Electric potential at a point P, r distance away due to a point charge q kept at point A is V. If twice of this charge is distributed uniformly on the surface of a hollow sphere of radius 4r with centre at point A the potential at P now is

Options

A. V
B. `V//2`
C. `V//4`
D. `V//8`

Explanation

To solve the problem, we need to determine the electric potential at point P due to a hollow sphere with a uniformly distributed charge. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Situation
Initially, we have a point charge \( q \) located at point A. The electric potential \( V \) at point P, which is at a distance \( r \) from charge \( q \), is given by the formula:
\[ V = k \frac{q}{r} \]
where \( k \) is Coulomb's constant.

### Step 2: Determine the New Charge Distribution
The problem states that twice the initial charge \( q \) (i.e., \( 2q \)) is now uniformly distributed over the surface of a hollow sphere with a radius of \( 4r \) centered at point A.

### Step 3: Calculate the Potential Due to the Hollow Sphere
For a hollow sphere with a uniform charge distribution, the electric potential at any point outside the sphere (or on its surface) is given by:
\[ V' = k \frac{Q}{R} \]
where \( Q \) is the total charge and \( R \) is the distance from the center of the sphere to the point of interest.

In this case, since point P is at a distance \( r \) from point A and the radius of the sphere is \( 4r \), point P is inside the hollow sphere.

### Step 4: Use the Formula for Potential Inside a Hollow Sphere
For any point inside a hollow charged sphere, the potential is constant and equal to the potential on the surface of the sphere. Thus, we need to calculate the potential at the surface of the sphere:
\[ V' = k \frac{2q}{4r} \]
Simplifying this gives:
\[ V' = k \frac{q}{2r} \]

### Step 5: Relate the New Potential to the Original Potential
From the original potential \( V = k \frac{q}{r} \), we can express the new potential \( V' \) in terms of \( V \):
\[ V' = \frac{1}{2} \left( k \frac{q}{r} \right) = \frac{V}{2} \]

### Conclusion
Thus, the potential at point P due to the hollow sphere is:
\[ V' = \frac{V}{2} \]

### Final Answer
The potential at point P now is \( \frac{V}{2} \).

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