6.4 Magnetic force on current-carrying wires

Magnetic fields and conductors

When electric current flows through a wire sitting in a magnetic field, the field pushes on that wire. This force is the operating principle behind electric motors, loudspeakers, and many other devices you encounter daily. To predict the force's strength and direction, you need to understand how magnetic fields and current-carrying conductors interact.

Properties of magnetic fields

Magnetic fields exert forces on moving charges and current-carrying conductors. They're represented by field lines that show both direction (the way the lines point) and relative strength (how closely the lines are packed together). The SI unit for magnetic field strength is the tesla (T).

Magnetic fields can be produced by:

• Permanent magnets
• Moving electric charges (including currents)
• Changing electric fields

Current-carrying conductors

A conductor carrying electric current creates its own magnetic field around it. The strength of this field is proportional to the current magnitude, and its direction wraps around the wire according to the right-hand rule: point your thumb in the direction of conventional current, and your fingers curl in the direction of the magnetic field.

When a current-carrying conductor sits inside an external magnetic field, the two fields interact, producing forces and torques on the conductor.

Lorentz force law

The Lorentz force law describes the total electromagnetic force on a charged particle. For magnetism specifically, it tells you how a magnetic field pushes on moving charges, which extends directly to current in a wire.

Force on moving charges

The magnetic force on a single moving charge is:

$\vec{F} = q\vec{v} \times \vec{B}$

where $q$ is the charge, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field. A few things to notice:

• The force magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$
• The force is always perpendicular to both the velocity and the field (that's what the cross product gives you)
• Because the force is perpendicular to velocity, it does no work on the charge. It changes direction, not speed
• A charge moving perpendicular to a uniform field traces a circular path

Force on current-carrying wires

Current in a wire is just a collection of moving charges, so the Lorentz force applies to the wire as a whole. The net force on a straight wire segment is:

$\vec{F} = I\vec{L} \times \vec{B}$

Here $I$ is the current, and $\vec{L}$ is a length vector pointing in the direction of conventional current flow. The direction of the resulting force follows the right-hand rule for cross products.

Magnetic force equation

Magnitude of magnetic force

Taking the magnitude of the cross product gives:

$F = ILB\sin\theta$

where $\theta$ is the angle between the wire and the magnetic field direction.

• When the wire is perpendicular to the field ($\theta = 90°$), $\sin\theta = 1$, and the force is at its maximum: $F = ILB$
• When the wire is parallel to the field ($\theta = 0°$ or $180°$), $\sin\theta = 0$, and there is no force at all

This sine dependence is worth remembering. If an exam problem doesn't specify the angle, check whether you need the full $ILB\sin\theta$ or if the geometry gives you a clean 90°.

Direction of magnetic force

You can find the force direction using the right-hand rule for cross products:

1. Point your fingers in the direction of the current ($\vec{L}$)
2. Curl your fingers toward the magnetic field ($\vec{B}$)
3. Your thumb points in the direction of the force ($\vec{F}$)

The force is always perpendicular to both the current direction and the magnetic field. Some textbooks use Fleming's left-hand rule instead (thumb = current, forefinger = field, middle finger = force), but both methods give the same result. Pick whichever one you're more comfortable with and stick with it.

Factors affecting magnetic force

The equation $F = ILB\sin\theta$ tells you exactly which variables control the force. Each one is worth understanding individually.

Current intensity

Force is directly proportional to current. Double the current, double the force. This is how many devices regulate the force they produce: by adjusting the current. Keep in mind that high currents also generate heat (due to resistance), which can damage conductors if not managed.

Wire length

In a uniform magnetic field, force increases linearly with the length of wire exposed to the field. A 2 m wire experiences twice the force of a 1 m wire, all else equal. In practice, longer wires have more resistance, so you need more voltage to maintain the same current.

Magnetic field strength

Force is directly proportional to field strength. Stronger magnets or electromagnets produce larger forces. Field strength typically drops off with distance from the source, so positioning matters.

Angle between wire and field

The $\sin\theta$ factor means the force varies smoothly from zero (parallel) to maximum (perpendicular). This gives you another way to control force: by adjusting the wire's orientation relative to the field.

Applications of magnetic force

Electric motors

Electric motors convert electrical energy into mechanical energy using the magnetic force on current-carrying loops.

1. A current-carrying coil sits in a magnetic field
2. The magnetic force on opposite sides of the coil pushes in opposite directions, creating a torque that rotates the coil
3. A commutator (in DC motors) reverses the current direction every half-turn so the torque always pushes the same rotational direction
4. The coil spins continuously, driving a shaft

Motors based on this principle appear in electric vehicles, power tools, fans, and countless other devices.

