The longer side of a rectangle is twice the length of its shorter side. A charge q is kept at one vertex. The maximum electric potential due to that charge at any other vertex is V, then the minimum electric potential at any other vertex will be

Options

A. 2V
B. `sqrt3V`
C. `V//sqrt5`
D. `sqrt5V`

Explanation

To solve the problem step by step, we will analyze the rectangle and the electric potential due to the charge \( q \) at different vertices.

### Step 1: Define the rectangle dimensions
Let the length of the shorter side of the rectangle be \( L \). Therefore, the longer side will be \( 2L \).

### Step 2: Identify the vertices
Assume the rectangle has vertices at:
- \( A(0, 0) \) (where the charge \( q \) is located)
- \( B(L, 0) \)
- \( C(L, 2L) \)
- \( D(0, 2L) \)

### Step 3: Calculate distances from charge \( q \) to other vertices
We need to find the distances from charge \( q \) at vertex \( A \) to the other vertices:
- Distance to vertex \( B \): \( r_{AB} = L \)
- Distance to vertex \( C \): \( r_{AC} = \sqrt{L^2 + (2L)^2} = \sqrt{L^2 + 4L^2} = \sqrt{5L^2} = L\sqrt{5} \)
- Distance to vertex \( D \): \( r_{AD} = 2L \)

### Step 4: Calculate electric potential at each vertex
The electric potential \( V \) due to a point charge \( q \) at a distance \( r \) is given by:
\[ V = \frac{kq}{r} \]
where \( k = \frac{1}{4\pi \epsilon_0} \).

1. **At vertex \( B \)**:
\[ V_B = \frac{kq}{L} \]

2. **At vertex \( C \)**:
\[ V_C = \frac{kq}{L\sqrt{5}} \]

3. **At vertex \( D \)**:
\[ V_D = \frac{kq}{2L} \]

### Step 5: Identify maximum and minimum potentials
From the calculations:
- The maximum potential \( V_{max} \) occurs at vertex \( B \):
\[ V_{max} = \frac{kq}{L} \]

- The minimum potential \( V_{min} \) occurs at vertex \( C \):
\[ V_{min} = \frac{kq}{L\sqrt{5}} \]

### Step 6: Relate \( V_{min} \) to \( V \)
Given that the maximum electric potential at any other vertex is \( V \):
\[ V = \frac{kq}{L} \]
Thus, we can express the minimum potential in terms of \( V \):
\[ V_{min} = \frac{V}{\sqrt{5}} \]

### Final Answer
The minimum electric potential at any other vertex is:
\[ V_{min} = \frac{V}{\sqrt{5}} \]

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