Force between parallel currents

Like currents attract

Narrated video. Transcript:

Two long straight wires, a and b, run parallel to each other, a distance d apart. Wire a carries a current I a, and wire b carries I b, in the same direction.

Each wire sits in the magnetic field of the other. So each must feel a force. Let us find the force on wire b.

Wire a makes circular field lines around itself. At the position of wire b, its field is B a equals μ₀ I a over 2πd.

By the right-hand thumb rule, this field at b is perpendicular to the plane of the two wires, and perpendicular to wire b.

A length L of wire b carrying I b in this field feels a force I b L B a. That is F equals μ₀ I a I b L over 2πd.

And the direction? Work out I b L cross B a: it points from b straight towards a. The wires pull towards each other.

Wire a feels an equal and opposite force, towards b, as Newton’s third law demands. So parallel currents attract.

Now reverse the current in b. The field of a is unchanged, but the force on b flips. Antiparallel currents repel.

Compare this with charges: like charges repel, but like currents attract. The force per unit length is f equals μ₀ I a I b over 2πd.

Defining the ampere

Narrated video. Transcript:

This force gave us a way to define the unit of current. Put 1 ampere in each of two long wires, 1 metre apart in vacuum.

Then f equals μ₀ over 2π, times 1 times 1, over 1. And μ₀ over 2π is 2 × 10⁻⁷. So the force is 2 × 10⁻⁷ newton on each metre of wire.

So the ampere was defined as that steady current which, in each of two very long, straight, parallel conductors of negligible cross-section placed 1 metre apart in vacuum, produces a force of 2 × 10⁻⁷ newton per metre of length.

Then the coulomb follows: it is the charge that flows in 1 second through a wire carrying a steady current of 1 ampere. Since 2019 the SI fixes the value of the electron’s charge instead, but the size of the ampere is the same.

Now an example. Wires A and B carry 8 amperes and 5 amperes in the same direction, 4 centimetres apart. What is the force on a 10 centimetre length of A?

f equals 2 × 10⁻⁷ times 8 times 5, over 0.04. That is 8 × 10⁻⁶ over 0.04, which is 2 × 10⁻⁴ newton per metre.

On 10 centimetres, the force is 2 × 10⁻⁴ times 0.1, which is 2 × 10⁻⁵ newton. The currents are in the same direction, so the force is attractive, towards B.

These forces are tiny for everyday currents. But a short circuit can carry thousands of amperes, and then the force can bend metal bars in a power station.

Try it: two parallel wires

Interactive simulation. Things to try:

• The wires are seen end-on: ⊙ means current out of the screen, ⊗ into it.
• With both currents the same way, the field lines between the wires cancel out and the wires attract.
• Press “Reverse I₂”: the field lines crowd between the wires and the wires repel.
• Halve d: the force per metre doubles. Double one current: it doubles too.

Force on a short length

Two long parallel wires A and B carry 8 A and 5 A in the same direction and are 4 cm apart. What is the force on a 10 cm length of wire A?

1. 2 × 10⁻⁵ N, towards B
2. 2 × 10⁻⁵ N, away from B
3. 2 × 10⁻⁴ N, towards B
4. 1.3 × 10⁻⁴ N, towards B

Double everything

Two long parallel wires repel each other with a force f per metre. Both currents are doubled and the distance between the wires is also doubled. What is the new force per metre?

1. f
2. 2f, still repulsive
3. 4f, still repulsive
4. 2f, but now attractive

Summary

• Field of wire a at wire b: B_a = μ₀ I_a / 2πd
• Force on length L of wire b: F = μ₀ I_a I_b L / 2πd
• Force per unit length: f = μ₀ I_a I_b / 2πd

• Each wire lies in the field of the other, so each feels a force F = I L × B.
• Parallel currents attract; antiparallel currents repel — the opposite of charges.
• The forces on the two wires are equal and opposite (Newton’s third law).
• 1 ampere: the current in each of two long parallel wires 1 m apart in vacuum that gives a force of 2 × 10⁻⁷ N per metre.