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Question
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.
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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`
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