Coaching Centre
Parallel wires and the ampere
The magnetic field around a long straight wire, why two parallel current-carrying wires attract or repel, the force per unit length F/l = μ₀I₁I₂/2πr, how Newton's third law applies, and the old and current SI definitions of the ampere.
When a fault shorts out a large substation, the current through its thick copper busbars can jump to tens of thousands of amperes for a fraction of a second. The bars then slam towards or away from each other with forces of hundreds of newtons per metre. Engineers brace them to survive it. The cause is two currents pushing on each other through their magnetic fields.
The field around a straight wire
A long straight wire carrying a current is surrounded by a circular magnetic field. To find its direction, use the right-hand grip rule: grip the wire with your right thumb along the conventional current, and your fingers curl in the direction of the field. The field gets weaker with distance r from the wire:
$$B = \frac{\mu_0 I}{2\pi r}$$
where $\mu_0 = 4\pi \times 10^{-7}$ T m A−1 is the permeability of free space. (This formula isn't on the HSC data sheet, but it is the step that explains the one that is.) At 1 cm from a wire carrying 10 A, B = 2 × 10−4 T, about three and a half times the strength of Earth's field in Sydney.
Two wires: a force on each
Put a second wire parallel to the first. Wire 2 is now a current sitting in wire 1's magnetic field, so it feels a motor-effect force, $F = lI_2B_1$. Substitute wire 1's field at distance r:
$$\frac{F}{l} = \frac{\mu_0}{2\pi}\frac{I_1 I_2}{r}$$
This is the force on each metre of either wire. Since $\mu_0/2\pi = 2 \times 10^{-7}$, you can write it as $F/l = 2 \times 10^{-7}\,I_1I_2/r$.
Work out the direction with the grip rule and then the palm rule. For currents in the same direction, wire 1's field at wire 2 points so that the force on wire 2 is towards wire 1: the wires attract. For currents in opposite directions, they repel. (This is the opposite of the "likes repel" rule for charges and poles, which catches a lot of students out.)
Two long parallel wires 2.0 cm apart carry currents of 5.0 A and 10 A in the same direction. Find the force on a 50 cm length of each wire.
$\dfrac{F}{l} = \dfrac{(2 \times 10^{-7})(5.0)(10)}{0.020} = 5.0 \times 10^{-4}$ N m−1.
$F = (5.0 \times 10^{-4})(0.50) = 2.5 \times 10^{-4}$ N.
Same direction, so they attract. The 5.0 A wire is pulled towards the 10 A wire with 2.5 × 10−4 N, and the 10 A wire is pulled towards the 5.0 A wire with the same 2.5 × 10−4 N, even though its current is twice as big.
Newton's third law
That last step is not a coincidence. The force on wire 2 is $lI_2B_1 = lI_2\,\mu_0I_1/2\pi r$, and the force on wire 1 is $lI_1B_2 = lI_1\,\mu_0I_2/2\pi r$. They are identical expressions. The two forces are an action–reaction pair: equal in magnitude, opposite in direction, acting on different objects. A bigger current in one wire makes its field stronger, but the other wire's smaller current feels that stronger field, and the product $I_1I_2$ is the same either way.
Defining the ampere
From 1948 to 2019, the SI unit of current was defined using exactly this interaction:
One ampere is the constant current which, if maintained in two straight parallel conductors of infinite length and negligible cross-section placed 1 metre apart in a vacuum, would produce between them a force of 2 × 10−7 newtons per metre of length.
Check it with the formula: $F/l = 2 \times 10^{-7} \times 1 \times 1 \div 1 = 2 \times 10^{-7}$ N m−1. This definition is what fixed μ0 at exactly 4π × 10−7. It relies on Newton's third law: the definition talks about "the force between them" because the force on each wire is the same, so it doesn't matter which wire you measure.
In May 2019 the SI was redefined so that every base unit rests on a fixed constant of nature. The ampere is now defined by fixing the elementary charge at exactly e = 1.602 176 634 × 10−19 C: one ampere is a flow of 1/e, about 6.24 × 1018, elementary charges per second. μ0 is now a measured quantity, but it still equals 4π × 10−7 to better than one part in a billion, so the formula and every calculation in this module are unchanged. For the HSC, know the parallel-wire definition and its link to Newton's third law, and be ready to mention that the modern definition is based on the charge of the electron.
- Set 5.0 A and 10 A, 2.0 cm apart, length 1.0 m. Check F/l with the formula. Are the two force arrows the same size?
- Reverse one current. What happens to the field between the wires in the end-on view, and to the forces?
- Double the separation, then double it again. Plot F/l against 1/r to show the inverse relationship.
- Press Show the old ampere definition. Read off F/l and connect it to the wording of the definition.
- Set one current to zero. Explain why neither wire feels a force, not just the one with no current.
Test yourself
Two parallel wires carry currents in opposite directions. The wires
Opposite currents repel.
Wire X carries 2 A and wire Y, parallel to it, carries 6 A. The force on X due to Y is F. The force on Y due to X is
Newton's third law: an action–reaction pair is equal and opposite.
The separation of two parallel wires is doubled and the current in each is doubled. The force per unit length
F/l ∝ I₁I₂/r = (2 × 2)/2 = 2.
Two long wires 5.0 cm apart each carry 15 A in the same direction. The force per metre on each is
F/l = 2 × 10−7 × 15 × 15 ÷ 0.050 = 9.0 × 10−4 N m−1; same direction attracts.
Since 2019 the SI ampere has been defined by
The 2019 SI fixes e = 1.602 176 634 × 10−19 C exactly; μ₀ is now measured.
Label the parallel-wire diagram.
Put the explanation of why parallel currents attract in order.
The field around a straight wire is found with the right-hand rule. Two parallel wires with currents in the same direction . The force per unit length is inversely proportional to the . The forces on the two wires form an action–reaction pair, as described by Newton's law. The old definition of the ampere used two wires 1 apart.
Explain how the pre-2019 SI definition of the ampere is related to Newton's third law of motion.
Before 2019 the ampere was defined as the constant current which, in two straight parallel conductors of infinite length and negligible cross-section placed 1 m apart in a vacuum, produces a force of 2 × 10−7 N per metre between them. Each wire carries a current in the magnetic field of the other, so each experiences a motor-effect force. The force per metre on wire 2 is I₂B₁ = μ₀I₁I₂/2πr, and on wire 1 it is I₁B₂ = μ₀I₂I₁/2πr: the same expression. The two forces are therefore equal in magnitude and opposite in direction and act on different objects, which makes them an action–reaction pair, as Newton's third law requires. This is why the definition can simply refer to "the force between" the conductors: it is the same on each wire, so it does not matter which one is measured.
Two parallel overhead conductors 30 cm apart each carry 800 A in opposite directions. Calculate the force on a 50 m span of one conductor and state its direction.
F/l = μ₀I₁I₂/2πr = (2 × 10−7 × 800 × 800) ÷ 0.30 = 0.43 N m−1. Over 50 m, F = 0.427 × 50 = 21 N. The currents are in opposite directions, so the force is repulsive: each conductor is pushed away from the other.