Question 255: Electrostatics & Capacitance

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 :

Accepted Answer

The question is about calculating the electric field at the surface of a thin spherical shell with a uniform surface charge density denoted by $$\sigma$$. To determine the electric field at any point on the surface of the shell, we can use Gauss's law, which is particularly useful for systems with high symmetry like a spherical shell.
Gauss's law in its integral form states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space ($$ \epsilon_0 $$), mathematically represented as:
$$ \Phi_E = \frac{Q_{	ext{enc}}}{\epsilon_0} $$
In the case of a spherical shell of radius $$R$$, the total charge $$Q_{	ext{enc}}$$ on the shell can be written in terms of the surface charge density $$\sigma$$ as:
$$ Q_{	ext{enc}} = \sigma 	imes 4\pi R^2 $$
Now, we apply Gauss's law using a Gaussian surface that coincides with the surface of the spherical shell. The electric field $$E$$ at the surface of the shell is uniform over the Gaussian surface, and the area of the Gaussian surface (which is also the area of the spherical shell) is $$4\pi R^2$$. Thus, the electric flux $$\Phi_E$$ through the Gaussian surface is:
$$ \Phi_E = E \cdot 4\pi R^2 $$
Using Gauss's law:
$$ E \cdot 4\pi R^2 = \frac{\sigma \cdot 4\pi R^2}{\epsilon_0} $$
Simplifying this equation gives us the electric field at the surface of the spherical shell:
$$ E = \frac{\sigma}{\epsilon_0} $$
Therefore, the correct answer is
Option B
:
$$ \frac{\sigma}{\epsilon_0} $$

Answer

$$\sigma / \epsilon_o R$$

Answer

$$\sigma / \in_o$$

Answer

$$\sigma / 2 \epsilon_o$$

Answer

$$\sigma / 4 \epsilon_o$$

Electrostatics & Capacitance

\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 :

Choose the correct answer: