The electric field at 2R from the centre of a uniformly charged non - conducting sphere of rarius R is E. The electric field at a distance `( R )/(2)` from the centre will be

Options

A. Zero
B. 2E
C. 4E
D. 16E

Answer

To solve the problem, we need to find the electric field at a distance \( \frac{R}{2} \) from the center of a uniformly charged non-conducting sphere of radius \( R \), given that the electric field at a distance \( 2R \) from the center is \( E \).

### Step 1: Understand the electric field outside the sphere
For a uniformly charged non-conducting sphere, the electric field outside the sphere (at a distance greater than \( R \)) is given by the formula:
\[ E_{\text{outside}} = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant, \( Q \) is the total charge of the sphere, and \( r \) is the distance from the center of the sphere.

### Step 2: Apply the formula at \( 2R \)
At a distance \( 2R \) from the center, we can substitute \( r = 2R \):
\[ E = \frac{kQ}{(2R)^2} = \frac{kQ}{4R^2} \]

### Step 3: Understand the electric field inside the sphere
For points inside the uniformly charged non-conducting sphere (at a distance less than \( R \)), the electric field is given by:
\[ E_{\text{inside}} = \frac{kQ}{R^3} \cdot r \]
where \( r \) is the distance from the center of the sphere.

### Step 4: Apply the formula at \( \frac{R}{2} \)
Now, we need to find the electric field at a distance \( \frac{R}{2} \) from the center:
\[ E_{\text{inside}} = \frac{kQ}{R^3} \cdot \left(\frac{R}{2}\right) = \frac{kQ}{R^3} \cdot \frac{R}{2} = \frac{kQ}{2R^2} \]

### Step 5: Relate the two electric fields
We have two expressions for the electric field:
1. At \( 2R \): \( E = \frac{kQ}{4R^2} \)
2. At \( \frac{R}{2} \): \( E_{\frac{R}{2}} = \frac{kQ}{2R^2} \)

### Step 6: Find the ratio of the electric fields
To find the relationship between \( E_{\frac{R}{2}} \) and \( E \):
\[ \frac{E_{\frac{R}{2}}}{E} = \frac{\frac{kQ}{2R^2}}{\frac{kQ}{4R^2}} = \frac{2R^2}{4R^2} = \frac{2}{4} = \frac{1}{2} \]

### Step 7: Conclude the electric field at \( \frac{R}{2} \)
Thus, we can express \( E_{\frac{R}{2}} \) in terms of \( E \):
\[ E_{\frac{R}{2}} = \frac{1}{2} E \]

### Final Answer
The electric field at a distance \( \frac{R}{2} \) from the center of the sphere is:
\[ E_{\frac{R}{2}} = \frac{1}{2} E \]