The potentaial function of an electrostatic field is given by `V = 2 x^2`. Determine the electric field strength at the point `(2 m, 0, 3 m)`.

Options

A. `vec E = 4 hat i(NC^-1)`
B. `vec E = -4 hat i(NC^-1)`
C. `vec E = 8 hat i(NC^-1)`
D. `vec E = -8 hat i(NC^-1)`

Explanation

To determine the electric field strength at the point (2 m, 0, 3 m) given the potential function \( V = 2x^2 \), 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 formula:
\[ \vec{E} = -\nabla V \]
where \( \nabla V \) is the gradient of the potential function.

### Step 2: Calculate the gradient of the potential function
The gradient in three dimensions is given by:
\[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]
For our potential function \( V = 2x^2 \), we need to calculate the partial derivatives.

1. **Partial derivative with respect to \( x \)**:
\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(2x^2) = 4x \]

2. **Partial derivative with respect to \( y \)**:
\[ \frac{\partial V}{\partial y} = 0 \quad (\text{since } V \text{ does not depend on } y) \]

3. **Partial derivative with respect to \( z \)**:
\[ \frac{\partial V}{\partial z} = 0 \quad (\text{since } V \text{ does not depend on } z) \]

Thus, the gradient of the potential is:
\[ \nabla V = (4x, 0, 0) \]

### Step 3: Substitute the coordinates into the gradient
Now, we need to evaluate the gradient at the point \( (2, 0, 3) \):
\[ \nabla V \bigg|_{(2, 0, 3)} = (4 \cdot 2, 0, 0) = (8, 0, 0) \]

### Step 4: Calculate the electric field
Now, we can find the electric field:
\[ \vec{E} = -\nabla V = -(8, 0, 0) = (-8, 0, 0) \]

### Step 5: Write the final answer in vector form
Thus, the electric field strength at the point \( (2, 0, 3) \) is:
\[ \vec{E} = -8 \hat{i} \text{ N/C} \]

### Summary of the Solution
The electric field strength at the point \( (2 m, 0, 3 m) \) is \( -8 \hat{i} \) N/C.

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