JEE Mains Physics · Moving Charges and Magnetism

Forces and Torques on Current-Carrying Wires

A wire in a field feels F = I L × B; two parallel currents attract or repel with μ₀I₁I₂/2πd per metre; and a coil of moment NIA feels a torque NIAB sin θ that turns its axis towards the field.

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Why this matters

Twenty-six PYQs, fourteen of them multiple choice, and four from 2026; nearly half ask for a number. Twelve find the force on a wire in a field: four on a straight piece, four on a wire held up or balanced against its weight, two on a loop in a field that grows along x, one on a bent wire and one on a flexible loop. Six are forces between parallel wires. Eight are a coil's magnetic moment or the torque on it.

Concept 1 of 3: Force on a current-carrying wire in a magnetic field

A current is moving charge, so a wire carrying it in a field feels the sum of q v × B on all its charges: I L × B. The force is across both the wire and the field. For a bent wire in a uniform field, only the straight line from one end to the other matters, so a closed loop in a uniform field feels no net force at all.

Definition

$\vec F = I\,\vec L \times \vec B$ , size $ILB\sin\theta$ , with $\theta$ the angle between the wire and B. Only the length inside the field counts.
Bent wire in a uniform field: use the straight vector from start to end. A semicircle of radius R acts like a straight wire of length 2R.
Closed loop in a uniform field: net force zero (it may still feel a torque).
Field that varies, $\vec B = B_0(1 + kx)\hat k$ , on a square loop of side L with sides along the axes: the two sides along y feel opposite forces of different size; the net force is $I L \times B_0 k L = I L^{2} B_0 k$ , along x. The sides along x cancel.
Wire held up by the field: $BIL = mg$ . Rod on a smooth incline of angle $\theta$ with a vertical field: the horizontal force balances when $BIL = mg\tan\theta$ .
A flexible loop carrying current in a field pulls itself into a circle, the shape of largest area, with its plane normal to the field.

Force on a wire

\vec F = I\,\vec L \times \vec B, \qquad F = ILB\sin\theta

Worked example

A wire bent into a semicircle of radius 10 cm carries 4 A. It lies in a plane perpendicular to a uniform field of 0.5 T. Find the force on it.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 5 Apr 2024 · Q22Moderate

Example 1 · Moving Charges and Magnetism · Forces and Torques on Current-Carrying Wires

A 2A current carrying straight metal wire of resistance

1\Omega

, resistivity

2 \times10^{- 6}\Omega m

, area of cross-section

10{\text{ }mm}^{2}

and mass

500\text{ }g

is suspended horizontally in mid air by applying a uniform magnetic field

\overrightarrow{B}

. The magnitude of

B

is _____

\times10^{- 1}\text{ }T

(given,

g = 10\text{ }m/s^{2}

)

Only the part in the field counts

When a wire runs partly through a field region, L in ILB is the length inside the region. Using the whole wire overstates the force.

A bent wire's force uses the chord

In a uniform field, a semicircle of radius R feels the same force as a straight wire of length 2R joining its ends, not πR.

θ is the angle between the wire and the field

F = ILB sin θ is largest when the wire is perpendicular to B and zero when the wire lies along B. A wire at 30° to the field feels half the largest force.

Drill 11 more on force on a current-carrying wire in a magnetic field

Concept 2 of 3: Force between two parallel currents

Each wire sits in the field of the other, so each feels I L × B. The result is a force per metre of μ₀I₁I₂/2πd. Unlike electric charges, like currents attract and opposite currents repel. By Newton's third law the two wires feel equal and opposite forces, even when the currents are different.

Definition

Force per unit length: $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ . For a length L, multiply by L.
Currents in the same direction attract; in opposite directions they repel.
The forces on the two wires are equal and opposite, over the length they share.
$F \propto \dfrac{I_1 I_2}{d}$ : with equal currents, $F \propto \dfrac{I^{2}}{d}$ .
Several wires: find the force from each neighbour on the wire in question, with its direction, and add.
The ampere was defined from this: two wires 1 m apart, each with 1 A, feel $2 \times 10^{-7}$ N per metre.

Force per metre between parallel wires

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

Worked example

Two long parallel wires 4 cm apart carry 6 A and 8 A in opposite directions. Find the force per metre and the force on a 50 cm length of either wire. Do they attract or repel? (

\mu_0/2\pi = 2 \times 10^{-7}

T m/A)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 24 Jan 2023 · Q3Moderate

Example 2 · Moving Charges and Magnetism · Forces and Torques on Current-Carrying Wires

Two long straight wires

P

and

Q

carrying equal current

10\text{ }A

each were kept parallel to each other at

5\text{ }cm

distance. Magnitude of magnetic force experienced by

10\text{ }cm

length of wire

P

F_{1}

- If distance between wires is halved and currents on them are doubled, force

F_{2}

10\text{ }cm

length of wire

P

will be:

Like currents attract

This is the opposite of charges: two wires with current in the same direction pull together. Opposite currents push apart.

