Parallel Currents & the Definition of the Ampere

Clip two slack jumper leads to a car battery, let them dangle side by side, and crank the starter: the leads twitch toward each other. Nothing touches them. Neither is a magnet. Yet a real, measurable force reaches across the air gap — and for seventy years, that exact force defined what an ampere was. Every ammeter on Earth was, in principle, calibrated against two parallel wires pulling on each other. The physics is a two-step chain built entirely from parts you already own. Each wire wraps itself in the circular magnetic field of Fields of Straight Wires & Circular Loops; the other wire sits in that field carrying current, and Force on a Current-Carrying Conductor says a current in a field feels $F = BIL$ . Chain the two and out drops one of the tidiest formulas in the chapter: force per metre equals $\mu_0 I_1 I_2 / 2\pi d$ . Same-direction currents attract; opposite currents repel — the reverse of what charge intuition whispers, and a trap examiners never tire of. This module builds the chain, fixes the attract-repel logic so it can't flip under exam pressure, and tells the ampere's story honestly — including the 2019 plot twist, when the SI quietly retired the two-wire definition and handed the job to the electron's charge. Along the way you'll meet the formula's violent side: the same $I^2$ force that defines units also blows apart busbars in short-circuit faults.

Key takeaways

Each wire creates a field; the other wire feels a force. Two known results chained: $B = \mu_0 I_1/2\pi d$ , then $F = B I_2 L$ .
The result per metre of wire: $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ — proportional to both currents, inversely proportional to separation.
Parallel currents attract; antiparallel currents repel — opposite to the like-charges rule. Work it out with the right hand once, then trust the slogan.
The force is mutual and equal on both wires (Newton's third law), even when the currents are unequal.
Two 1 A currents 1 m apart pull with exactly $2\times10^{-7}$ N per metre — the number that defined the ampere from 1948 to 2019; since 2019 the ampere is defined by fixing $e = 1.602\,176\,634\times10^{-19}$ C.
KEY IDEA: a current-carrying wire is both a field source and a field detector — put two side by side and each sits in the other's field, giving F/L = μ₀I₁I₂/2πd, attraction for same-direction currents.

The core insight

Two long parallel wires exert equal and opposite forces per unit length $F/L = \mu_0 I_1 I_2/2\pi d$ on each other — attractive when the currents run the same way, repulsive when opposite — because each wire carries current through the magnetic field made by the other.

Where this lives today

The formula's gentle side runs metrology's history; its violent side runs power engineering's present. Every substation busbar, every switchboard, every transformer winding is braced against equation (2) evaluated at fault current — the $I^2$ scaling means a fault at twenty times operating current multiplies the magnetic force four hundred fold, and "short-circuit withstand" certification is this module's arithmetic performed by controlled explosion. Inside every high-field magnet, adjacent windings carry parallel currents and crush inward on each other; superconducting magnet designers speak of "hoop stress" budgets the way bridge engineers speak of load limits. And the definition story completed its arc in 2019: the Kibble balance — the instrument that now realises the kilogram from the fixed Planck constant — is philosophically a descendant of Weber's current balance, weighing electromagnetic force against gravity. The handshake between mechanical and electrical units that the two parallel wires brokered for seventy years didn't disappear; it moved into quantum standards, where charge is counted electron by electron.

The physical picture

The deepest thing in this module is not the formula — it is the mechanism. Wire 2 does not "know" about wire 1 directly. What wire 2 knows is the magnetic field at its own location. Wire 1's job is to create that field; wire 2's job is to respond to it. Every current-on-current force in physics is this two-step transaction, and setting it up as two steps is what makes the direction logic foolproof. Run both currents upward, wires a distance $d$ apart. Wire 1's field circles it — by the right-hand grip rule, at wire 2's location (to the right of wire 1) that field points into the page. Now wire 2 is an upward current sitting in an into-the-page field: Fleming's left hand (or $\vec F = I\vec L\times\vec B$ ) gives a force pointing left — toward wire 1. Run the same argument from wire 2's side and you get a force on wire 1 pointing right — toward wire 2. Equal, opposite, attractive. Flip one current and one step of the chain flips sign: repulsion.

