If the potential function is define as `V = ( - 3 x + 4y + 12 z)V`, then magnitude of electric field `E( x,y,z)` is

Options

A. `16 V //m`
B. ` 12 V //m`
C. `sqrt(14) V//m`
D. `13 V //m`

Answer

To find the magnitude of the electric field \( E(x,y,z) \) from the given potential function \( V = -3x + 4y + 12z \), we will follow these steps:

### Step 1: Understand the relationship between electric field and potential
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 gradient of the potential
The gradient \( \nabla V \) in three dimensions is given by:
\[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]

### Step 3: Find the partial derivatives
Now, we will compute the partial derivatives of \( V \):
1. **Partial derivative with respect to \( x \)**:
\[ \frac{\partial V}{\partial x} = -3 \]
2. **Partial derivative with respect to \( y \)**:
\[ \frac{\partial V}{\partial y} = 4 \]
3. **Partial derivative with respect to \( z \)**:
\[ \frac{\partial V}{\partial z} = 12 \]

### Step 4: Write the electric field vector
Using the results from the partial derivatives, we can write the electric field vector:
\[ \mathbf{E} = -\left( -3, 4, 12 \right) = (3, -4, -12) \]

### Step 5: 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 components of \( \mathbf{E} \):
\[ |\mathbf{E}| = \sqrt{3^2 + (-4)^2 + (-12)^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \, \text{V/m} \]

### Final Answer
The magnitude of the electric field \( E(x,y,z) \) is:
\[ |\mathbf{E}| = 13 \, \text{V/m} \]

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