A magnetic needle suspended, at a point where there are two crossed magnetic fields acting at right angles to each other, will come to rest in the direction of the resultant of the two fields.

**Tangent law**

*A magnetic needle suspended, at a point where there are two
crossed magnetic fields acting at right angles to each other, will come to rest
in the direction of the resultant of the two fields.*

B_{1}
and B_{2} are two uniform magnetic fields acting at right angles to
each other. A magnetic needle

placed
in these two fields will be subjected to two torques tending to rotate the
magnet in opposite

directions.
The torque τ_{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 τ_{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 ? with B_{2} as shown in
the Fig.

The deflecting torque due to the
forces *mB*_{1} and *mB*_{1}

τ_{1} = mB_{1} ?
NA

= mB_{1} ? NS cos θ

= mB_{1} ? 2*l* cos θ

= 2*l* mB_{1} cos θ

τ_{ 1} = MB_{1}cos θ

Similarly the restoring torque
due to the forces mB_{2} and mB_{2}

τ_{ 2} = mB_{2} ?
SA

= mB_{2} ? 2*l* sin θ

= 2*l*m ? B_{2} sin θ

= τ_{ 2} = MB_{2}
sin θ

At equillibrium,

τ_{ 1} = τ_{ 2}

MB_{1} cos θ = MB_{2} sin θ

B_{1} = B_{2} tan θ

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}.

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