Question

\(\sigma\) is the uniform surface charge density of a thin spherical shell of radius \(R\). The electric field at any point on the surface of the spherical shell is:

• A. \( \frac{\sigma}{\epsilon_0 R} \)
• B. \( \frac{\sigma}{2\epsilon_0} \)
• C. \( \frac{\sigma}{\epsilon_0} \)
• D. \( \frac{\sigma}{4\epsilon_0} \)

Answer 1

The Correct Option is C

By Gauss's Law:

\[ \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{in}}}{\varepsilon_0}. \]

For a spherical shell, the electric field \( E \) is constant across the surface area:

\[ E \cdot 4\pi R^2 = \frac{\sigma \cdot 4\pi R^2}{\varepsilon_0}. \]



\[ E = \frac{\sigma}{\varepsilon_0}. \]

Thus, the electric field at the surface of the spherical shell is:

\[ \boxed{\frac{\sigma}{\varepsilon_0}}. \]

Answer 2

To solve the problem of finding the electric field at any point on the surface of a thin spherical shell with uniform surface charge density \(\sigma\), let us apply the concepts of Gauss's law in electrostatics.

According to Gauss's law, the electric flux \(\Phi\) through a closed surface is given by:

\(\Phi = \frac{Q}{\epsilon_0}\)

where:

• \(\Phi\) is the electric flux through the surface.
• \(Q\) is the total charge enclosed by the surface.
• \(\epsilon_0\) is the permittivity of free space.

For a spherical shell, consider a Gaussian surface that is also a sphere of the same radius \(R\) as the shell. By symmetry, the electric field \(E\) is constant over the surface and is directed radially outward.

The electric flux through the Gaussian surface is then given by:

\(\Phi = E \cdot 4\pi R^2\)

The total charge \(Q\) on the shell is:

\(Q = \sigma \cdot 4\pi R^2\)

Applying Gauss's law:

\(E \cdot 4\pi R^2 = \frac{\sigma \cdot 4\pi R^2}{\epsilon_0}\)

Solving for \(E\), we get:

\(E = \frac{\sigma}{\epsilon_0}\)

Therefore, the electric field at any point on the surface of the spherical shell is given by:

\(\boxed{\frac{\sigma}{\epsilon_0}}\)

Thus, the correct answer is:

\( \frac{\sigma}{\epsilon_0} \)

This result is consistent with Gauss's law and the symmetry of the problem, leading us to the conclusion that the correct option is:

• \(\frac{\sigma}{\epsilon_0}\)