Let ρ(r)=((Q)/(πR^4) r) be the charge density distribution...

Let $\rho(r)=\left(\frac{Q}{\pi R^{4}} r\right)$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q .$ For a point $P$ inside the sphere at distance $r_{1}$ from the centre of the sphere, the magnitude of electric field is (a) zero (b) $\frac{Q}{4 \pi \varepsilon_{o} r_{1}^{2}}$ (c) $\frac{Q r_{1}^{2}}{4 \pi \varepsilon_{o} R^{4}}$ (d) $\frac{Q r_{1}^{2}}{3 \pi \varepsilon_{o} R^{4}}$

Let $\rho(r)=\left(\frac{Q}{\pi R^{4}} r\right)$ be the charge density distribution for a solid sphere of radius $R$ and total charge $Q .$ For a point $P$ inside the sphere at distance $r_{1}$ from the centre of the sphere, the magnitude of electric field is
(a) zero
(b) $\frac{Q}{4 \pi \varepsilon_{o} r_{1}^{2}}$
(c) $\frac{Q r_{1}^{2}}{4 \pi \varepsilon_{o} R^{4}}$
(d) $\frac{Q r_{1}^{2}}{3 \pi \varepsilon_{o} R^{4}}$

Key Concepts

Gauss's Law in Electrostatics

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is a fundamental principle used to compute electric fields from charge distributions, especially when symmetry allows for a simplification of the integral form of the law.

Non-uniform Charge Distribution

In problems involving a non-uniform charge distribution, the charge density varies with position. Understanding how to integrate the charge density over a volume is crucial to determine the total charge enclosed within a certain region, which then influences the resulting electric field.

Spherical Symmetry

Spherical symmetry implies that the charge distribution and resulting electric field depend only on the radial distance from the center. This symmetry simplifies the application of Gauss's Law, as the electric field on a spherical Gaussian surface is constant in magnitude, making the flux calculation straightforward.

Enclosed Charge Calculation

Computing the enclosed charge involves integrating the charge density over the volume contained by the Gaussian surface. This step is essential when applying Gauss's Law, as it directly determines the magnitude of the electric field inside the distribution by relating the total enclosed charge to the field's radial dependence.

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