An insulated sphere of radius R has charge density ρ. The electric field at a distance r from th

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An insulated sphere of radius R has charge density ρ. The...

An insulated sphere of radius $R$ has charge density $\rho .$ The electric field at a distance $r$ from the centre of the sphere $(r<R)$ (a) $\frac{\rho r}{3 \varepsilon_{0}}$ (b) $\frac{\rho R}{3 \varepsilon_{0}}$ (c) $\frac{\rho r}{\varepsilon_{0}}$ (d) $\frac{\rho R}{\varepsilon_{0}}$

An insulated sphere of radius $R$ has charge density $\rho .$ The electric field at a distance $r$ from the centre of the sphere $(r<R)$
(a) $\frac{\rho r}{3 \varepsilon_{0}}$
(b) $\frac{\rho R}{3 \varepsilon_{0}}$
(c) $\frac{\rho r}{\varepsilon_{0}}$
(d) $\frac{\rho R}{\varepsilon_{0}}$

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Key Concepts

Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. This principle is crucial for calculating electric fields in systems with high symmetry, such as a uniformly charged sphere, where the law simplifies the integration process.

Uniform Charge Density

Uniform charge density means that the charge is distributed evenly throughout the volume of an object. This uniformity allows for straightforward computation of the charge enclosed within any given volume by multiplying the density by the volume, which is essential when applying Gauss's Law.

Spherical Symmetry and Gaussian Surface

Spherical symmetry implies that the properties of the system are identical in every direction from the center. In such cases, choosing a spherical Gaussian surface centered on the charge distribution simplifies the analysis because the electric field has a consistent magnitude on the surface and is directed radially. This makes it easier to derive expressions for the electric field inside the sphere.

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