Figure

Each wire sits in the other's field — same-direction currents attract, opposite repel

Here is the intuition anchor for why the rule is opposite to charges. Two parallel same-direction currents are like two lanes of traffic moving together — their circular field lines between the wires run in opposite senses and cancel, while outside they cooperate. Field lines behave like stretched elastic that also pushes sideways on its neighbours: weak field between the wires, strong outside, and the wires get squeezed together. For antiparallel currents the field piles up between the wires and pries them apart. If you remember one picture, remember this pressure picture; if you remember one slogan: like currents attract, unlike currents repel — the anti-charges rule.

The formula, chained together

The setup: two long straight parallel wires, currents $I_1$ and $I_2$ , separated by $d$ (much less than their length, so the infinite-wire field applies). Wire 1's field at distance $d$ is the result owned by Fields of Straight Wires & Circular Loops:

B_1 = \frac{\mu_0 I_1}{2\pi d} \qquad (1)

where: $\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m/A}$ , and the direction circles wire 1 by the right-hand grip rule — at wire 2's location, perpendicular to wire 2. That perpendicularity is a gift: when Force on a Current-Carrying Conductor's law $F = BIL\sin\theta$ is applied, $\sin\theta = 1$ automatically, always, for parallel wires.

Derivation AThe force per unit length between parallel wires

We want $F/L = \mu_0 I_1 I_2/2\pi d$ and its direction rules, by chaining the two owned results — the field of a wire and the force on a current in a field. Nothing new is derived from scratch; the skill being drilled is the chaining, because this exact two-step is how every current-current force problem (and half of the induction chapter) is set up. Watch where each formula's direction information enters.

Step 1. Wire 1 makes a field at wire 2's location. Magnitude from equation (1): $B_1 = \mu_0 I_1/2\pi d$ . Direction by the right-hand grip on wire 1: with both currents upward and wire 2 to the right, $B_1$ at wire 2 points into the page.

Step 2. Wire 2 carries current through that field. A length $L$ of wire 2 feels the force from Force on a Current-Carrying Conductor:

F = B_1 I_2 L \sin 90° = B_1 I_2 L

Why the angle is exactly 90°: a straight wire's circular field is always perpendicular to any wire parallel to it — the geometry pins $\sin\theta = 1$ with no decision needed.

Step 3. Substitute and divide by $L$ :

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \qquad (2)

Step 4. Direction. Upward current ( $I_2$ , up) in an into-page field: $\vec F = I\vec L\times\vec B$ points toward wire 1 — attraction. Rerun Steps 1–3 from wire 2's side: wire 2's field at wire 1 points out of the page (grip rule from the other position), and the force on wire 1 points toward wire 2. Equal magnitudes, opposite directions — Newton's third law honoured, even if $I_1 \neq I_2$ : the product $I_1 I_2$ is symmetric, so both wires feel the same $F/L$ . Verified: $I_1 = I_2 = 1\ \mathrm{A}$ , $d = 1\ \mathrm{m}$ : $F/L = (4\pi\times10^{-7}\times1\times1)/(2\pi\times1) = 2\times10^{-7}\ \mathrm{N/m}$ — the SI's old calibration number, recomputed. Units: $\mathrm{(T\,m/A)\,A^2/m} = \mathrm{T\,A} = \mathrm{N/m}$ ✓. Matches NCERT Ch. 4 and the double-integral treatment in the UG literature.

