The pattern of iron filings, i.e., the magnetic field lines gives us an approximate idea of the magnetic field B. We may at times be required to determine the magnitude of B accurately. This is done by placing a small compass needle of known magnetic moment m and moment of inertia I and allowing it to oscillate in the magnetic field. This arrangement is shown in Fig.
The torque on the needle is τ = m × B
In magnitude τ = mB sin θ
Here τ is restoring torque and θ is the angle between m and B. Therefore, in equilibrium
Negative sign with mB sinθ implies that restoring torque is in opposition to deflecting torque.
For small values of θ in radians, we approximate sin θ ≈ θ and get
This represents a simple harmonic motion. The square of the angular frequency is ω2 = mB/I and the time period is,
An expression for magnetic potential energy can also be obtained on lines similar to electrostatic potential energy.
The magnetic potential energy Um is given by
