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GATE 2024 EC – Question 16

Electromagnetics · Maxwell's Equations · 1 mark · Multiple choice

Let $\hat i$ and $\hat j$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is true for the vector fields $\vec F_1=A(\hat iy+\hat jx)$ and $\vec F_2=A(\hat iy-\hat jx)$?

  1. Both $\vec F_1$ and $\vec F_2$ are electrostatic fields.
  2. Only $\vec F_1$ is an electrostatic field.
  3. Only $\vec F_2$ is an electrostatic field.
  4. Neither $\vec F_1$ nor $\vec F_2$ is an electrostatic field.

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Correct answer: (B) Only $\vec F_1$ is an electrostatic field.

Explanation

An electrostatic field must be curl-free. For $\vec F=F_x\hat i+F_y\hat j$, $(\nabla\times\vec F)_z=\partial F_y/\partial x-\partial F_x/\partial y$. For $\vec F_1$: $A-A=0$, so it is curl-free. For $\vec F_2$: $-A-A=-2A\ne0$, so it is not. Only $\vec F_1$ is an electrostatic field.