Q.A rectangular coil ABCD of $100$ turns hangs vertically from one arm of a beam balance, with its lower horizontal side CD (length $1\ \text{cm}$) at the bottom. A $500\ \text{g}$ mass placed on the other pan exactly balances the weight of the coil. A current of $4.9\ \text{A}$ is then sent through the coil while a uniform magnetic field of $0.2\ \text{T}$, directed horizontally into the region and perpendicular to CD, is switched on so that only the arm CD lies in the field. Find the additional mass $m$ that must be added to the other pan to restore the balance.

Magnetic Force Balance

When a current-carrying wire or coil sits in a magnetic field, it feels a force $F = BIL\sin\theta$ (or, for a point charge, $F=qvB\sin\theta$). On its own that force just pushes the conductor - but in many real situations the push is deliberately set up to CANCEL another force, so the whole system sits in equilibrium. That equilibrium condition - magnetic force balanced against weight, against another wire's magnetic force, or against a mechanical counterweight - is what "magnetic force balance" means, and it is also historically how the ampere itself was defined.

The balance condition

Whenever a conductor is in equilibrium under a magnetic force and one other force, the two must be equal and opposite:

Solving this equation for whichever quantity is unknown ($B$, $I$, $L$, or the other force) is the entire skill in this class of problem - the only new step, beyond the force law itself, is correctly identifying what the magnetic force is opposing.

Case 1: a wire suspended against gravity

A straight horizontal wire of mass $m$ and length $l$, carrying current $I$, can be held up ("floated") in mid-air by a horizontal magnetic field perpendicular to it. The upward magnetic force must equal the downward weight:

For example, a $200\,\text{g}$, $1.5\,\text{m}$ wire carrying $2\,\text{A}$ needs $B=\dfrac{(0.2)(9.8)}{(2)(1.5)}\approx0.65\,\text{T}$ to stay suspended.

Case 2: two wires balancing each other

Two long parallel wires carrying currents $I_1,I_2$ exert a force per unit length on each other of $\dfrac{\mu_0 I_1I_2}{2\pi d}$ (attractive if the currents run the same way, repulsive if opposite). If one wire is free to move, this magnetic force can itself balance that wire's weight:

This is exactly how a "current balance" apparatus works, and historically it is how the ampere was defined: the current that, flowing in two infinite parallel wires one metre apart, produces a force of exactly $2\times10^{-7}\,\text{N}$ per metre of length.

Case 3: balancing on a beam

A current-carrying coil arm hanging from one pan of a beam balance feels an extra force $F=NBIl$ when only that arm sits in an external field. Re-balancing the beam means adding a mass $m$ so that $mg = NBIl$.