Question

Electric field in a region is given by \[ \vec{E} = A x\,\hat{i} + B y\,\hat{j}, \] where \( A = 10 \,\text{V/m}^2 \) and \( B = 5 \,\text{V/m}^2 \). If the electric potential at a point \( (10, 20) \) is \(500\ \text{V}\), then the electric potential at origin is __________ V.

• A. \(1000\)
• B. \(500\)
• C. \(2000\)
• D. \(0\)

Hint:

Electric potential can be obtained by integrating the electric field components with proper constants of integration.

Answer 1

The Correct Option is A

The electric field is related to potential by \[ \vec{E} = -\nabla V. \]
Step 1: Write component-wise relations.
\[ E_x = -\frac{\partial V}{\partial x} = A x, \quad E_y = -\frac{\partial V}{\partial y} = B y. \] Thus, \[ \frac{\partial V}{\partial x} = -A x, \quad \frac{\partial V}{\partial y} = -B y. \]
Step 2: Integrate to find potential.
Integrating with respect to \(x\), \[ V = -\frac{A x^2}{2} + f(y). \] Differentiating with respect to \(y\), \[ \frac{\partial V}{\partial y} = f'(y) = -B y. \] Integrating, \[ f(y) = -\frac{B y^2}{2} + C. \] Hence, \[ V(x,y) = -\frac{A x^2}{2} - \frac{B y^2}{2} + C. \]
Step 3: Use given potential value.
At \( (10,20) \), \[ 500 = -\frac{10(10)^2}{2} - \frac{5(20)^2}{2} + C. \] \[ 500 = -500 - 1000 + C. \] \[ C = 2000. \] Thus, potential at origin is \[ V(0,0) = 2000 - 0 - 0 = 1000\ \text{V}. \]
Final Answer: \[ \boxed{1000} \]