Question

The ratio of magnetic moments of two short magnets that give null deflection in tan B position at 12cm and 18cm from the center of a deflection magnetometer is?
A) 2:3
B) 8:27
C) 27:8
D) 4:9

Answer 1

Hint: Magnetometer is an instrument to measure the value of magnetic moment for bar magnets. At the tan B position, the strength of the magnetic moment at the magnetometer is proportional to the cube of the distance between the magnetometer and center of deflection of the magnetometer.

Formula used:
Relation between magnetic moment strength and distance is given by:
$\dfrac{M}{{{d^3}}} = Constant$ …………...(1)
Where,
$M$ is the magnetic moment of the magnet,
$d$ is the distance of a magnet from the magnetometer.

Complete step by step solution:
Given:
Distance of the first magnet is \[{d_1} = 12cm\].
Distance of the second magnet is ${d_2} = 18cm$.
To find the ratio of magnetic strength for both magnets.
Step 1:
Let, the magnetic strength of the two magnets are \[{M_1}\]and ${M_2}$. Since, the magnetometer is showing no deflection hence, using eq.(1) we get the ratio of \[\dfrac{{{M_1}}}{{{M_2}}}\]as:
\[
  \dfrac{{{M_1}}}{{d_1^3}} = \dfrac{{{M_2}}}{{d_2^3}} = Constant \\
  \therefore \dfrac{{{M_1}}}{{{M_2}}} = \dfrac{{d_1^3}}{{d_2^3}} = {\left( {\dfrac{{{d_1}}}{{{d_2}}}} \right)^3} \\
 \] (2)
Step 2:
Substitute the values of \[{d_1}\]and \[{d_2}\]into eq.(2) to get the ratio of \[\dfrac{{{M_1}}}{{{M_2}}}\]as:
\[
  \dfrac{{{M_1}}}{{{M_2}}} = {\left( {\dfrac{{12cm}}{{18cm}}} \right)^3} \\
  \therefore \dfrac{{{M_1}}}{{{M_2}}} = {\left( {\dfrac{2}{3}} \right)^3} = \dfrac{8}{{27}} \\
 \]

The ratio of magnetic moments of the two short magnets will be 8:27. So, option (B) is correct.

Note:
Notice, this problem is one of those problems that can be solved without even a single line of calculation. You just have to keep the fact in mind that the magnetic moment is proportional to the cube of the distance. Hence, the ratio of the magnetic moment will be the cube of the ratio of the two distances given. Just be careful that nothing is inversely proportional here, hence, the answer will be 8:27, not 27:8.