Two structural readings of equation (2) repay a pause. First, the product $I_1 I_2$ : double both currents and the force quadruples — the $I^2$ scaling that turns a gentle unit-defining nudge into the busbar-bending violence of a short-circuit fault (a 100 kA fault in switchgear with 10 cm spacing exerts $2\times10^{-7}\times10^{10}/0.1 = 20{,}000$ N on every metre — two tonnes of force, arriving in milliseconds, which is why high-power switchboards are braced like bridges). Second, the $1/d$ : gentle by inverse-square standards. Halve the gap and the force merely doubles — a signature worth recognising in ratio questions.

The ampere: a unit built from this force

Equation (2) is unit-historic. From 1948 to 2019, the SI defined: one ampere is the current which, flowing in two infinitely long parallel wires one metre apart in vacuum, produces a force of exactly $2\times10^{-7}$ newtons per metre between them. Read equation (2) backwards and you'll see the definition is really a decision about $\mu_0$ : fixing the force at $2\times10^{-7}$ N/m for 1 A at 1 m is fixing $\mu_0 = 4\pi\times10^{-7}$ T·m/A exactly. The mechanical unit (newton) anchored the electrical unit (ampere) through this module's formula. No lab ever strung infinite wires, of course — real calibrations used precisely wound coils and force balances — but the two-wire form was the definition because it is exact and geometry-free. Then 2019. The SI was rebuilt on fixed constants of nature, and the ampere's new definition became: fix the elementary charge at $e = 1.602\,176\,634\times10^{-19}$ C exactly; one ampere is one coulomb per second of that charge. In the new system $\mu_0$ is no longer exactly $4\pi\times10^{-7}$ — it became a measured quantity (agreeing with the old value to better than a part per billion, so every exam calculation is untouched). The honesty note worth carrying: before 2019, equation (2) defined the unit and could not be "tested"; after 2019, the two-wire force is a measurable consequence — a genuine experimental check of electromagnetism.

Limiting cases

case	result	meaning
$I_1 = I_2 = 1$ A, $d = 1$ m	$F/L = 2\times10^{-7}$ N/m	the old ampere definition, verbatim
same-direction currents	attraction	like currents attract — anti-charges rule
opposite currents	repulsion	field piles up between, pries apart
double both currents	force ×4	$I^2$ scaling — fault forces are savage
double the separation	force halved	$1/d$ , not inverse-square
unequal currents	same $F/L$ on both wires	$I_1I_2$ symmetric; Newton's third law
one wire's current off	zero force	no source field, no transaction

Common trapTreating currents like charges — likes repel, so parallel currents must repel

The single most-marked wrong answer in this topic. Charge intuition says "like repels like," and students port it straight onto currents: parallel currents repel? No — parallel (same-direction) currents attract; antiparallel currents repel. The chain shows why the analogy fails: the force on wire 2 comes from $I\vec L\times\vec B$ , and the cross product's direction at wire 2's position points toward the source wire when the currents cooperate. If you prefer a picture: same-direction currents cancel their field in the gap between them (the circles run opposite ways there) and field pressure squeezes them together; opposite currents double the field in the gap and it pushes them apart. Either derive the direction fresh with your right hand each time, or engrave the slogan — but never borrow it from electrostatics.

Worked examples

Board level — force per length, then total force (CBSE/ISC pattern)

Two long straight parallel wires are 5.0 cm apart and carry currents of 8.0 A and 12.0 A in the same direction. Find (a) the force per unit length between them, (b) the force on a 30 cm segment of either wire, and (c) state the nature of the force.

Step 1. List and convert: $I_1 = 8.0$ A, $I_2 = 12.0$ A, $d = 0.050$ m, $L = 0.30$ m. (The centimetre-to-metre conversion is where this problem loses most of its marks.)

Step 2. Apply equation (2), using the $\mu_0/2\pi = 2\times10^{-7}$ shortcut:

\frac{F}{L} = \frac{2\times10^{-7} \times 8.0 \times 12.0}{0.050} = \frac{2\times10^{-7}\times96}{0.050}

so $\boxed{F/L = 3.84\times10^{-4}\ \mathrm{N/m}}$ .

