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

Step 1: Understanding the Question:
We need to find the electric field vector \(\vec{E}\) at a specific coordinate from the potential function \(V(x, y)\).
Step 2: Key Formula or Approach:
The electric field is related to potential as:
\[ \vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} \right) \]
Step 3: Detailed Explanation:
Given \(V = 5x^2 - 5y^2\).
Partial derivative with respect to \(x\):
\[ \frac{\partial V}{\partial x} = 10x \]
Partial derivative with respect to \(y\):
\[ \frac{\partial V}{\partial y} = -10y \]
Electric field vector:
\[ \vec{E} = -(10x \hat{i} - 10y \hat{j}) = -10x \hat{i} + 10y \hat{j} \]
At point \((2, 3)\):
\[ \vec{E} = -10(2)\hat{i} + 10(3)\hat{j} = (-20\hat{i} + 30\hat{j}) \text{ V/m} \]
Step 4: Final Answer:
The electric field vector is \((-20\hat{i} + 30\hat{j})\).