The potential V is varying with x as `V = (1)/(2) (y^2 - 4 x)` volt. The field at `x = 1m, y = 1 m,` is.

Options

A. 1. `2 hat i + hat j Vm^-1`
B. 2. `- 2 hat i + hat j Vm^-1`
C. 3. `2 hat i - hat j Vm^-1`
D. 4. `-2 hat i + 2 hat j Vm^-1`

Explanation

To find the electric field at the point \( (x, y) = (1 \, \text{m}, 1 \, \text{m}) \) given the potential \( V = \frac{1}{2} (y^2 - 4x) \) volts, we will follow these steps:

### Step 1: Calculate the electric field in the x-direction \( E_x \)

The electric field in the x-direction is given by the negative partial derivative of the potential \( V \) with respect to \( x \):

\[ E_x = -\frac{\partial V}{\partial x} \]

Substituting the expression for \( V \):

\[ E_x = -\frac{\partial}{\partial x} \left( \frac{1}{2} (y^2 - 4x) \right) \]

### Step 2: Differentiate \( V \) with respect to \( x \)

When differentiating, treat \( y \) as a constant:

\[ E_x = -\frac{1}{2} \cdot (-4) = 2 \, \text{V/m} \]

Thus,

\[ E_x = 2 \, \hat{i} \, \text{V/m} \]

### Step 3: Calculate the electric field in the y-direction \( E_y \)

The electric field in the y-direction is given by the negative partial derivative of the potential \( V \) with respect to \( y \):

\[ E_y = -\frac{\partial V}{\partial y} \]

Substituting the expression for \( V \):

\[ E_y = -\frac{\partial}{\partial y} \left( \frac{1}{2} (y^2 - 4x) \right) \]

### Step 4: Differentiate \( V \) with respect to \( y \)

When differentiating, treat \( x \) as a constant:

\[ E_y = -\frac{1}{2} \cdot (2y) = -y \]

Given that \( y = 1 \, \text{m} \):

\[ E_y = -1 \, \hat{j} \, \text{V/m} \]

### Step 5: Combine the electric fields

The total electric field \( \mathbf{E} \) at the point \( (1 \, \text{m}, 1 \, \text{m}) \) is given by:

\[ \mathbf{E} = E_x \hat{i} + E_y \hat{j} \]

Substituting the values we found:

\[ \mathbf{E} = 2 \hat{i} - 1 \hat{j} \, \text{V/m} \]

### Final Answer

Thus, the electric field at the point \( (1 \, \text{m}, 1 \, \text{m}) \) is:

\[ \mathbf{E} = 2 \hat{i} - 1 \hat{j} \, \text{V/m} \]

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