Step 3. Force on 30 cm: $F = 3.84\times10^{-4} \times 0.30 = \boxed{1.15\times10^{-4}\ \mathrm{N}}$ — about the weight of a mosquito, which is why you've never noticed your desk lamp's flex attracting itself.

Step 4. Nature: currents are in the same direction, so the force is attractive — and the marks-scheme sentence: "each wire lies in the magnetic field of the other; for same-direction currents the force $I\vec L\times\vec B$ points toward the other wire."

Competitive level — a rectangular loop beside a wire (JEE-Main pattern, ~3 minutes)

A long straight wire carries $I_1 = 10$ A upward. A square loop of side 10 cm carrying $I_2 = 2.0$ A lies in the same plane, its nearest side parallel to the wire at 5.0 cm, its farthest at 15 cm, with the near-side current running the same direction as $I_1$ . Find the net force on the loop.

Figure

Square loop beside a long wire — near side attracted, far side repelled, perpendicular sides cancel

Step 1. Decompose. The two sides perpendicular to the wire feel equal and opposite forces (each sits in the same field profile, carrying opposite current directions as the loop circulates) — they cancel. The battle is between the near and far parallel sides.

Step 2. Near side (current parallel to $I_1$ , attracted): treat as a parallel-wires problem with $L = 0.10$ m at $d = 0.050$ m:

F_{\text{near}} = \frac{2\times10^{-7}\times10\times2.0\times0.10}{0.050} = 8.0\times10^{-6}\ \mathrm{N} \;\;\text{(toward the wire)}

Step 3. Far side: the loop's circulation makes its current run antiparallel to $I_1$ , so this side is repelled — pushed away from the wire — at $d = 0.15$ m:

F_{\text{far}} = \frac{2\times10^{-7}\times10\times2.0\times0.10}{0.15} = 2.67\times10^{-6}\ \mathrm{N} \;\;\text{(away from the wire)}

Step 4. Net: both effects point the same way relative to the wire? Check signs carefully — near side is pulled toward the wire, far side is pushed away from the wire; "toward" and "away" here are opposite directions in space. Net force $= 8.0\times10^{-6} - 2.67\times10^{-6} = \boxed{5.3\times10^{-6}\ \mathrm{N}\ \text{toward the wire}}$ . The loop as a whole is attracted, because the attracted side sits in the stronger field. Timing note: 3 minutes. The 30-second version: cancel the perpendicular sides by symmetry, write both parallel-side forces with the $2\times10^{-7}$ shortcut, subtract. If your two forces came out in the same spatial direction, you've mis-tracked the loop's circulation — the classic sub-error.

Exam traps

The attract/repel flip. Same-direction = attract. Write it in the margin before the clock starts. (Anatomy in the warning box.)

Charges vs currents in the same problem. Some JEE items mix electrostatic repulsion of the wires (if charged) with magnetic attraction (if current-carrying). Keep the two ledgers separate; only the magnetic one is equation (2)'s business.

Forgetting the force is mutual. With $I_1 = 8$ A and $I_2 = 12$ A, both wires feel the same force per metre — the product $I_1I_2$ doesn't care which wire you ask. "The stronger current pushes harder" is a Newton's-third-law violation and a favourite wrong option.

Using $d$ in centimetres. $F/L$ answers come out $10^{-4}$ -ish for bench values; if yours landed $10^{-6}$ -ish, hunt the unconverted 5 cm.

$\sin\theta$ reflexes. For genuinely parallel wires the field at one wire due to the other is automatically perpendicular to the current — $\sin\theta = 1$ always. Inserting a spurious angle from the wires' positions is a confusion between geometry in the plane and the field-current angle.

A-Level scope note. CIE 9702 asks you to explain the origin of the forces and determine their directions — the qualitative chain — without the $F/L$ formula. AP Physics 2 similarly builds this from field-of-a-wire plus force-on-a-wire (exactly this module's chain) rather than quoting equation (2). If you sit those boards, the two-step mechanism and direction logic are the assessed content; the formula is your bonus speed.

