Electric potential at any point is `V = - 5 x + 3y + sqrt(15) z`, then the magnitude of the electric field is

Options

A. `3 sqrt(2)`
B. `4 sqrt(2)`
C. `5sqrt(2)`
D. `7`

Explanation

To find the magnitude of the electric field from the given electric potential \( V = -5x + 3y + \sqrt{15}z \), we will follow these steps:

### Step 1: Identify the electric field components
The electric field \( \mathbf{E} \) is related to the electric potential \( V \) by the following equations:
\[ E_x = -\frac{\partial V}{\partial x}, \quad E_y = -\frac{\partial V}{\partial y}, \quad E_z = -\frac{\partial V}{\partial z} \]

### Step 2: Calculate the partial derivatives
1. **Calculate \( E_x \)**:
\[ E_x = -\frac{\partial V}{\partial x} = -\frac{\partial}{\partial x}(-5x + 3y + \sqrt{15}z) = -(-5) = 5 \]

2. **Calculate \( E_y \)**:
\[ E_y = -\frac{\partial V}{\partial y} = -\frac{\partial}{\partial y}(-5x + 3y + \sqrt{15}z) = -3 \]

3. **Calculate \( E_z \)**:
\[ E_z = -\frac{\partial V}{\partial z} = -\frac{\partial}{\partial z}(-5x + 3y + \sqrt{15}z) = -\sqrt{15} \]

### Step 3: Write the electric field vector
Now we can express the electric field vector \( \mathbf{E} \):
\[ \mathbf{E} = (E_x, E_y, E_z) = (5, -3, -\sqrt{15}) \]

### Step 4: Calculate the magnitude of the electric field
The magnitude of the electric field \( |\mathbf{E}| \) is given by:
\[ |\mathbf{E}| = \sqrt{E_x^2 + E_y^2 + E_z^2} \]
Substituting the values we found:
\[ |\mathbf{E}| = \sqrt{5^2 + (-3)^2 + (-\sqrt{15})^2} \]
Calculating each term:
\[ = \sqrt{25 + 9 + 15} = \sqrt{49} = 7 \]

### Final Answer
The magnitude of the electric field is \( 7 \).

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