A conducting sphere of radius 0.1 m has a uniform positive charge density of `1.8 mu C//m^(2)` on its surface. The electric field in V/m in free space at a radial distance of 0.2m from a point on the surface of the sphere is given by

Options

A. `6.0xx10^(-8)//epsi_(0)`
B. `2.0xx10^(-7)//epsi_(0)`
C. `4.5xx10^(-7)//epsi_(0)`
D. `6.0xx10^(-6)//epsi_(0)`

Answer

To solve the problem, we need to calculate the electric field at a distance of 0.2 m from the surface of a conducting sphere with a given surface charge density. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a conducting sphere of radius \( R = 0.1 \, \text{m} \) with a uniform surface charge density \( \sigma = 1.8 \, \mu \text{C/m}^2 \). We need to find the electric field at a distance of \( 0.2 \, \text{m} \) from a point on the surface of the sphere.

### Step 2: Calculate the Total Charge on the Sphere
The total charge \( Q \) on the sphere can be calculated using the formula:
\[ Q = \sigma \times A \]
where \( A \) is the surface area of the sphere given by \( A = 4\pi R^2 \).

Substituting the values:
\[ A = 4\pi (0.1)^2 = 4\pi (0.01) = 0.04\pi \, \text{m}^2 \]
Now, substituting \( \sigma \):
\[ Q = (1.8 \times 10^{-6} \, \text{C/m}^2) \times (0.04\pi) \approx 2.26 \times 10^{-7} \, \text{C} \]

### Step 3: Determine the Distance from the Center of the Sphere
The distance from the center of the sphere to the point where we want to calculate the electric field is:
\[ d = R + 0.2 = 0.1 + 0.2 = 0.3 \, \text{m} \]

### Step 4: Use the Formula for Electric Field Due to a Point Charge
For a point charge, the electric field \( E \) at a distance \( d \) is given by:
\[ E = \frac{kQ}{d^2} \]
where \( k = \frac{1}{4\pi \epsilon_0} \).

### Step 5: Substitute the Values into the Electric Field Formula
Substituting \( Q \) and \( d \):
\[ E = \frac{(1/(4\pi \epsilon_0)) \cdot (2.26 \times 10^{-7})}{(0.3)^2} \]
Calculating \( (0.3)^2 = 0.09 \):
\[ E = \frac{(2.26 \times 10^{-7})}{0.09 \cdot (4\pi \epsilon_0)} \]

### Step 6: Simplify the Expression
Since we are looking for the electric field in terms of \( \epsilon_0 \), we can simplify:
\[ E = \frac{2.26 \times 10^{-7}}{0.09} \cdot \frac{1}{4\pi \epsilon_0} \]
Calculating \( \frac{2.26 \times 10^{-7}}{0.09} \approx 2.51 \times 10^{-6} \):
\[ E \approx \frac{2.51 \times 10^{-6}}{4\pi \epsilon_0} \]

### Step 7: Final Result
Thus, the electric field at a distance of 0.2 m from the surface of the sphere is:
\[ E \approx \frac{2.51 \times 10^{-6}}{4\pi \epsilon_0} \, \text{V/m} \]