Question

When the magnetic moment of two magnets is compared using the equal distance method, the deflections produced are \[{45^0}\] and \[{30^0}\]. If the lengths of magnets are in the ratio 1:2, find the ratio of their pole strength.
A. \[3:1\]
B. \[3:2\]
C. \[\sqrt 3 :1\]
D. \[2\sqrt 3 :1\]

Answer 1

Hint: Before we start addressing the problem, we need to know the data provided. Here the deflections produced by the two magnets and also the lengths of the magnets are given. Using this we are going to find the ratio of pole strength of the two magnets. The pole strength of a magnet is referred to as the strength with which the materials get attracted to the magnet.

 Formula Used:
The formula to find the magnetic moment is given by,
\[M = mL\]
Where, m is pole strength of the magnet and L is the length of the magnet.

Complete step by step solution:
In order to find the magnetic moment, we have,
\[M = mL\]
Now the magnetic moment of first magnet is given by,
\[{M_1} = {m_1}{L_1}\]………….. (1)
Similarly, for the second magnet is given by,
\[{M_2} = {m_2}{L_2}\]……………. (2)
We know that ,
\[\dfrac{{{M_1}}}{{{M_2}}} = \dfrac{{\tan {\theta _1}}}{{\tan {\theta _2}}}\]……..(3)

Substitute the value of equations (1) and (2) in equation (3) we get,
\[\dfrac{{{m_1}{L_1}}}{{{m_2}{L_2}}} = \dfrac{{\tan \left( {{{45}^0}} \right)}}{{\tan \left( {{{30}^0}} \right)}} \\ \]
\[\Rightarrow \dfrac{{{m_1}{L_1}}}{{{m_2}{L_2}}} = \dfrac{1}{{\dfrac{1}{{\sqrt 3 }}}} \\ \]
\[\Rightarrow \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{{L_2}}}{{{L_1}}} \times \dfrac{1}{{\dfrac{1}{{\sqrt 3 }}}} \\ \]
\[\Rightarrow \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{2}{1} \times \dfrac{{\sqrt 3 }}{1} \\ \]
\[\Rightarrow \dfrac{{{m_1}}}{{{m_2}}} = \dfrac{{2\sqrt 3 }}{1} \\ \]
\[\therefore {m_1}:{m_2} = 2\sqrt 3 :1\]
Therefore, the ratio of their pole strengths is \[2\sqrt 3 :1\].

Hence, option D is the correct answer.

Note:The strongest magnetic field of a bar magnet is found at its poles. The field starts strongest at the poles and keeps reducing in strength and reaches the minimum at the centre and again keeps increasing as we move towards the other end. Hence, it is strongest at the poles and is considered equally strong at both poles.