Question

Electric potential due to a space is given by: \[ \phi(x, y, z) = \phi_0 \cdot \frac{x_0}{x} \] where \( x_0 = 5\,\text{m}, \phi_0 = 8\,\text{V} \). Find the electric field at the point (10 m, 5 m, 5 m).

• A. \( 0.40 \, \text{Vm}^{-1} \hat{i} \)
• B. \( -0.40 \, \text{Vm}^{-1} \hat{i} \)
• C. \( 4.0 \, \text{Vm}^{-1} \hat{i} \)
• D. \( -4.0 \, \text{Vm}^{-1} \hat{i} \)

Hint:

Use \( \vec{E} = -\nabla \phi \), and carefully take derivative of scalar potential with respect to position.

Answer 1

The Correct Option is A

Electric field is the negative gradient of potential: \[ \vec{E} = -\nabla \phi = -\frac{d\phi}{dx} \hat{i} \] Given: \[ \phi(x) = \phi_0 \cdot \frac{x_0}{x} = 8 \cdot \frac{5}{x} \Rightarrow \frac{d\phi}{dx} = -\frac{40}{x^2} \Rightarrow \vec{E} = -(-\frac{40}{x^2}) \hat{i} = \frac{40}{x^2} \hat{i} \] At \( x = 10 \), \[ \vec{E} = \frac{40}{100} \hat{i} = 0.40 \, \text{Vm}^{-1} \hat{i} \]