Investigate quantitatively and analyse the interaction between current-carrying conductors and uniform magnetic fields F/l = I B sin theta, including parallel current-carrying wires F/l = mu_0 I_1 I_2 / (2 pi r)

What this dot point is asking

NESA wants you to apply $F = BIL \sin \theta$ to a straight wire carrying current $I$ of length $L$ in a uniform field $B$, derive the parallel-wire force $F/L = \mu_0 I_1 I_2 / (2 \pi r)$ as a special case, work out direction with the right-hand rule, and connect the parallel-wire result to the historical definition of the ampere.

The answer

Force on a straight wire in a uniform field

A wire of length $L$ carrying current $I$ in a uniform magnetic field $\vec{B}$ experiences a force:

where $\theta$ is the angle between the current direction and the field. Per unit length:

This force comes from the magnetic force on each moving charge in the wire: $\vec{F} = q \vec{v} \times \vec{B}$ summed over $N$ charges gives $\vec{F} = I \vec{L} \times \vec{B}$ in vector form.

Direction is given by the right-hand rule:

1. Point the fingers of the right hand in the direction of the conventional current.
2. Curl them toward $\vec{B}$.
3. The thumb gives the force direction.

Equivalently: flat right hand, fingers along $\vec{B}$, thumb along $I$, palm pushes the force out.

Special angles

• IMATH_20 : maximum force, $F = BIL$.
• IMATH_22 : zero force (current parallel to field).
• IMATH_23 : zero force (current antiparallel to field).

Force between two long parallel wires

Two long, straight, parallel wires carrying currents $I_1$ and $I_2$ separated by a distance $r$ exert magnetic forces on each other. The magnetic field at wire 2 due to wire 1 (at distance $r$) is:

This field is perpendicular to wire 2, so the force per unit length on wire 2 is:

By Newton's third law, wire 1 feels the same magnitude of force per unit length.

The constant $\mu_0 = 4 \pi \times 10^{-7}$ T m/A is the permeability of free space. With this value, $\mu_0 / (2 \pi) = 2 \times 10^{-7}$ T m/A exactly, which simplifies a lot of arithmetic.

Attraction and repulsion

• Same direction currents. Each wire sits in the magnetic field of the other; the right-hand rule shows the force on each wire points toward the other wire. The wires attract.
• Opposite direction currents. The forces reverse. The wires repel.

This is sometimes summarised as "parallel currents attract, antiparallel currents repel," the reverse of the rule for electric charges.

Historical definition of the ampere

The pre-2019 SI definition of the ampere used parallel wires. One ampere was defined as the current in each of two infinite, parallel wires 1 m apart in vacuum that would produce a force per unit length of:

This defined $\mu_0 = 4 \pi \times 10^{-7}$ T m/A exactly. Since the 2019 SI redefinition the ampere is defined via the fixed value of the electronic charge $e$, and $\mu_0$ is measured rather than defined, but the value is essentially unchanged for HSC work.

Worked example: rail gun (qualitative)

A conducting bar of length $L = 0.30$ m slides along two rails carrying a current $I = 200$ A. It sits in a field $B = 0.50$ T perpendicular to both the bar and the rails. The force on the bar is:

$F = BIL = 0.50 \times 200 \times 0.30 = 30$ N.

This force accelerates the bar along the rails. Rail guns scale this idea up to thousands of amperes and tesla-class fields to launch projectiles.

Try it: Lorentz force calculator for the force on moving charges, the same physics that gives $F = BIL\sin\theta$ when summed over a current.

Common traps

Forgetting $\sin \theta$ in the single-wire formula. Quoting $F = BIL$ when the wire is at an angle other than 90 degrees costs marks.

Getting the direction wrong for negative charges. This formula uses conventional current (positive charge flow). The right-hand rule applies directly; you do not need to flip it for the electron-flow direction.

Forgetting Newton's third law for parallel wires. Both wires feel the same magnitude of force, even if their currents have different magnitudes. Both wires have the same $F/L = \mu_0 I_1 I_2 / (2 \pi r)$.

Mixing up $\mu_0 / 2 \pi$ and $\mu_0 / 4 \pi$. In the parallel-wire formula the denominator is $2 \pi r$, so the useful constant is $\mu_0 / 2 \pi = 2 \times 10^{-7}$ T m/A.

Treating parallel-wire force as Coulomb-like. The parallel-wire force is magnetic and depends on currents (charge flow), not on net charge. Two stationary wires with no current exert no force on each other, regardless of how much net charge they carry per unit length (ignoring electrostatic effects).

Direction confusion for antiparallel currents. Antiparallel parallel wires repel. The right-hand rule confirms: reversing one current flips the force on that wire.

In one sentence

A current $I$ in a length $L$ of wire at angle $\theta$ to a magnetic field $B$ feels a force $F = BIL \sin \theta$ perpendicular to both, and two long parallel wires separated by $r$ exert $F/L = \mu_0 I_1 I_2 / (2 \pi r)$ on each other (attractive for parallel currents, repulsive for antiparallel).