A magnetic compass needle oscillates times per minute at a place where the dip is , and times per minute where the dip is . If B1 and B2 are respectively the total magnetic field due to the earth at the two places, then the ratio B1/ B2 is best given by:
Step 1: Given data
For the first magnetic needle frequency = and
And, for the second magnetic needle frequency and
Step 2: Formula used
The following expression describes the relationship between the frequency of motion of a magnetic needle and the magnetic field at a given location:
Here is the magnetic moment of the magnetic needle, denotes the horizontal component earth’s magnetic field while represent the moment of inertia of the magnetic needle about the axis about which it oscillates.
Step 3: Relation between frequency and magnetic field
The frequency of oscillations of a magnetic needle in the earth’s magnetic field is given as:
, and , where is known as the angle of dip which the magnetic field (B) makes with the horizontal axis. Thus,
We can observe from the above equation that
Step 4: Expression for both the needles
For the first needle,
For the second needle,
Diving equation (i) with equation (ii),
Inserting the known values,
Squaring both sides,
Hence, the correct answer is option (C).