Question

The electric potential as a function of \(x,y\) is given by \(V=5(x^2-y^2)\,\text{V}\). The electric field at the point \((2,3)\) m is _____ V/m.

• A. \( (-20\hat{i}+30\hat{j}) \)
• B. \( (20\hat{i}-30\hat{j}) \)
• C. \( (20\hat{i}+45\hat{j}) \)
• D. \( (-4\hat{i}+6\hat{j}) \)

Answer 1

The Correct Option is A

Concept: Electric field is the negative gradient of potential. \[ \vec{E}=-\nabla V \] \[ \vec{E}=-\left(\frac{\partial V}{\partial x}\hat{i}+\frac{\partial V}{\partial y}\hat{j}\right) \] Step 1: {Compute partial derivatives.} Given \[ V=5(x^2-y^2) \] \[ \frac{\partial V}{\partial x}=10x \] \[ \frac{\partial V}{\partial y}=-10y \] Step 2: {Apply the negative gradient.} \[ \vec{E}=-(10x\hat{i}-10y\hat{j}) \] \[ =-10x\hat{i}+10y\hat{j} \] Step 3: {Substitute the point \((2,3)\).} \[ \vec{E}=-20\hat{i}+30\hat{j} \]