The two forces are equal even if the currents differ

A 2 A wire beside a 10 A wire feels the same size of force as the 10 A wire does, over the same length. The force depends on the product I₁I₂, which is the same for both.

The force goes as the product, not the sum

F/L = μ₀I₁I₂/2πd. Doubling both currents multiplies the force by four, and halving the distance doubles it again.

Drill 5 more on force between two parallel currents

Concept 3 of 3: Magnetic moment of a coil and the torque on it

A current loop behaves like a small magnet. Its strength is the magnetic moment m = NIA, pointing along the loop's axis by the right-hand rule. In a uniform field the forces on its sides cancel, but they form a couple that turns the axis towards the field. The turning effect is largest when the coil's plane lies along the field and zero when its axis lies along the field.

Definition

Magnetic moment $\vec m = N I \vec A$ , size NIA, along the coil's axis (curl the right-hand fingers along the current; the thumb gives $\vec m$ ).
Torque $\vec \tau = \vec m \times \vec B$ , size $NIAB\sin\theta$ , where $\theta$ is the angle between the AXIS (the normal) and B, not the plane.
Plane of the coil parallel to B: $\theta = 90^{\circ}$ , largest torque NIAB. Plane perpendicular to B: zero torque.
Concentric loops with currents in opposite senses: their moments point opposite ways and subtract.
A given length of wire made into a coil: more turns means smaller area per turn. For a circle of N turns from a length L, $A = \pi\left(\dfrac{L}{2\pi N}\right)^{2}$ .
Vector form: with $\vec m$ and $\vec B$ in components, find $\vec \tau$ by the cross product.

Moment and torque

\vec m = NI\vec A, \qquad \vec \tau = \vec m \times \vec B, \qquad \tau = NIAB\sin\theta

Worked example

A rectangular coil of 50 turns, 4 cm by 5 cm, carries 2 A in a uniform field of 0.3 T. The plane of the coil makes

30^{\circ}

with the field. Find the magnetic moment and the torque.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 4 Apr 2026 Shift 2 · Q24Moderate

Example 3 · Moving Charges and Magnetism · Forces and Torques on Current-Carrying Wires

A circular coil of radius 2 cm and 125 turns carries a current of 1 A . The coil is placed in a uniform magnetic field of magnitude 0.4 T . The axis of the coil makes an angle of

30^{\circ}

with the direction of the magnetic field. The torque acting on the coil is

\alpha \times 10^{- 4}\text{ }N.m

. The value of

\alpha

\_\_\_\_

. (

\pi = 3.14

)

θ is measured from the axis, not the plane

In τ = NIAB sin θ, θ is between the coil's normal and B. If a question gives the angle between the plane and the field, use 90° minus it.

Opposite currents give opposite moments

Two concentric loops with currents in opposite senses have moments pointing opposite ways. The net moment is the difference, not the sum.

Do not forget N

A coil of N turns has N times the moment of one turn. The torque, NIAB sin θ, carries the same factor N.

Drill 7 more on magnetic moment of a coil and the torque on it

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Force on a current-carrying wire in a magnetic field
Force on a wire
$\vec F = I\,\vec L \times \vec B, \qquad F = ILB\sin\theta$
Force between two parallel currents
Force per metre between parallel wires
$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$
Magnetic moment of a coil and the torque on it
Moment and torque
$\vec m = NI\vec A, \qquad \vec \tau = \vec m \times \vec B, \qquad \tau = NIAB\sin\theta$

Watch out for (9)

Only the part in the field counts
→ Force on a current-carrying wire in a magnetic field
A bent wire's force uses the chord
→ Force on a current-carrying wire in a magnetic field
θ is the angle between the wire and the field
→ Force on a current-carrying wire in a magnetic field
Like currents attract
→ Force between two parallel currents
The two forces are equal even if the currents differ
→ Force between two parallel currents
The force goes as the product, not the sum
→ Force between two parallel currents
θ is measured from the axis, not the plane
→ Magnetic moment of a coil and the torque on it
Opposite currents give opposite moments
→ Magnetic moment of a coil and the torque on it
Do not forget N
→ Magnetic moment of a coil and the torque on it

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