Animated Solution for Physics - Magnetism and Matter: A magnetic needle is kept in a non-uniform magnetic field. It experiences

A Needle in a Non-Uniform Field

Imagine a region of space where the magnetic field is not perfectly uniform. In a uniform field, the magnetic field lines are parallel and equally spaced, meaning the field strength is identical everywhere. However, in a non-uniform magnetic field, the field lines are curved and the spacing between them varies. This means the magnitude and direction of the magnetic field vector $\vec{B}$ change from point to point.

Now, let's place a small magnetic needle into this environment. A magnetic needle is essentially a tiny bar magnet. In physics, we model this as a magnetic dipole consisting of a North pole with pole strength $+m$ and a South pole with pole strength $-m$, separated by a small distance.

The Tug of War

When placed in a magnetic field, each pole of the needle experiences a magnetic force. The fundamental equation for the force on a magnetic pole is $\vec{F}=m\vec{B}$.

For the North pole, the force is ${\vec{F}}_N=m{\vec{B}}_N$, acting in the direction of the local magnetic field at that specific point. For the South pole, the force is ${\vec{F}}_S=-m{\vec{B}}_S$, acting in the opposite direction of the local magnetic field at its respective location.

Why the Net Force is Non-Zero

Here is where the non-uniformity of the field becomes crucial. Because the field is non-uniform, the magnetic field strength at the location of the North pole (${\vec{B}}_N$) is strictly different from the magnetic field strength at the location of the South pole (${\vec{B}}_S$).

Mathematically, since $\vec{B}_N eq \vec{B}_S$, it directly implies that the magnitudes of the forces are unequal: $|\vec{F}_N| eq |\vec{F}_S|$.

When we calculate the net translational force on the needle, we sum these two forces:

Because the forces are unequal in magnitude (and potentially not perfectly anti-parallel), they do not cancel each other out. Therefore, $\vec{F}_{net} eq 0$. The needle will experience a net force, causing it to accelerate translationally through space.

The Turning Effect

What about rotation? The two forces ${\vec{F}}_N$ and ${\vec{F}}_S$ are acting at two different points on the rigid body of the needle. Unless the needle is perfectly aligned along a straight field line (which is a highly specific and unstable equilibrium in a non-uniform field), the lines of action of these two forces will not coincide.

Whenever you have forces acting at different points with non-coinciding lines of action, they generate a turning effect, or torque. The net torque is given by:

Since the forces are unequal and misaligned, $\vec{\tau}_{net} eq 0$. The needle will experience a net torque, causing it to rotate.

In conclusion, a magnetic needle placed in a non-uniform magnetic field will experience both a net force and a net torque. It will be pulled towards the region of stronger magnetic field while simultaneously rotating to align itself with the local field lines.

A quick tip for JEE: Always contrast this with a uniform magnetic field. In a perfectly uniform field, ${\vec{B}}_N={\vec{B}}_S$, so the forces are exactly equal and opposite, resulting in zero net force. However, it can still experience a torque if it is placed at an angle to the field!