Two infinitely long straight wires lie in the $xy$-plane along the lines $x= \pm R$. The wire located at $x=+R$ carries a constant current $I_{1}$ and the wire located at $x=-R$ carries a constant current $I_{2}$. A circular loop of radius $R$ is suspended with its centre at $(0,0, \sqrt{3} R)$ and in a plane parallel to the $xy$-plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\hat{\jmath}$ direction. Which of the following statements regarding the magnetic field $\vec{B}$ is (are) true?

(A) If $I_{1}=I_{2}$, then $\vec{B}$ cannot be equal to zero at the origin $(0,0,0)$; (B) If $I_{1}>0$ and $I_{2}<0$, then $\vec{B}$ can be equal to zero at the origin $(0,0,0)$; (D) If $I_{1}=I_{2}$, then the $z$-component of the magnetic field at the centre of the loop is $\left(-\frac{\mu_{0} I}{2 R}\right)$

(C) If $I_{1} 0$, then $\vec{B}$ can be equal to zero at the origin $(0,0,0)$

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Magnetic Effects of Current

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Two infinitely long straight wires lie in the

x y

-plane along the lines

x = \pm R

. The wire located at

x = + R

carries a constant current

I_{1}

and the wire located at

x = - R

carries a constant current

I_{2}

. A circular loop of radius

R

is suspended with its centre at

(0, 0, 3 R)

and in a plane parallel to the

x y

-plane. This loop carries a constant current

I

in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the

+ \overset{}{^}

direction. Which of the following statements regarding the magnetic field

B

is (are) true?

(A)If $I_{1} = I_{2}$ , then $B$ cannot be equal to zero at the origin $(0, 0, 0)$ ✓ Correct
(B)If $I_{1} > 0$ and $I_{2} < 0$ , then $B$ can be equal to zero at the origin $(0, 0, 0)$ ✓ Correct
(C)If $I_{1} < 0$ and $I_{2} > 0$ , then $B$ can be equal to zero at the origin $(0, 0, 0)$
(D)If $I_{1} = I_{2}$ , then the $z$ -component of the magnetic field at the centre of the loop is $(- \frac{μ _{0} I}{2 R})$ ✓ Correct

Answer: (A), (B), (D)

Solution

(A) If

I_{1} = I_{2}

, the fields from the two wires cancel at the origin, but the loop's field persists, so

B \neq = 0

— correct. (B) If

I_{1} > 0

and

I_{2} < 0

, both wires and the loop contribute fields in the same direction, so

B

can be zero by adjusting magnitudes — correct. (C) If

I_{1} < 0

and

I_{2} > 0

, the wire fields and loop field are all in the same direction, so

B

cannot be zero — incorrect. (D) If

I_{1} = I_{2}

, the wire contributions cancel, leaving only the loop's field at its center:

B_{z} = - \frac{μ _{0} I}{2 R}

— correct.

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