Question

The force between two magnetic poles reduces to $aN$, if the distance between them is increased to $n$ times and it increases to $bN$ if the distance between them is $\dfrac{1}{{{n^{th}}}}$ of the original value. Then $a:b$ is
A. $1:{n^2}$
B. ${n^2}:1$
C. ${n^4}:1$
D. $1:{n^4}$

Answer 1

Hint: We know that the force between two magnetic poles is classically given by the equation $F = \dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {r^2}}}$ . Use this equation for when the distance increases by $n$ times and when the distance reduces by $\dfrac{1}{{{n^{th}}}}$ times. Then compare these equations to reach the solution.

Complete step by step answer:
If both poles are small enough to be described as single points, then magnetic charges can be regarded as point charges. The force between two magnetic poles is classically given by the equation
$F = \dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {r^2}}}$
Here, $F = $ The magnetic force between the two magnetic poles
${\mu _0} = $ Magnetic susceptibility of free space
${m_1} = $ The magnetic moment of one magnetic pole
${m_2} = $ The magnetic moment of the second magnetic pole
$r = $ The distance between the two magnetic poles
This equation is an example of an inverse square law
So, let the initial distance between the two magnetic poles be $d$ and the initial force be $F$
So, for the initial position of the magnetic poles
$F = \dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {d^2}}}$
When the distance between the two magnetic poles is increased by $n$ times then the magnetic force between the two magnetic poles becomes $aN$
${F_1} = aN = \dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {{\left( {nd} \right)}^2}}}$ (Equation 1)
When the distance between the two magnetic poles is reduced by $\dfrac{1}{n}$ times then the magnetic force between the two magnetic poles becomes $bN$
${F_2} = bN = \dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {{\left( {\dfrac{d}{n}} \right)}^2}}}$ (Equation 2)
Dividing equation 2 by equation 1, we get
$\dfrac{{aN}}{{bN}} = \dfrac{{\dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {{\left( {nd} \right)}^2}}}}}{{\dfrac{{{\mu _0}{m_1}{m_2}}}{{4\pi {{\left( {\dfrac{d}{n}} \right)}^2}}}}}$
$\dfrac{{aN}}{{bN}} = \dfrac{1}{{{n^4}}}$
$\therefore a:b = 1:{n^4}$

So, the correct answer is “Option D”.

Note:
Due to their motion, magnetic force, attraction, or repulsion that occurs between electrically charged particles. It is the underlying force responsible for phenomena such as the operation of electric motors and the iron attraction of magnets. Among stationary electric charges, electric forces exist; among moving electric charges, both electric and magnetic forces exist.