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A solid sphere of radius R has charge q ...

A solid sphere of radius R has charge q uniformly distributed over its volume. The distance from it surfce at which the electrostatic potential is equal to half of the potential at the centre is

A

R

B

`R//2`

C

`R//3`

D

`2R`

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The correct Answer is:
To solve the problem, we need to determine the distance from the surface of a uniformly charged solid sphere where the electrostatic potential is equal to half of the potential at the center of the sphere. ### Step-by-Step Solution: 1. **Understand the Potential at the Center and Surface:** - The potential \( V \) at the center of a uniformly charged solid sphere of radius \( R \) with total charge \( q \) is given by: \[ V_{\text{center}} = \frac{3}{2} \frac{kq}{R} \] - The potential \( V \) at the surface of the sphere is given by: \[ V_{\text{surface}} = \frac{kq}{R} \] 2. **Set Up the Equation for Half the Center Potential:** - According to the problem, we need to find the distance \( r \) from the surface where the potential is half of the potential at the center: \[ V = \frac{1}{2} V_{\text{center}} = \frac{1}{2} \left( \frac{3}{2} \frac{kq}{R} \right) = \frac{3}{4} \frac{kq}{R} \] 3. **Potential Outside the Sphere:** - For a point located at a distance \( r \) from the center (where \( r > R \)), the potential is given by: \[ V = \frac{kq}{r} \] 4. **Equate the Two Potentials:** - Set the expression for the potential outside the sphere equal to \( \frac{3}{4} \frac{kq}{R} \): \[ \frac{kq}{r} = \frac{3}{4} \frac{kq}{R} \] - Cancel \( kq \) from both sides: \[ \frac{1}{r} = \frac{3}{4R} \] 5. **Solve for \( r \):** - Rearranging gives: \[ r = \frac{4R}{3} \] 6. **Find the Distance from the Surface:** - The distance from the surface of the sphere to this point is: \[ \text{Distance from surface} = r - R = \frac{4R}{3} - R = \frac{4R}{3} - \frac{3R}{3} = \frac{R}{3} \] ### Final Answer: The distance from the surface at which the electrostatic potential is equal to half of the potential at the center is: \[ \frac{R}{3} \]

To solve the problem, we need to determine the distance from the surface of a uniformly charged solid sphere where the electrostatic potential is equal to half of the potential at the center of the sphere. ### Step-by-Step Solution: 1. **Understand the Potential at the Center and Surface:** - The potential \( V \) at the center of a uniformly charged solid sphere of radius \( R \) with total charge \( q \) is given by: \[ V_{\text{center}} = \frac{3}{2} \frac{kq}{R} ...
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