Magnetics 6 Question 16
16. 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 $(0,0, \sqrt{3} R)$ 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 $+\hat{j}$-direction. Which of the following statements regarding the magnetic field $B$ is (are) true?
(2018 Adv.)
(a) If $I _1=I _2$, then $B$ cannot be equal to zero at the origin $(0,0,0)$
(b) If $I _1>0$ and $I _2<0$, then $B$ can be equal to zero at the origin $(0,0,0)$
(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 $-\frac{\mu _o I}{2 R}$
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Solution:
(a) At origin, $\mathbf{B}=0$ due to two wires if $I _1=I _2$, hence $\left(\mathbf{B} _{\text {net }}\right)$ at origin is equal to $\mathbf{B}$ due to ring. which is non-zero.
(b) If $I _1>0$ and $I _2<0, B$ at origin due to wires will be along $+\hat{k}$. Direction of $B$ due to ring is along $-\hat{k}$ direction and hence $B$ can be zero at origin.
(c) If $I _1<0$ and $I _2>0$, B at origin due to wires is along $-\hat{k}$ and also along $-\hat{k}$ due to ring, hence $B$ cannot be zero.
(d)
At centre of ring, $B$ due to wires is along $x$-axis.
Hence, $z$-component is only because of ring which $B=\frac{\mu _0 i}{2 R}(-\hat{k})$.
