Question

A circular current-carrying coil has a radius R. The distance from the center of the coil, on the axis, where B will be $\dfrac{1}{8}$ of its value at the center of the coil is:
A) $\dfrac{R}{{\sqrt 3 }}$
B) $\sqrt 3 R$
C) $2\sqrt 3 R$
D) $\dfrac{{2R}}{{\sqrt 3 }}$

Answer 1

Hint: We know that a circular current-carrying coil will have a magnetic field along its axis. For any circular current carrying current, the magnetic field along its axis will be given by \[B = \dfrac{{{\mu _0}I{R^2}}}{{2{{\left( {{R^2} + {a^2}} \right)}^{\dfrac{3}{2}}}}}\] . In the given question, first, we need to find the magnetic field at the center which can be given by simply putting the value of $a = 0$ . Now, we can calculate the distance $a$ by equating the condition given in the question mathematically.

Complete step by solution:
According to the question, we are given:
$R$ = radius of circular current-carrying coil
Now, the magnetic field along the axis of the circular coil is given by:
\[B = \dfrac{{{\mu _0}I{R^2}}}{{2{{\left( {{R^2} + {a^2}} \right)}^{\dfrac{3}{2}}}}}\]
Where,
${\mu _0}$ is the permeability of free space
$I$ is the current flowing in the coil
$R$ is the radius of circular current-carrying coil
$a$ is the distance from the center of the coil on the axis where B is calculated
Let this be equation 1.
We need to find the magnetic field at the center of the circular coil i.e ${B_0}$ .
Substituting the $a = 0$ in equation 1, we get
\[{B_0} = \dfrac{{{\mu _0}I}}{{2R}}\]
According to the given data in the question, $B = \dfrac{1}{8}{B_0}$
Now, putting values of $B$ and ${B_0}$, we get
\[
\Rightarrow \dfrac{{{\mu _0}I{R^2}}}{{2{{\left( {{R^2} + {a^2}} \right)}^{\dfrac{3}{2}}}}} = \dfrac{1}{8}\dfrac{{{\mu _0}I}}{{2R}} \\
 \Rightarrow 8{R^3} = {\left( {{R^2} + {a^2}} \right)^{\dfrac{3}{2}}} \\
\Rightarrow 64{R^6} = {\left( {{R^2} + {a^2}} \right)^3} \\
\Rightarrow {\left( {4{R^2}} \right)^3} = {\left( {{R^2} + {a^2}} \right)^3} \\
\Rightarrow 4{R^2} = {R^2} + {a^2} \\
\Rightarrow a = \sqrt 3 R \\
 \]
Therefore, the distance from the center of the coil, on the axis, where B will be $\dfrac{1}{8}$ of its value at the center of the coil is $\sqrt 3 R$ .

Hence, option (B) is correct.

Note: These types of questions can be solved by using the direct formula for the magnetic field on a circular current-carrying coil. We should always be cautious doing such calculations because a small calculation mistake can lead us to an incorrect answer. Also, we need to be precise with the details given in the question.