Animated Solution for Physics - Magnetic Effects of Current: A galvanometer coil has 500 turns and each turn has an average area of $3\times 10^{-4}{\text{ m}}^2$. If a torque of $1.5\text{ N-m}$ is required to keep this coil parallel to a magnetic field when a current of $0.5\text{ A}$ is flowing through it, the strength of the field (in T) is ........... .

When a current-carrying coil is placed in a magnetic field, it experiences a magnetic torque. This is the fundamental principle behind electric motors and moving coil galvanometers.

The problem states that the coil is held parallel to the magnetic field.

This phrasing is a classic trap! The angle $\theta$ in our torque formula is not the angle between the coil's plane and the field, but rather the angle between the coil's area vector $\mathbf{A}$ and the magnetic field $\mathbf{B}$. Since the area vector is always perpendicular to the plane of the coil, a coil parallel to the field means its area vector is perpendicular to the field. Therefore, $\theta =90^{\circ }$.

The magnetic torque $\tau$ acting on the coil is given by the cross product of its magnetic dipole moment $\mathbf{m}$ and the magnetic field $\mathbf{B}$:

Since the magnitude of the magnetic moment is $m=NIA$, the magnitude of the torque becomes:

Because $\theta =90^{\circ }$, $\sin 90^{\circ }=1$, and the torque is at its maximum value:

We are given the following values: - Torque, $\tau =1.5\text{ N-m}$ - Number of turns, $N=500$ - Current, $I=0.5\text{ A}$ - Area, $A=3\times 10^{-4}{\text{ m}}^2$

We need to find the magnetic field strength $B$. Rearranging our master equation to solve for $B$:

Substituting the known values into the equation:

Let's simplify the denominator. $500\times 0.5=250$, and $250\times 3=750$. So we have:

Bringing the $10^{-4}$ to the numerator makes it $10^4$ or $10000$:

The required magnetic field strength is an astonishing 20 Tesla! In reality, such a strong magnetic field is typically only found in massive superconducting electromagnets, like those used in MRI machines or particle accelerators. A typical galvanometer operates on a fraction of a Tesla.