In the previous question, if `V` is the electric potential of the first sphere, what would be the electric potential of the second sphere ?

Options

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

Explanation

To find the electric potential of the second sphere given that the electric potential of the first sphere is \( V \), we can follow these steps:

### Step 1: Understand the relationship between electric field and potential
The electric potential \( V \) at a distance \( r \) from a charged sphere is related to the electric field \( E \) by the formula:
\[ V = -\int E \cdot dr \]
For a uniformly charged sphere, the electric field outside the sphere is given by:
\[ E = \frac{\sigma}{\epsilon_0} \]
where \( \sigma \) is the surface charge density and \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Determine the potential at the first sphere
Let’s denote the potential at the first sphere (radius \( r \)) as \( V_1 \). The potential at this radius can be expressed as:
\[ V_1 = \frac{Q}{4\pi \epsilon_0 r} \]
where \( Q \) is the total charge on the sphere.

### Step 3: Calculate the potential at the second sphere
Now, if the radius of the second sphere is \( 2r \), the potential \( V_2 \) at this radius can be calculated similarly:
\[ V_2 = \frac{Q}{4\pi \epsilon_0 (2r)} = \frac{1}{2} \cdot \frac{Q}{4\pi \epsilon_0 r} = \frac{1}{2} V_1 \]

### Step 4: Relate \( V_2 \) to \( V \)
Since we know \( V_1 = V \) (the electric potential of the first sphere), we can express the potential of the second sphere as:
\[ V_2 = \frac{1}{2} V \]

### Conclusion
Thus, the electric potential of the second sphere is:
\[ V_2 = \frac{1}{2} V \]

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