Tangent law

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.

B₁ and B₂ 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 τ₁ due to the two equal and opposite parallel forces mB₁ and mB₁ tend to set the magnet parallel to B₁. Similarly the torque τ₂ due to the two equal and opposite parallel forces mB₂ and mB₂ tends to set the magnet parallel to B₂. In a position where the torques balance each other, the magnet comes to rest. Now the magnet makes an angle ? with B₂ as shown in the Fig.

The deflecting torque due to the forces mB₁ and mB₁

τ₁ = mB₁ ? NA

=  mB₁ ? NS cos θ

=  mB₁ ? 2l cos θ

=  2l mB₁ cos θ

τ ₁ = MB₁cos θ

Similarly the restoring torque due to the forces mB₂ and mB₂

             τ ₂ = mB₂ ? SA

=  mB₂ ? 2l sin θ

=  2lm ? B₂ sin θ

=  τ ₂ = MB₂ sin θ

At equillibrium,

τ ₁ = τ ₂

MB₁ cos θ = MB₂ sin θ

B₁ = B₂ tan θ

This is called Tangent law

Invariably, in the applications of tangent law, the restoring magnetic field B₂ is the horizontal component of Earth?s magnetic field B_h.