Uniform electric field exists in a region and is given by `vecE = E_0hati + E_0hatj`. There are four points `A(-a,0), B(0,-a), C(a,0), and D(0,a)` in the xy plane. Which of the following is the correct relation for the electirc potential?

Options

A. `V_A=V_CgtV_B=V_D`
B. `V_A=V_BgtV_C=V_D`
C. `V_AgtV_CgtV_B=V_D`
D. `V_AltV_CltV_BltV_D`

Explanation

To solve the problem, we need to analyze the electric potential at the four points A, B, C, and D in the given uniform electric field \( \vec{E} = E_0 \hat{i} + E_0 \hat{j} \). The relationship between electric field and electric potential is given by:

\[ \vec{E} = -\nabla V \]

This implies that the change in electric potential \( dV \) can be expressed as:

\[ dV = -\vec{E} \cdot d\vec{r} \]

Where \( d\vec{r} = dx \hat{i} + dy \hat{j} \).

### Step 1: Calculate the electric potential at each point

1. **Point A (-a, 0)**:
\[ V_A = -\int_{(0,0)}^{(-a,0)} \vec{E} \cdot d\vec{r} \]
\[ = -\int_0^{-a} E_0 \hat{i} \cdot dx \hat{i} - \int_0^{0} E_0 \hat{j} \cdot dy \hat{j} \]
\[ = -(-E_0 a) = E_0 a \]

2. **Point B (0, -a)**:
\[ V_B = -\int_{(0,0)}^{(0,-a)} \vec{E} \cdot d\vec{r} \]
\[ = -\int_0^{0} E_0 \hat{i} \cdot dx \hat{i} - \int_0^{-a} E_0 \hat{j} \cdot dy \hat{j} \]
\[ = -(-E_0 a) = E_0 a \]

3. **Point C (a, 0)**:
\[ V_C = -\int_{(0,0)}^{(a,0)} \vec{E} \cdot d\vec{r} \]
\[ = -\int_0^{a} E_0 \hat{i} \cdot dx \hat{i} - \int_0^{0} E_0 \hat{j} \cdot dy \hat{j} \]
\[ = -E_0 a \]

4. **Point D (0, a)**:
\[ V_D = -\int_{(0,0)}^{(0,a)} \vec{E} \cdot d\vec{r} \]
\[ = -\int_0^{0} E_0 \hat{i} \cdot dx \hat{i} - \int_0^{a} E_0 \hat{j} \cdot dy \hat{j} \]
\[ = -E_0 a \]

### Step 2: Summarize the potentials

From our calculations, we have:
- \( V_A = E_0 a \)
- \( V_B = E_0 a \)
- \( V_C = -E_0 a \)
- \( V_D = -E_0 a \)

### Step 3: Establish the relations

From the results:
- \( V_A = V_B \)
- \( V_C = V_D \)
- \( V_A > V_C \) and \( V_B > V_D \)

Thus, the correct relations for the electric potential are:
\[ V_A = V_B > V_C = V_D \]

### Final Answer

The correct relation for the electric potential is:
\[ V_A = V_B > V_C = V_D \]