Let there be a spherically symmetric charge distribution...

Let there be a spherically symmetric charge distribution with charge density varying as $\rho(r)=\rho_{0}\left(\frac{5}{4}-\frac{r}{R}\right)$ upto $r=R$, and $\rho(r)=0$ for $r>R$ where, $r$ is the distance from the origin. The electric field at a distance $r(r<R)$ from the origin is given by [AIEEE 2010] (a) $\frac{4 \pi \rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)$ (b) $\frac{\rho_{0} r}{4 \varepsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)$ (c) $\frac{4 \rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{4}-\frac{r}{R}\right)$ (d) $\frac{\rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{4}-\frac{r}{R}\right)$

Let there be a spherically symmetric charge distribution with charge density varying as $\rho(r)=\rho_{0}\left(\frac{5}{4}-\frac{r}{R}\right)$ upto $r=R$, and $\rho(r)=0$ for $r>R$
where, $r$ is the distance from the origin. The electric field at a distance $r(r<R)$ from the origin is given by [AIEEE 2010]
(a) $\frac{4 \pi \rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)$
(b) $\frac{\rho_{0} r}{4 \varepsilon_{0}}\left(\frac{5}{3}-\frac{r}{R}\right)$
(c) $\frac{4 \rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{4}-\frac{r}{R}\right)$
(d) $\frac{\rho_{0} r}{3 \varepsilon_{0}}\left(\frac{5}{4}-\frac{r}{R}\right)$

Key Concepts

Volume Integration in Electrostatics

Integration over a volume is essential when dealing with non-uniform charge distributions. By integrating the charge density function over the appropriate limits, one obtains the total enclosed charge, which is then used in Gauss's Law to determine the electric field inside the distribution.

Non-Uniform Charge Distribution

A non-uniform charge distribution is one where the charge density varies with position. In this context, the charge density depends on the radial distance from the center. Calculating the electric field requires integrating this variable charge density over the volume enclosed by the Gaussian surface to determine the total charge present.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful in systems with high symmetry, such as spherical charge distributions, for calculating electric fields by simplifying the integration process over the surface.

Spherical Symmetry

Spherical symmetry means that the physical properties of the system, like the charge density and resulting electric field, depend only on the radial distance from the center. This symmetry allows the use of spherical Gaussian surfaces and significantly simplifies the calculation of the electric field.

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