The surface density on a copper sphere is `sigma`. The electric field strength on the surface of the sphere is

Options

A. `sigma`
B. `sigma//2`
C. `sigma//2epsilon_0`
D. `sigma//epsilon_0`

Explanation

To find the electric field strength on the surface of a copper sphere with a surface charge density \( \sigma \), we can use Gauss's Law. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a copper sphere with a uniform surface charge density \( \sigma \). We need to calculate the electric field strength \( E \) at the surface of the sphere.

### Step 2: Apply Gauss's Law
Gauss's Law states that the electric flux \( \Phi_E \) through a closed surface is equal to the charge \( Q_{\text{enc}} \) enclosed by that surface divided by the permittivity of free space \( \epsilon_0 \):
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]

### Step 3: Choose a Gaussian Surface
For a sphere, we can choose a Gaussian surface that is also a sphere with the same radius as the copper sphere. The electric field \( E \) is uniform and radial at the surface of the sphere.

### Step 4: Calculate the Electric Flux
The electric flux through the Gaussian surface is given by:
\[ \Phi_E = E \cdot A \]
where \( A \) is the surface area of the Gaussian sphere. The surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 5: Calculate the Enclosed Charge
The total charge \( Q_{\text{enc}} \) enclosed by the Gaussian surface is given by the surface charge density multiplied by the surface area of the sphere:
\[ Q_{\text{enc}} = \sigma \cdot A = \sigma \cdot 4\pi r^2 \]

### Step 6: Substitute into Gauss's Law
Substituting the expressions for electric flux and enclosed charge into Gauss's Law:
\[ E \cdot (4\pi r^2) = \frac{\sigma \cdot (4\pi r^2)}{\epsilon_0} \]

### Step 7: Simplify the Equation
We can cancel \( 4\pi r^2 \) from both sides of the equation (assuming \( r \neq 0 \)):
\[ E = \frac{\sigma}{\epsilon_0} \]

### Conclusion
Thus, the electric field strength \( E \) on the surface of the copper sphere is:
\[ E = \frac{\sigma}{\epsilon_0} \]