The needle of a deflection magnetometer deflects through `45^@` from north to south when the instrument is used in Tan-A position with a magnet of length 10cm placed at a distance of 25cm. (a) Find the magnetic moment of the magnet if the earth's horizontal magnetic field is `20 muT`. (b) If the magnetometer is used in `Tan-B` position with the same magnet at the same separation from the needle, what will be the deflection?

Explanation

(a) In the Tan-A position, the needle is in end-on position of the magnet so that the field at the needle due to the magnet is :
`B = (mu_0)/(4pi) (2Md)/((d^2 - l)^2)^2)` .
Thus, `mu_0/(4pi) (2Md)/(d^2 - l^2)^2) = B_H tantheta`
or,
M = (4pi/(mu_0) (B_H tantheta(d^2 - l^2)^2)/(2d)`
`= 10^7 A m^(-1) T^(-1) xx (20 xx 10^(-16) T xx 1 xx (625 - 25)^2 xx 10^(-18) m^4)/(2 xx 25 xx 10^(-2) m`
`= 1.44 A m^2`.
(b) In `Tan-B` position, the needle is in broadside on position of the manget, so that
`B = (mu_0)/4pi) (M)/((d^2 + l^2)^(3-2) = B_H tantheta`
or, `tantheta = (10^(-7) T m A^(-1) xx 1.44 A m^2)/((625 + 25)^(3/2) xx 10^(-6) m^3 xx 20 xx 10^(-6) T)`
`=0.43`
or, `theta = 23.5^5@`.