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Forces between parallel currents

Key idea: Parallel currents in the same direction attract; opposite currents repel. Each wire feels the other's field, with equal and opposite forces proportional to I₁I₂ and inversely to separation.

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  • GCE A-Level H2 Physics 2027
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Two wires carrying currents side by side pull together or push apart, even with no magnet nearby. Each wire sits in the magnetic field of the other, so each feels a force. Putting together the field of a straight wire and the force on a current explains the direction of these forces and how big they are.

Check your understanding 1 Before drawing the forces

Two long parallel wires carry currents in the same direction, one twice as large as the other. Predict whether the wires attract or repel, and compare the force magnitudes on equal lengths of the wires.

The wires
Force magnitudes on equal lengths
Show answer

The wires attract. Their force magnitudes are equal on equal lengths, even though the currents differ. Work through the field directions below to see why the force points towards the other wire.

Why parallel currents attract or repel

To find the force on one wire, find the field that the other wire produces where it is, then apply Fleming’s left-hand rule to its own current. For this interaction, use the neighbour’s field as the external field acting on the wire.

Take two long parallel wires, both carrying current out of the page, with wire 1 on the left. By the right-hand grip rule, wire 1’s field circles anticlockwise, so at wire 2 it points up the page. With the current in wire 2 out of the page and the field up, Fleming’s left-hand rule gives a force on wire 2 to the left, towards wire 1. Repeating this for wire 1 in the field of wire 2 gives a force to the right. The wires attract.

Fields and forces between parallel currentsTwo cross-sections show wire 1 on the left and wire 2 on the right. Wire 1 carries current out of the page in both. With wire 2 current outwards, the neighbour field points down at wire 1 and up at wire 2, and forces point together. With wire 2 current inwards, both neighbour fields point up and the forces point apart. B₂ at wire 1 and B₁ at wire 2 are neighbour fields, not self-fields. Equal and opposite forces act on different wires, even with unequal currents.Same-direction currents attractI₁ outI₂ outF₁F₂B₂ at 1B₁ at 2Opposite-direction currents repelI₁ outI₂ inF₁F₂B₂ at 1B₁ at 2
Fields and forces between parallel currentsTwo cross-sections show wire 1 on the left and wire 2 on the right. Wire 1 carries current out of the page in both. With wire 2 current outwards, the neighbour field points down at wire 1 and up at wire 2, and forces point together. With wire 2 current inwards, both neighbour fields point up and the forces point apart. B₂ at wire 1 and B₁ at wire 2 are neighbour fields, not self-fields. Equal and opposite forces act on different wires, even with unequal currents. Same-direction currents attractI₁ outI₂ outF₁F₂B₂ at 1B₁ at 2 Opposite-direction currents repelI₁ outI₂ inF₁F₂B₂ at 1B₁ at 2
Parallel currents in the same direction attract; opposite currents repel. Dots mean outwards and crosses inwards; fields come from the other wire; not to scale.
  • Currents in the same direction attract.
  • Currents in opposite directions repel.

The two forces are a Newton’s third law pair: equal in size and opposite in direction, even when the currents are different. Note that the rule is the opposite of the one for electric charges, where like charges repel.

Check your understanding 2 Which field, and where?

Two long wires 0.080 m apart each carry 4.0 A in the same direction. Their resultant field at the midpoint is zero. A student concludes that they exert no force on one another. Find the field that one wire produces at the other, then calculate the force magnitude on a 0.50 m length of each wire.

Show the steps Optional working
Unit: T
Unit: N
Show answer

Each wire experiences its neighbour’s non-zero field at the wire, not the resultant field at the midpoint. That field is B = (4π × 10⁻⁷)(4.0)/(2π × 0.080) = 1.0 × 10⁻⁵ T. Each wire feels F = BIl = (1.0 × 10⁻⁵)(4.0)(0.50) = 2.0 × 10⁻⁵ N towards the other wire. The equal and opposite forces act on different wires and do not cancel on either one.

Check your understanding 3 Why zero at the midpoint is not enough

For those two wires, explain why zero resultant field at the midpoint does not make the force on either wire zero. Explain also why the equal and opposite forces do not cancel on either wire.

Show answer

Check that you identify where the field is needed and which body each force acts on. Each wire lies in its neighbour’s non-zero field at the full separation d, not in the zero field at the midpoint. The magnetic forces point towards each other but act on different wires. They cancel when both wires are included in one system; they do not cancel on either wire separately.