Constants: $\mu_0 = 4\pi\times10^{-7}$ T·m/A, and burn in the shortcut $\mu_0/2\pi = 2\times10^{-7}$ — nearly every parallel-wires answer starts there. Old ampere anchor: $2\times10^{-7}$ N/m at 1 A, 1 A, 1 m. New SI anchor: $e = 1.602\,176\,634\times10^{-19}$ C exact (use $1.6\times10^{-19}$ in calculations).

Origins · 1 minWilhelm Eduard Weber and how this was discovered

Wilhelm Eduard Weber

1804–1891 · Göttingen

Weber is the reason "the force between currents" became a measurement rather than a demonstration. Recruited to Göttingen by Gauss in 1831, he co-built with him the world's first working electromagnetic telegraph (1833) — a kilometre of double wire strung over the town's rooftops between the physics lab and the observatory — and then set about doing for electricity what Gauss had done for magnetism: expressing every electrical quantity in absolute mechanical units of length, mass and time. His electrodynamometer measured a current by the force between two coils carrying it — this module's physics, built into brass and silk thread — and his precision current-balance measurements anchored the entire chain of electrical units. In 1856, with Kohlrausch, he measured the ratio of electrostatic to electromagnetic charge units and found it suspiciously equal to the speed of light — a number Maxwell seized on as evidence that light itself is electromagnetic. When the SI needed a name for the unit of magnetic flux, it chose his. The 2019 redefinition retired the force-based ampere Weber's instruments had made possible — but every current balance and every "absolute" electrical measurement for 150 years was Weber's programme running.

History

1820

— September: within a week of hearing of Ørsted's discovery, Ampère shows parallel currents attract and antiparallel currents repel

1833

— Gauss and Weber's telegraph wires cross Göttingen; electricity joins the absolute system of units

1846

— Weber's electrodynamometer measures current by the force between coils — the parallel-wire force as an instrument

1948

— The SI defines the ampere via the two-wire force: exactly 2×10⁻⁷ N/m at one ampere and one metre

2019

— The SI redefinition fixes the elementary charge instead; the two-wire force becomes a testable consequence

Going further

Texts. NCERT Physics Class 12, Ch. 4 §4.5 (the force between parallel currents and the pre-2019 ampere) · H C Verma, Concepts of Physics II, Ch. 35 · BIPM SI Brochure, 9th ed. (the 2019 ampere) · Griffiths, Introduction to Electrodynamics §5.1–5.2 for the general circuit-circuit force (UG deep dive).

Provenance. Standard results per NCERT Ch. 4; the ampere history per the BIPM SI Brochure. A verifier should recompute: $F/L = 2\times10^{-7}\times8\times12/0.05 = 3.84\times10^{-4}$ N/m and $F = 1.15\times10^{-4}$ N on 0.30 m; the loop example $F_{\text{near}} = 8.0\times10^{-6}$ N, $F_{\text{far}} = 2.67\times10^{-6}$ N, net $5.3\times10^{-6}$ N toward the wire; the definition anchor $2\times10^{-7}$ N/m at 1 A, 1 A, 1 m; the fault-force figure $2\times10^{-7}\times(10^5)^2/0.1 = 2\times10^{4}$ N/m.

Prerequisites. Fields of Straight Wires & Circular Loops · Force on a Current-Carrying Conductor. Leads to. Net Fields of Wire Systems & Zero-Field Points · Floating Wires & Magnetic Balances (coming up later in this chapter's gym) · Magnetic Forces Between Currents (UG deep dive).

Why do parallel currents attract when parallel (like) charges repel? Both are "like" things.

Because the force laws have different geometry. Charges push along the line joining them via a radial field. Currents interact through a circulating field crossed with a current direction — $I\vec L\times\vec B$ — and when you turn that crank for same-direction currents, the cross product points toward the other wire. There's also a deeper relativistic answer worth a glimpse: ride along with the drifting electrons and the attraction can be re-read as an electrostatic effect of length-contracted charge densities — magnetism is what electrostatics looks like when sources move. At exam level: derive with the right hand, or trust the slogan "like currents attract."

