Question

In a space having electric field \( \vec{E} = A(x\hat{i} + y\hat{j}) \), the potential at point (10 m, 20 m) is zero. Then the potential at the origin is: (Given \( A = 10 \, \text{Vm}^{-2} \))

• A. \( 500 \, \text{V} \)
• B. \( 2000 \, \text{V} \)
• C. \( 2500 \, \text{V} \)
• D. \( 1500 \, \text{V} \)

Hint:

To find potential from a known point, integrate \( -\vec{E} \cdot d\vec{r} \) along a path between the two points.

Answer 1

The Correct Option is C

Given: \[ \vec{E} = A(x\hat{i} + y\hat{j}), \quad A = 10 \, \text{Vm}^{-2} \] Electric potential is given by: \[ V = -\int \vec{E} \cdot d\vec{r} \] Let us calculate potential at origin with reference to point (10, 20), where \( V = 0 \). Then: \[ V_O = -\int_{(10,20)}^{(0,0)} A(x\, dx + y\, dy) \] \[ = -A \left[ \int_{10}^{0} x\, dx + \int_{20}^{0} y\, dy \right] = -10 \left[ \frac{1}{2}(0^2 - 10^2) + \frac{1}{2}(0^2 - 20^2) \right] \] \[ = -10 \left[ -50 - 200 \right] = 2500 \, \text{V} \]