The combination of two bar magnets makes 10 oscillations per second in an oscillation magnetometer when like poles are tied together and 2 oscillations per second when unlike poles are tied together. Find the ratio of the magnetic moments of the magnets. Neglect any induced magnetism.

#### Solution

Given :

Number of oscillations per second made by the combination of bar magnets with like poles, `V_1 = 10 s^-1`

Number of oscillations per second made by the combination of bar magnets with unlike poles, `V_2 = 2 s^-1`

The frequency of oscillations in the magnetometer `(V)` is given by

`V = 1/(2pi) sqrt((MB_H)/I)`

When like poles are tied together, the effective magnetic moment is `M = M_1 - M_2`

When unlike poles are tied together, the effective magnetic moment is `M = M_1 + M_2`

As the frequency of oscillations is directly proportional to the magnetic moment ,

`V_1/V_2 = sqrt((M_1 - M_2)/(M_1 + M_2))`

⇒ `(10/2)^2 = (M_1 - M_2)/(M_1 + M_2)`

⇒ `25/1 = (M_1 - M_2)/(M_1 + M_2)`

⇒ `(25 + 1)/(25 - 1) = (M_1 - M_2+M_1 + M_2)/(M_1 - M_2-M_1-M_2)`

⇒ `26/24 = (2M_1)/(-2M_2)`

⇒ `M_1/M_2 = -26/24 = -13/12`

Hence, the ratio of the effective magnetic moment is `-13/12`