho _0}\left( {{5 \over 4} - {x \over R}} ight).4\pi {x^2}dx$$
$$	herefore$$ Total charge in the spherical region from center to $$r$$  $$\left( {r < R} ight)$$ is
$$q = \int {dq = 4\pi {ho _0}\int\limits_0^r {\left( {{5 \over 4} - {x \over R}} ight)} } {x^2}dx$$
$$ = 4\pi {ho _0}\left[ {{5 \over 4}.{{{r^3}} \over 3} - {1 \over R}.{{{r^4}} \over 4}} ight]$$
$$ = \pi {ho _0}{r^3}\left( {{5 \over 3} - {r \over R}} ight)$$
$$	herefore$$ Electric field at $$r,$$ $$E = {1 \over {4\pi { \in _0}}}.{q \over {{r^2}}}$$
$$ = {1 \over {4\pi { \in _0}}}.{{\pi {ho _0}{r^3}} \over {{r^2}}}\left( {{5 \over 3} - {r \over R}} ight)$$
$$ = {{{ho _0}r} \over {4{ \in _0}}}\left( {{5 \over 3} - {r \over R}} ight)$$

Answer

$${{{\rho _0}r} \over {4{\varepsilon _0}}}\left( {{5 \over 3} - {r  \over R}} \right)$$

Answer

$${{4\pi {\rho _0}r} \over {3{\varepsilon _0}}}\left( {{5 \over 3} - {r  \over R}} \right)$$

Answer

$${{4{\rho _0}r} \over {4{\varepsilon _0}}}\left( {{5 \over 4} - {r  \over R}} \right)$$

Answer

$${{{\rho _0}r} \over {3{\varepsilon _0}}}\left( {{5 \over 4} - {r  \over R}} \right)$$

Electrostats & Capacitance

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