If the two currents are different, which wire feels the bigger force?

Neither — and the formula says so before Newton does. $F/L = \mu_0 I_1I_2/2\pi d$ contains the product of the currents; swap the labels and nothing changes. The physical chain also balances: the weaker current sits in the stronger wire's field, and the stronger current sits in the weaker wire's field — the two asymmetries cancel exactly. "The 10 A wire pushes the 2 A wire harder than vice versa" is a Newton's-third-law violation dressed as intuition, and it appears as a distractor on NEET and JEE-Main with remarkable regularity.

Does the formula work for short wires, like the two leads of my multimeter?

As an approximation, and you should know when it degrades. Equation (2) assumes each wire is effectively infinite — meaning the separation $d$ is small compared with the wire lengths. For leads 30 cm long and a couple of centimetres apart, it's excellent. For wire segments whose length is comparable to $d$ , the field of a finite segment is weaker than the infinite-wire value and the honest calculation needs the finite-segment field (a UG exercise). Exam problems signal the safe regime with the word "long." If a question gives comparable length and separation without that word, it usually wants only an estimate.

How did the old definition make the ampere exact if nobody can build infinite wires?

The definition fixed the theory's constant, not a laboratory artefact. Declaring "1 A produces exactly 2×10⁻⁷ N/m at 1 m" is mathematically equivalent to declaring $\mu_0 = 4\pi\times10^{-7}$ T·m/A exactly — and with $\mu_0$ fixed, any computable geometry becomes a realisation route. Real current balances used coils whose force could be calculated from their measured winding geometry via the same physics, then weighed that force against calibrated masses. The infinite-wire wording was chosen precisely because it is clean and geometry-free — a calibration standard on paper, realised through buildable proxies in metal.

What changed for problem-solving after the 2019 redefinition?

Numerically, nothing you will ever detect at school or in JEE: $\mu_0$ is still $4\pi\times10^{-7}$ T·m/A to better than a part per billion, and every textbook answer is unchanged. Conceptually, two things flipped. The ampere is now realised by counting charge ( $e$ is exact), not by weighing forces; and $\mu_0$ was demoted from defined to measured — so the two-wire force became a genuine experimental test of electromagnetic theory rather than a tautology. If an exam asks for the current definition of the ampere: it is the current corresponding to one coulomb per second, with the coulomb fixed by $e = 1.602\,176\,634\times10^{-19}$ C exactly.

Do the perpendicular sides of my loop-near-a-wire really contribute nothing?

They contribute forces — just not net force, and only along the wire's direction. Each perpendicular side runs through the wire's field, which weakens with distance, so each feels a force (directed parallel to the long wire, by the cross product). But the loop's circulation makes the currents in the two perpendicular sides run oppositely through the same field profile, so their forces are equal and opposite: they cancel pairwise, and they also act along the wire's axis, perpendicular to the attract/repel axis you're computing. What they do contribute in harder problems is a stretching or compressing stress on the loop — and a torque, if the loop is tilted out of the plane, which is the next module's territory (Current Loops as Magnetic Dipoles).

Is this force what makes electromagnets hum and transformers buzz?

Substantially, yes. In AC equipment the currents reverse a hundred times a second (50 Hz mains), but the force goes as the product of currents: reverse both and $I_1I_2$ stays positive, so parallel conductors are tugged at twice the mains frequency — a 100 Hz mechanical drumming (120 Hz on 60 Hz grids). Windings flex microscopically at that rate, laminations rattle (aided by magnetostriction of the iron), and the sum is the hum you hear near transformers and chargers. When engineers "vacuum-impregnate" windings with varnish, a big part of the job is bracing the conductors against exactly the force in equation (2), cycling forever.