The GATE Grind

GATE 2015 EE – Question 22

Electromagnetic Fields · Biot-Savart law, Ampere law, Curl, Faraday law, Lorentz force · 1 mark · Multiple choice

A steady current $I$ is flowing in the $-x$ direction through each of two infinitely long wires at $y = \pm\frac{L}{2}$ as shown in the figure. The permeability of the medium is $\mu_0$. The $\vec{B}$-field at $(0, L, 0)$ is

Two long wires lying along the $x$-axis direction, at $y = -L/2$ and $y = +L/2$, each carrying current $I$ in the $-x$ direction.
  1. $-\frac{4\mu_0 I}{3\pi L}\hat{z}$
  2. $+\frac{4\mu_0 I}{3\pi L}\hat{z}$
  3. 0
  4. $-\frac{3\mu_0 I}{4\pi L}\hat{z}$

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Correct answer: (A) $-\frac{4\mu_0 I}{3\pi L}\hat{z}$

Explanation

The point $(0, L, 0)$ is a distance $\frac{L}{2}$ from the wire at $y = \frac{L}{2}$ and $\frac{3L}{2}$ from the wire at $y = -\frac{L}{2}$. For a current in the $-x$ direction, the field at points above the wire points along $-\hat{z}$. The magnitudes are $\frac{\mu_0 I}{2\pi (L/2)} = \frac{\mu_0 I}{\pi L}$ and $\frac{\mu_0 I}{2\pi (3L/2)} = \frac{\mu_0 I}{3\pi L}$. The total is $-\frac{4\mu_0 I}{3\pi L}\hat{z}$.