Explain 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. The tangent law of magnetism is a way to contrast the strengths of two magnetic fields that are perpendicular to each other. The law of tangents describes the relationship between the tangent of two angles of a triangle and the lengths of the opposite sides.

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 mB₂tend 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 θ with B₂ as shown in the Figure.

The deflecting torque due to the forces mB₁ and mB₁

τ₁ = mB₁ x NA

= mB₁ x NS cos θ

= mB₁ x 2Ɩ cos θ

= 2Ɩ mB₁ cos θ

So, τ₁ = MB₁ cos θ

Similarly the restoring torque due to the forces mB₂ and mB₂

τ₂ = mB₂ x SA

= mB₂ x 2Ɩ sin θ

= 2Ɩ m x B₂ sin θ

so, τ₂ = MB₂ sin θ

At equilibrium, τ₁ = τ₂

MB₁ cos θ = MB₂ sin θ

so, 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. If a compass or other magnet is subjected to a magnetic field B that is perpendicular to the horizontal magnetic field of the Earth (Bh), it will rest at an angle theta to the Earth’s magnetic field. The relationship between the two magnetic fields is:

B = Bh(tan theta).

Because the tangent of the angle is the relevant factor here, the law is called the tangent law of magnetism. It represents the relationship between the tangent of two angles of a triangle and the length of the opposite sides.

To understand the law of tangents in an enhanced way, we will need three pieces of information for a common triangle even it may or may not be a triangle. The four cases involved are: