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 the electric field are not true, let's analyze each option step by step.

### Step 1: Understanding the Relationship between Electric Field and Electric Potential
The electric field (E) is related to the electric potential (V) by the equation:
\[ E = -\frac{dV}{dx} \]
This means that the electric field is the negative gradient of the electric potential.

### Step 2: Analyzing Each Statement
1. **Option 1: If E = 0, V must be 0.**
- This statement is **not true**. If the electric field is zero, it means there is no change in potential across that region, but the potential (V) can be a constant value (not necessarily zero).

2. **Option 2: If V = 0, E must be 0.**
- This statement is also **not true**. The potential can be zero at a point, but the electric field can still exist. For example, at the midpoint between two equal and opposite charges, the potential is zero, but the electric field is not.

3. **Option 3: If E ≠ 0, V cannot be 0.**
- This statement is **not true**. The electric field can be non-zero while the potential is zero at certain points, such as the midpoint between two equal and opposite charges.

4. **Option 4: If V ≠ 0, E cannot be 0.**
- This statement is **not true**. The electric field can be zero while the potential is a non-zero constant, such as inside a conductor where the potential is constant.

### Conclusion
All four statements are not true based on the relationships between electric field and electric potential.

### Summary of Findings:
- **Option 1:** Not true
- **Option 2:** Not true
- **Option 3:** Not true
- **Option 4:** Not true

### Final Answer:
All four options are incorrect regarding the relationship between electric field and electric potential.

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