4 bar magnet, held horizontally, is set into angular oscillations in the Earth's magnetic field. It has time periods `T_(1)` and `T_(2)` at two places, where the angles of dip are `theta_(1)` and `theta_(2)` respectively. Deduce an expression for the ratio of the resultant magnetic fields at the two places.

Explanation

(1) Suppose, the resultant magnetic fields is to be compared at two places A and B.
(2) A barmagnet, field horizontally at A and which is set into angular oscillatins in the Earth's magnetic field.
(3) Let time period of a bar magnet at place 'A' is `T_(1)` and angular displacement or angle of dip is `theta_(1)`.
(4) As the bar magnet is free to rotate horizontally, it does not remain vertical component `(B_(1) sin theta_(1))`. It can have only horizontal component `(B_(1) cos theta_(1))`
(5) The time period of a bar magnet in uniform magnetic field is given by `T = 2pi sqrt((I)/(mB_(H)))`
(6) Now, in this case `T = T_(1)` and `B_(H) = B_(1) cos theta_(1)`
(7) Therefore time period of a bar magnet at place 'A' is given by
`T_(1) = 2pi sqrt((I)/(mB_(1) cos theta_(1)))` - (1) where I is moment of Inerfia of a barmagnet and m is magnitude of magnetic moment.
(8) Similarly, the same magnet is placed at B and which is set into angular oscillations in the earth's magnetic field.
(9) Let time period of a bar magnet at place B is `T_(2)` and angle of dip is `theta_(2)`.
(10) Since horizontal component of earth's field at B is `B_(H) = B_(2) cos theta_(2)`, time period, `T_(2) = 2pi sqrt((I)/(mB_(2) cos theta_(2)))` - (2)
(11) Dividing equation (1) by equation (2), we get `(T_(1))/(T_(2)) = sqrt((mB_(2) cos theta_(2))/(mB_(1) cos theta_(1)))`
Squaring on both sides, we have `(T_(1)^(2))/(T_(2)^(2)) = (B_(2) cos theta_(2))/(B_(1) cos theta_(1))`
(12) But `B_(1) = mu_(0)H_(1)` and `B_(2) = mu_(0) H_(2)`
`(T_(2)^(1))/(T_(2)^(2)) = (mu_(0) H_(2) cos theta_(2))/(mu_(0) H_(1) cos theta_(1))`
(13) Therefore, `(H_(1))/(H_(2)) = (T_(2)^(2) cos theta_(2))/(T_(1)^(2) cos theta_(1))`
(14) By knowing `T_(1), T_(2)` and `theta_(1), theta_(2)` at different places A and B, we can find the ratio of resultant magnetic fields.