Tangent law :
(i) A magnetic needle suspended, at a point where there are two crossed magnetic frelds acting at right angles to each other, will come to rest in the direction of the resultant of the two fields.
(ii) `B_(1) " and " B_(2)` are two uniform magnetic fields acting at right angles to each other.
(iii) A magnetic needle placed in these two fields will be subjected to two torques tending to ritate the magnet in opposite directions.
(iv) The torque `tau_(1)` due to the two equal and opposite parallel forces `mB_(1)" and " mB_(1)` tend to set the magnet parallel to B 1. Similarly the torque `tau_(2)` due to the two equal and opposite parallel forces `mB_(2)" and " mB_(2)` tends to set the magnet parallel to `B_(2)`. In a position where the torques balance each other, the magnet comes to rest. Now the magnet makes an angle `theta ` with `B_(2)` as shown in the Fig.
The deflecting torque due to the forces, `mB_(1)" and " mB_(1)`
`tau_(1) = mB_(1) xx NA`
` = mB_(1) xx NS cos theta `
` = mB_(1) xx 2l cos theta `
` = 2l mB_(1) cos theta `
` :. " " tau_(1) = MB_(1) cos theta`
Similarly the restoring torque due to the forces `mB_(2)" and "mB_(2)`
`tau_(2) = mB_(2) xx SA`
` = mB_(2) xx 2l sin theta `
` = 2lm xx B_(2) sin theta`
` tau_(2) = MB_(2) sin theta `
At equillibrium ,
`tau_(1) = tau_(2)`
` :. " " MB_(1) cos theta = MB_(2) sin theta `
` :. " " B_(1) = B_(2) tan theta `
This is called Tangent law
Invariably , in the applications of tangent law, the restoring magnetic field `B_(2)` is the horizontal component of Earth's magnetic field `B_(h) `.