Loudspeakers

A loudspeaker turns electrical audio signals into sound:

1. A coil of wire is attached to a flexible diaphragm and sits inside a permanent magnet's field
2. When varying current flows through the coil, the magnetic force pushes the coil (and diaphragm) back and forth
3. The moving diaphragm compresses and rarefies the air, producing sound waves that match the electrical signal

Magnetic levitation

Magnetic levitation (maglev) uses magnetic forces to suspend objects without physical contact. Maglev trains, for example, float above the track using carefully controlled magnetic repulsion, eliminating friction between the train and rail. The same principle appears in magnetic bearings, which reduce wear in rotating machinery.

Torque on current loops

When a current loop sits in a uniform magnetic field, the net force on the loop is zero (forces on opposite sides cancel), but there is a net torque that tends to rotate the loop. This is the principle that makes motors spin.

Magnetic dipole moment

The magnetic dipole moment captures how strongly a current loop interacts with an external field:

$\vec{\mu} = IA\hat{n}$

where $I$ is the current, $A$ is the area enclosed by the loop, and $\hat{n}$ is the unit vector perpendicular to the loop's plane (direction found by the right-hand rule: curl fingers in the direction of current, thumb points along $\hat{n}$).

For a coil with $N$ turns, the dipole moment becomes $\mu = NIA$.

Torque equation

The torque on a current loop in a uniform magnetic field is:

$\vec{\tau} = \vec{\mu} \times \vec{B}$

The magnitude is:

$\tau = \mu B \sin\theta$

where $\theta$ is the angle between $\vec{\mu}$ and $\vec{B}$.

• Maximum torque occurs when the loop plane is parallel to the field ($\theta = 90°$, so $\vec{\mu}$ is perpendicular to $\vec{B}$)
• Zero torque when $\vec{\mu}$ is aligned with $\vec{B}$ ($\theta = 0°$). This is the equilibrium position the loop naturally rotates toward

Parallel vs perpendicular wires

Force between parallel wires

Two parallel wires carrying current exert forces on each other because each wire sits in the magnetic field created by the other.

• Currents in the same direction: the wires attract
• Currents in opposite directions: the wires repel

The force per unit length between two long, parallel wires separated by distance $r$ is:

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

where $\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A}$ is the permeability of free space. This relationship was historically used to define the ampere in SI units.

Force on perpendicular wires

Two wires crossing at right angles exert no net force on each other. The magnetic field from one wire, at the location of the other, produces forces that cancel out over the length of the wire. This is a useful fact for circuit design: routing wires perpendicularly minimizes unwanted magnetic interactions between them.

Magnetic force in solenoids

A solenoid is a coil of wire wound in a helix. It produces a strong, nearly uniform magnetic field inside when current flows through it.

Magnetic field inside solenoids

For a long solenoid (length much greater than diameter), the interior field is approximately uniform and given by:

$B = \mu_0 n I$

where $n$ is the number of turns per unit length and $I$ is the current. The field points along the solenoid's axis, and outside the solenoid it's approximately zero.

Force on solenoid windings

The current-carrying turns of a solenoid sit in each other's magnetic fields, so they experience forces:

• Adjacent turns carry current in the same direction, so they attract each other (just like parallel wires with same-direction currents)
• Turns near the ends experience an inward force due to the fringing fields at the solenoid's edges
• These internal forces can create significant mechanical stress in high-current solenoids, which must be accounted for in engineering design

Experimental demonstrations

Jumping wire experiment

This classic demo shows the magnetic force in action:

1. Place a loosely supported wire between the poles of a strong magnet
2. Send a pulse of current through the wire
3. The wire jumps out from between the poles due to the $\vec{F} = I\vec{L} \times \vec{B}$ force
4. Reversing either the current direction or the magnet orientation reverses the direction of the jump

Magnetic rail gun

A rail gun demonstrates how magnetic force can accelerate a projectile:

1. Two parallel conducting rails are connected to a power source
2. A conducting bar (the projectile) bridges the gap between the rails
3. Current flows through one rail, across the bar, and back through the other rail
4. The current in the bar, combined with the magnetic field generated by the current in the rails, produces a Lorentz force that accelerates the bar along the rails

Practical considerations

Wire configurations

The shape and arrangement of wires affect how magnetic forces are distributed. Straight wires, loops, and helical coils each produce different force patterns. In complex circuits, multiple wires can be arranged so their magnetic effects either reinforce or cancel each other. Proper wire routing also minimizes electromagnetic interference with nearby components, and thermal effects from current flow always need to be considered in the design.

Magnetic shielding

Sensitive equipment often needs protection from stray magnetic fields. Magnetic shielding uses materials with high magnetic permeability (such as mu-metal) to redirect magnetic flux around the protected region. Active shielding takes a different approach: it generates opposing magnetic fields to cancel the unwanted ones. Both techniques are critical in precision instruments and medical devices like MRI machines.