Question

Two concentric circular loops, one of radius $R$ and the other of radius $2 R$, lie in the $x y$-plane with the origin as their common centre, as shown in the figure. The smaller loop carries current $I_{1}$ in the anti-clockwise direction and the larger loop carries current $I_{2}$ in the clock wise direction, with $I_{2}>2 I_{1}, \vec{B}(x, y)$ denotes the magnetic field at a point $(x, y)$ in the $x y$-plane. Which of the following statement (s) is ( are ) correct?

• A. $\vec{ B }( x , y )$ is perpendicular to the $xy$-plane at any point in the plane
• B. $|\vec{ B }( x , y )|$ depends on $x$ and $y$ only through the radial distance $r =\sqrt{ x ^{2}+ y ^{2}}$
• C. $|\vec{ B }( x , y )|$ is non-zero at all points for $r < R$
• D. $\vec{ B }( x , y )$ points normally outward from the $xy$-plane for all the points between the two loops

Answer 1

Answer: (A), (B)

Consider a circular loop of radius $r$ in $x - y$ plane and having centre at origin
$\oint \vec{ B } \cdot \overrightarrow{ d \ell}=0$
$B \oint d \ell \cos \theta=0$
$\because B \neq 0 \,\,\,$ for given $r$
$\Rightarrow \cos \theta=0$
$\theta=90^{\circ}$

Here $d \ell$ is in $xy$ plane $\Rightarrow B$ is normal to plane (B can't be in $xy$ plane as its magnetic lines would have been in radial direction)
Also, for given $r , B$ must be same in magnitude for all points on loop of radius $r$.
At centre $B =\left(\frac{\mu_{0} i _{1}}{2 R }-\frac{\mu_{0} i _{2}}{4 R }\right)$
(inwards)

For point $P$, Let field of inner loop increases $x_{1}$ times and that of outer loop increases $x _{2}$ times
$\Rightarrow $ magnetic field at $P$
$B _{ P }=\left( x _{1} \frac{\mu_{0} i _{1}}{2 R }- x _{2} \frac{\mu_{0} i _{2}}{4 R }\right)$
For $B _{ P }=0, i _{2}=\left(\frac{ x _{1}}{ x _{2}}\right) \cdot\left(2 i _{1}\right)$
$\because B$ changes more rapidly as point $P$ come closer to circumference.
$\Rightarrow x _{1} > x _{2}$
Or $i _{2} >2 i _{1}$ (which is given condition)
So, there are points inside inner loop where magnetic field will be zero.