In a region where E = 0, the potential (V) varies with distance r as -

Options

A. `V alpha (1)/(r )`
B. `V alpha r`
C. `V alpha (1)/(r^(2))`
D. V = const. independent of (r )

Answer

To solve the question regarding how the electric potential (V) varies with distance (r) in a region where the electric field (E) is zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Electric Field and Potential:**
The electric field (E) is related to the electric potential (V) by the equation:
\[ E = -\frac{dV}{dr} \]
This equation indicates that the electric field is the negative gradient of the electric potential.

2. **Set the Electric Field to Zero:**
In the given scenario, we know that the electric field is zero:
\[ E = 0 \]

3. **Substitute E into the Relationship:**
Substituting \(E = 0\) into the equation gives:
\[ 0 = -\frac{dV}{dr} \]

4. **Analyze the Implication of the Equation:**
The equation \(0 = -\frac{dV}{dr}\) implies that:
\[ \frac{dV}{dr} = 0 \]
This means that the derivative of the potential with respect to distance (r) is zero.

5. **Integrate the Equation:**
Since the derivative of V with respect to r is zero, it indicates that V does not change with r. Therefore, we can conclude that:
\[ V = \text{constant} \]
This means that the potential V is independent of the distance r in a region where the electric field is zero.

6. **Conclusion:**
In a region where the electric field \(E = 0\), the potential \(V\) remains constant regardless of the distance \(r\).

### Final Answer:
The potential \(V\) varies with distance \(r\) as a constant value.

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