Question

Two short magnets (having magnetic moments in the ratio $27:{\text{ }}8$. When placed on the opposite sides of a deflection magnetometer causes no deflection. If the distance of the weaker magnetic is $0.12\,m$ from the centre of deflection magnetometer, the distance of the stronger magnet from the centre will be:
A. $0.17$
B. $0.19$
C. $0.18$
D. $0.16$

Answer 1

Hint:The method of electrical measurements known as deflection uses the deflection of measuring the instrument's index to calculate the current or other element under consideration. It is distinct from and the inverse of the zero or null method. The null deflection method must be used to calculate the distance of the stronger magnet from the centre.

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

Complete step by step solution:
In the question, we have given the ratio of two short magnets is $27:{\text{ }}8$ and the distance of the weaker magnet from the centre of the deflection magnetometer is $0.12\,m$. As we know that $\tan \,A$ is the location of the deflection magnetometer.

A bar magnet with magnetic moment ${M_2} = 8$ and the distance of the weak magnet is ${d_2} = 0.12\,m$ from the magnetic needle's centre. The second bar magnet, with magnetic moment ${M_1} = 27$. The second magnet is adjusted to cancel out the deflection caused by the first magnet.

Let assume the distance of the strong magnet from the magnetic needle’s centre be ${d_2}$. Because the magnetic fields produced by the two bar magnets in the magnetic needle's centre are equal in magnitude but opposite in direction ${B_1} = {B_2}$, the null deflection is given by:
$\dfrac{{{M_1}}}{{{M_2}}} = \dfrac{{d_1^3}}{{d_2^3}}$

Substitute the given information in the above formula, then we have:
$\dfrac{{27}}{8} = \dfrac{{d_1^3}}{{{{(0.12)}^3}}} \\$
$\Rightarrow {d_1} = \sqrt[3]{{\dfrac{{27 \times {{(0.12)}^3}}}{8}}} \\$
$\Rightarrow {d_1} = \dfrac{{3 \times 0.12}}{2} \\$
$\therefore {d_1} = 0.18\,m \\$
Therefore, the distance of the stronger magnet from the centre will be $0.18\,m$.

Thus, the correct option is C.

Note: It should be noted that we can be solved without even writing a single line of code. Simply remember that the magnetic moment is proportional to the distance cubed. As a result, the magnetic moment ratio will be the cube of the distance ratio given.