Animated Solution for Physics - Magnetic Effects of Current: A magnetic needle lying parallel to a magnetic field requires $W$ unit of work to turn it through $60^{\circ }$. The torque needed to maintain the needle in this position will be

Aligning with the Field

Imagine a magnetic needle resting peacefully, perfectly aligned with a uniform magnetic field. In this state, it is in stable equilibrium. The magnetic moment $\mathbf{M}$ and the magnetic field $\mathbf{B}$ are parallel, meaning the angle between them is ${\theta }_1=0^{\circ }$.

Nature loves equilibrium. To disturb this peace and rotate the needle, we must act against the magnetic forces. This requires us to do external work.

Calculating Work Done

The work done $W$ in rotating a magnetic dipole in a uniform magnetic field from an initial angle ${\theta }_1$ to a final angle ${\theta }_2$ is given by the master equation:

In our scenario, we are rotating the needle from ${\theta }_1=0^{\circ }$ to ${\theta }_2=60^{\circ }$. Let's substitute these angles into our equation:

We know the standard trigonometric values: $\cos 0^{\circ }=1$ and $\cos 60^{\circ }=\frac{1}{2}$. Plugging these in, we get:

This gives us a beautiful relationship between the maximum magnetic energy term $MB$ and the work done $W$. By rearranging, we find:

We will hold onto this crucial piece of information for the next phase of our problem.

Restoring Torque

Now that the needle is held at $60^{\circ }$, the magnetic field is relentlessly trying to pull it back to its stable position. It exerts a restoring torque on the needle. The magnitude of this torque $\tau$ is given by the cross product of the magnetic moment and the magnetic field, which simplifies to:

We need to find the torque required to maintain the needle at this exact position, which must be equal and opposite to the restoring torque at $\theta =60^{\circ }$.

We know that $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$. Let's substitute this, along with our previously derived relation $MB=2W$, into the torque equation:

The $2$ in the numerator and denominator cancel out perfectly, leaving us with our final, elegant result:

This is the exact torque required to hold the needle steady at a $60^{\circ }$ angle. The physics beautifully connects the energy spent (work) to the force applied (torque) through the geometry of the rotation!