The electric potential in a region is given by `V = (2x^(2) - 3y)` volt where `x` and `y` are in meters. The electric field intensity at a point `(0, 3m, 5m)` is

Options

A. `-6 hati N//C`
B. `3 hatjN//C`
C. `-3 hatj N//C`
D. Zero

Explanation

To find the electric field intensity at the point (0, 3m, 5m) given the electric potential \( V = 2x^2 - 3y \), we will follow these steps:

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

### Step 2: Calculate the partial derivatives of \( V \)
We need to find the partial derivatives of \( V \) with respect to \( x \), \( y \), and \( z \).

1. **Calculate \( \frac{\partial V}{\partial x} \)**:
\[ V = 2x^2 - 3y \]
Differentiating with respect to \( x \):
\[ \frac{\partial V}{\partial x} = 4x \]

2. **Calculate \( \frac{\partial V}{\partial y} \)**:
\[ \frac{\partial V}{\partial y} = -3 \]

3. **Calculate \( \frac{\partial V}{\partial z} \)**:
Since \( V \) does not depend on \( z \):
\[ \frac{\partial V}{\partial z} = 0 \]

### Step 3: Write the components of the electric field
Using the results from the partial derivatives, we can express the components of the electric field:
\[ E_x = -\frac{\partial V}{\partial x} = -4x \]
\[ E_y = -\frac{\partial V}{\partial y} = 3 \]
\[ E_z = -\frac{\partial V}{\partial z} = 0 \]

### Step 4: Evaluate the electric field at the point (0, 3m, 5m)
Now we substitute \( x = 0 \), \( y = 3 \), and \( z = 5 \) into the expressions for the electric field components:
1. **Calculate \( E_x \)**:
\[ E_x = -4(0) = 0 \]

2. **Calculate \( E_y \)**:
\[ E_y = 3 \]

3. **Calculate \( E_z \)**:
\[ E_z = 0 \]

### Step 5: Write the electric field vector
The electric field vector at the point (0, 3m, 5m) is:
\[ \mathbf{E} = E_x \hat{i} + E_y \hat{j} + E_z \hat{k} = 0 \hat{i} + 3 \hat{j} + 0 \hat{k} = 3 \hat{j} \, \text{N/C} \]

### Final Answer
The electric field intensity at the point (0, 3m, 5m) is:
\[ \mathbf{E} = 3 \hat{j} \, \text{N/C} \]

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