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 components of the electric field
The components of the electric field can be found using the following formulas:
- \( E_x = -\frac{\partial V}{\partial x} \)
- \( E_y = -\frac{\partial V}{\partial y} \)
- \( E_z = -\frac{\partial V}{\partial z} \)

### Step 3: Compute \( E_x \)
We start with the potential function:
\[ V = -3x + 4y + 12z \]
Now, we differentiate \( V \) with respect to \( x \):
\[ E_x = -\frac{\partial V}{\partial x} = -\frac{\partial}{\partial x}(-3x + 4y + 12z) = -(-3) = 3 \]

### Step 4: Compute \( E_y \)
Next, we differentiate \( V \) with respect to \( y \):
\[ E_y = -\frac{\partial V}{\partial y} = -\frac{\partial}{\partial y}(-3x + 4y + 12z) = -4 = 4 \]

### Step 5: Compute \( E_z \)
Now, we differentiate \( V \) with respect to \( z \):
\[ E_z = -\frac{\partial V}{\partial z} = -\frac{\partial}{\partial z}(-3x + 4y + 12z) = -12 = 12 \]

### Step 6: 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:
\[ |\mathbf{E}| = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \]

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

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