About an electric field which of the following statements are not true ?

Options

A. If `E=0,V` must be zero
B. If `V=0,E` must be zero
C. If `E ne 0,V` cannot be zero
D. If `V ne 0,E` cannot be zero

Answer

To determine which statements about electric fields are not true, we need to analyze the relationship between electric potential (V) and electric field (E). Let's break down the statements one by one.

### Step-by-Step Solution:

1. **Understanding Electric Field and Potential**:
- The electric field (E) is defined as the negative gradient of electric potential (V):
\[ E = -\frac{dV}{dr} \]
- This means that if the electric field is zero at a point, the potential must be constant at that point, but it does not necessarily have to be zero.

2. **Analyzing the Statements**:
- **Statement 1**: "If E = 0, then V must be 0."
- This statement is **not true**. If the electric field is zero, it indicates that the potential is constant, but it can be any value (not necessarily zero).

- **Statement 2**: "If V = 0, then E must be 0."
- This statement is also **not true**. The potential can be zero at a point while the electric field can still be non-zero. For example, at a point in space where the potential is zero, there can still be a gradient causing an electric field.

- **Statement 3**: "If E ≠ 0, then V cannot be 0."
- This statement is **not true**. It is possible for the electric field to be non-zero while the potential at a specific point is zero. The potential can be zero at one point while the electric field exists in the surrounding space.

- **Statement 4**: "If V ≠ 0, then E cannot be 0."
- This statement is **true**. If the potential is not constant (i.e., not equal to zero), then the electric field must be non-zero.

3. **Conclusion**:
- The statements that are not true are:
- Statement 1: "If E = 0, then V must be 0."
- Statement 2: "If V = 0, then E must be 0."
- Statement 3: "If E ≠ 0, then V cannot be 0."

### Summary of Findings:
- **Not True Statements**: 1, 2, and 3.
- **True Statement**: 4.