Consider a uniform electric field in the `hat (z)` direction. The potential is a constant.

Options

A. for any x for a given z
B. for any y for a given z
C. on the x-y plane for a given z
D. All of these

Explanation

To solve the problem regarding the uniform electric field in the \(\hat{z}\) direction and the behavior of electric potential, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electric Field**:
- We have a uniform electric field directed along the \(\hat{z}\) axis. This means that the electric field \(\vec{E}\) can be expressed as:
\[ \vec{E} = E \hat{z} \]
where \(E\) is a constant magnitude of the electric field.

2. **Electric Potential and Electric Field Relationship**:
- The relationship between electric potential \(V\) and electric field \(\vec{E}\) is given by:
\[ \vec{E} = -\nabla V \]
This indicates that the electric field is the negative gradient of the electric potential. In simpler terms, the potential decreases in the direction of the electric field.

3. **Potential Change in the z-direction**:
- If we move in the direction of the electric field (which is along the \(\hat{z}\) direction), the potential will change. Specifically, if we move a distance \(dz\) in the \(\hat{z}\) direction, the change in potential \(dV\) can be expressed as:
\[ dV = -E \, dz \]
This shows that the potential changes when we move in the \(\hat{z}\) direction.

4. **Potential Change in the x and y Directions**:
- If we move in the \(x\) or \(y\) directions while keeping \(z\) constant, the potential does not change. This is because there is no displacement in the direction of the electric field. Therefore, for any movement in the \(xy\) plane (where \(z\) remains constant), the potential remains constant:
\[ dV = 0 \quad \text{(for movement in the x or y direction)} \]

5. **Conclusion**:
- Since the potential only changes when there is movement along the direction of the electric field (the \(\hat{z}\) direction), and remains constant for movements in the \(x\) or \(y\) directions, we can conclude that the potential is constant for any motion that does not involve a change in the \(z\) coordinate.

### Final Answer:
The potential is constant for any motion in the \(xy\) plane (i.e., when \(z\) is held constant).