Question

If the electric potential is \(V = 500\) volts at the point \((10,\,20)\) and the electric field is given by \[ \vec{E} = 10x\,\hat{i} + 5y\,\hat{j}\ \text{N/C}, \] find the potential at the origin.

• A. \(1000\) volt
• B. \(2000\) volt
• C. \(1500\) volt
• D. \(3000\) volt

Hint:

If electric field components depend only on coordinates, first find the potential function using \(\vec{E} = -\nabla V\), then apply the given boundary condition to determine the constant.

Answer 1

The Correct Option is B

Concept:  
The electric field and electric potential are related as follows: 
\[ \vec{E} = -\nabla V \] or, \[ dV = -\vec{E} \cdot d\vec{r} \] If the electric field is conservative (as is the case here, since it depends only on position), a scalar potential function \(V(x, y)\) exists. 
Step 1: Determine the potential function. Given: \[ \vec{E} = 10x\, \hat{i} + 5y\, \hat{j} \] From \(\vec{E} = -\nabla V\), we can write: \[ \frac{\partial V}{\partial x} = -10x \Rightarrow V = -5x^2 + f(y) \] Next, taking the partial derivative with respect to \(y\): \[ \frac{\partial V}{\partial y} = -5y \Rightarrow f'(y) = -5y \Rightarrow f(y) = -\frac{5}{2}y^2 + C \] Thus, the potential function is: \[ V(x,y) = -5x^2 - \frac{5}{2}y^2 + C \] 
Step 2: Use the given potential at the point \((10, 20)\). We are given: \[ 500 = -5(10)^2 - \frac{5}{2}(20)^2 + C \] Simplifying: \[ 500 = -500 - 1000 + C \] Solving for \(C\): \[ C = 2000 \] 
Step 3: Find the potential at the origin. At the origin \((0,0)\), the potential is: \[ V(0,0) = C = 2000\ \text{V} \] Thus, the potential at the origin is: \[ \boxed{V_{\text{origin}} = 2000\ \text{V}} \]