The size of the force

At a distance d, the field of wire 1 is B₁ = μ₀I₁/(2π d), at right angles to wire 2. So the force on a length l of wire 2 is F = B₁I₂l, and the force per unit length is

F/l = μ₀I₁I₂/(2π d)

Swapping the labels gives the same expression for the force on wire 1, which is why the forces are equal: both contain the product I₁I₂.

Worked example 1

Two equal currents

Problem

Two long parallel wires 0.050 m apart each carry 6.0 A in the same direction. Find the force on a 0.20 m length of either wire.

Show full solution

  1. Field of one wire at the other

    Method

    Use B = μ₀I/(2π d) at the position of the other wire.

    Reason

    The force on a wire comes from its neighbour’s field, not its own.

    Working

    Answer

    B₁ = ((4π × 10⁻⁷)(6.0))/2π(0.050) = 2.4 × 10⁻⁵ T

  2. Force

    Working

    Answer

    F = B₁I₂l = (2.4 × 10⁻⁵)(6.0)(0.20) = 2.9 × 10⁻⁵ N, towards the other wire

Guided practice 2

Wider apart, one current reversed

About 3 min

Problem

Two parallel wires attract each other with a force of 1.2 × 10⁻⁴ N per metre. The separation is doubled and one of the currents is reversed, keeping its size. Find the new force per metre, and say whether the wires now attract or repel.

Find the new force per metre

Unit: N m⁻¹
The wires now

Hints

Hint 1

size and direction separately

Which change affects the field strength, and which changes its direction?

Show full solution

  1. New force

    Working

    Answer

    Doubling d halves the force per metre to 6.0 × 10⁻⁵ N m⁻¹. With one current reversed, the currents are opposite, so the wires repel.

Try it yourself 3

Unequal currents

Minimal support

Problem

Two long parallel wires 0.040 m apart carry currents of 8.0 A and 2.0 A in opposite directions. Find the force per metre on the wire carrying 2.0 A, state its direction relative to the other wire, and compare its magnitude with the force per metre on the other wire.

Find and compare the forces

Show the steps Optional working
Unit: T
Unit: N m⁻¹
Force on the 2.0 A wire
Force per metre on the 8.0 A wire is

Hints

Hint 1

which field

Find the field acting on each wire: which current produces it?

Show solution step by step

  1. Force on the 2.0 A wire

    Reason

    It is in the field of the 8.0 A wire.

    Working

    Answer

    F/l = ((4π × 10⁻⁷)(8.0)(2.0))/2π(0.040) = 8.0 × 10⁻⁵ N m⁻¹, away from the other wire

  2. Comparison

    Working

    Answer

    The same. The 8.0 A wire sits in a field four times weaker, but carries four times the current, so the product I₁I₂ is unchanged.

Exam-style question 1

Opposite currents

5 marks

Examination question

Two long parallel wires are seen end-on. Wire 1, on the left, carries a current into the page. Wire 2, on the right, carries a current out of the page.

(a) State the direction of the magnetic field of wire 1 at wire 2, and hence the direction of the force on wire 2. [2 marks]

(b) State the direction of the force on wire 1, and explain why it has the same size as the force on wire 2. [2 marks]

(c) The current in wire 1 is doubled. State the effect on the force on each wire. [1 mark]

Write your answer before viewing the mark scheme

Show solution step by step

  1. (a) Field and force at wire 2

    2 marks

    Method

    Use the grip rule for wire 1, then the left-hand rule for wire 2.

    Reason

    The force on wire 2 comes from wire 1’s field at wire 2.

    Working

    Answer

    Wire 1’s field circles clockwise, so at wire 2 it points down the page. With current out of the page, the force on wire 2 is to the right, away from wire 1.

  2. (b) Force on wire 1

    2 marks

    Reason

    The forces are a third-law pair, and both equal μ₀I₁I₂l/(2π d).

    Working

    Answer

    To the left, away from wire 2: the wires repel, with forces of equal size.

  3. (c) Doubling current 1

    1 mark

    Working

    Answer

    Both forces double, because both depend on I₁I₂.

Common mistakes

  • Using the field at the midpoint, at d/2, instead of the field of one wire at the other.
  • Applying the electrostatic rule: parallel currents in the same direction attract, not repel.
  • Thinking the wire with the larger current feels the larger force.
  • Adding the two forces on the two wires as if they acted on one object.

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