If `vec(E)=2y hat(i)+2x hat(j)`, then find `V (x, y, z)`

Answer

To find the electric potential \( V(x, y, z) \) given the electric field \( \vec{E} = 2y \hat{i} + 2x \hat{j} \), we can follow these steps:

### Step 1: Understand the relationship between electric field and potential
The electric field \( \vec{E} \) is related to the electric potential \( V \) by the equation:
\[ \vec{E} = -\nabla V \]
This means that the electric field is the negative gradient of the electric potential.

### Step 2: Write the expression for the electric field
Given:
\[ \vec{E} = 2y \hat{i} + 2x \hat{j} \]
We can express this in terms of its components:
\[ E_x = 2y, \quad E_y = 2x, \quad E_z = 0 \]

### Step 3: Relate the electric field components to the potential
From the relationship \( \vec{E} = -\nabla V \), we can write:
\[ E_x = -\frac{\partial V}{\partial x}, \quad E_y = -\frac{\partial V}{\partial y}, \quad E_z = -\frac{\partial V}{\partial z} \]
Substituting the components of the electric field:
\[ -\frac{\partial V}{\partial x} = 2y \quad \text{(1)} \]
\[ -\frac{\partial V}{\partial y} = 2x \quad \text{(2)} \]
\[ -\frac{\partial V}{\partial z} = 0 \quad \text{(3)} \]

### Step 4: Integrate to find the potential \( V \)
From equation (1):
\[ \frac{\partial V}{\partial x} = -2y \]
Integrating with respect to \( x \):
\[ V = -2xy + f(y, z) \]
where \( f(y, z) \) is an arbitrary function of \( y \) and \( z \).

From equation (2):
\[ \frac{\partial V}{\partial y} = -2x \]
Substituting \( V = -2xy + f(y, z) \):
\[ \frac{\partial}{\partial y}(-2xy + f(y, z)) = -2x \]
This gives:
\[ -2x + \frac{\partial f}{\partial y} = -2x \]
Thus, \( \frac{\partial f}{\partial y} = 0 \), which implies that \( f(y, z) \) does not depend on \( y \). Therefore, we can write:
\[ f(y, z) = g(z) \]
where \( g(z) \) is another arbitrary function of \( z \).

From equation (3):
\[ \frac{\partial V}{\partial z} = 0 \]
This implies that \( g(z) \) is a constant, say \( C \).

### Step 5: Write the final expression for the potential
Combining all parts, we have:
\[ V(x, y, z) = -2xy + C \]

### Final Answer:
\[ V(x, y, z) = -2xy + C \]