Find the potential `V` of an electrostatic field `vec E = a(y hat i + x hat j)`, where `a` is a constant.

Options

A. `axy + C`
B. `-axy + C`
C. `axy`
D. `-axy`

Explanation

To find the potential \( V \) of the given electrostatic field \( \vec{E} = a(\hat{y} \hat{i} + \hat{x} \hat{j}) \), we will follow these steps:

### Step 1: Understand the relationship between electric field and potential
The relationship between the electric field \( \vec{E} \) and the electric potential \( V \) is given by the equation:
\[ dV = -\vec{E} \cdot d\vec{r} \]
where \( d\vec{r} \) is the differential position vector.

### Step 2: Identify the electric field and position vector
Given:
\[ \vec{E} = a(y \hat{i} + x \hat{j}) \]
The position vector \( d\vec{r} \) can be expressed as:
\[ d\vec{r} = dx \hat{i} + dy \hat{j} \]

### Step 3: Calculate the dot product
Now, we need to calculate the dot product \( \vec{E} \cdot d\vec{r} \):
\[ \vec{E} \cdot d\vec{r} = a(y \hat{i} + x \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) \]
This expands to:
\[ = a(y \cdot dx + x \cdot dy) \]

### Step 4: Substitute into the potential equation
Now substituting this back into the equation for \( dV \):
\[ dV = -a(y \cdot dx + x \cdot dy) \]

### Step 5: Separate the variables
We can separate the variables:
\[ dV = -ay \, dx - ax \, dy \]

### Step 6: Integrate both terms
Now we will integrate both terms separately:
1. Integrate \( -ay \, dx \):
\[ \int -ay \, dx = -ayx + C_1 \]
2. Integrate \( -ax \, dy \):
\[ \int -ax \, dy = -axy + C_2 \]

### Step 7: Combine the results
Combining these results gives:
\[ V = -axy + C \]
where \( C \) is a constant of integration.

### Final Answer
Thus, the potential \( V \) is given by:
\[ V = -a xy + C \]