A sphere of radius r has a electric charge uniformly distributed in its entire volume. At a distance d from the center inside the sphere (d?

Question

Answer

Electric Field Inside a Charged Sphere

When an electric charge is uniformly distributed throughout the volume of a sphere, the electric field inside the sphere can be calculated using Gauss's Law.

Gauss's Law

Gauss's Law states that the flux of the electric field through a closed surface is equal to the net charge enclosed by the surface divided by the permittivity of free space.

Calculation of Electric Field Inside the Sphere

Consider a sphere of radius r with a charge Q distributed uniformly throughout its volume. Let us assume that we are interested in finding the electric field at a distance d from the center inside the sphere.

We can choose a Gaussian surface in the form of a sphere of radius d with its center at the center of the original sphere.

Since the charge is uniformly distributed throughout the volume of the sphere, the charge enclosed by the Gaussian surface is proportional to the volume of the sphere enclosed by the surface.

The electric field at any point on the Gaussian surface is perpendicular to the surface, and hence, the electric flux through the surface is given by:

Flux = E x 4πd²

where E is the electric field at a point on the Gaussian surface.

Using Gauss's Law, we can write:

Flux = Q / ε₀

where ε₀ is the permittivity of free space.

Equating the two expressions for Flux, we get:

E x 4πd² = Q / ε₀

which gives the electric field inside the sphere as:

E = Q / (4πε₀d²)

Thus, the electric field inside a uniformly charged sphere is inversely proportional to the square of the distance from the center.

Conclusion

The electric field inside a uniformly charged sphere is inversely proportional to the square of the distance from the center. This relationship is similar to the inverse square law that governs the behavior of gravity.

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