IB Physics SLTopic 4 — Force FieldsPaper 1 & 2F ÷ L = μ0I1I2 ÷ 2πr~8 min read
Force Between Parallel Wires
Put two current-carrying wires side by side and something surprising happens: they reach out and grab each other. Depending on which way the currents flow, the wires either pull together or push apart — with no magnet in sight. This is the last page’s F = BIL used twice over, because each wire is quietly sitting in the magnetic field made by the other.
📘 What you need to know
Every current-carrying wire wraps a magnetic field around itself. Place a second wire nearby and it sits inside that field, so it feels a force — and pushes back just as hard
Currents in the same direction → the wires attract; currents in opposite directions → the wires repel
The two forces are always equal in size and opposite in direction (Newton’s third law)
The force per unit length is F ÷ L = μ0I1I2 ÷ 2πr, where r is the separation and μ0 is the permeability of free space
μ0 = 4π × 10−7 N A−2 — a fixed constant of nature, given in the data booklet
The force gets stronger with bigger currents or longer wires, and weaker as the wires are moved apart (it falls off as 1 ÷ r)
Why Two Wires Feel Each Other
Nothing new is going on here — just the last page’s idea applied twice. Take wire 1: its current wraps circular field lines around itself (right-hand grip rule). Wire 2, sitting a short distance away, is now inside wire 1’s field, so it experiences a force F = BIL. But wire 2 makes its own field too, and wire 1 sits in that — so wire 1 feels an equal and opposite push. The wires act on each other as a matched pair.
Whether that pair pulls in or pushes out comes down to the field in the gap between them. When the currents run the same way, their fields point in opposite directions in the gap and partly cancel, leaving a weak spot between the wires — and the wires are drawn into it, so they attract. When the currents run opposite ways, their fields reinforce in the gap, crowding the field lines together, and that crowded region pushes the wires apart, so they repel.
Black lines are the wires, with their current directions; red arrows are the forces. Same-direction currents pull the wires together; opposite-direction currents push them apart. The two forces are always equal and opposite.
How Strong Is the Force?
We don’t usually quote a single number, because a longer pair of wires simply feels more force. Instead we give the force per metre of length — the force per unit length:
Force per unit length between parallel wiresF ÷ L = μ0I1I2 ÷ 2πr
Here F is the force on each wire (N), L is the length of the wires (m), I1 and I2 are the two currents (A), r is the separation between the wires (m), and μ0 = 4π × 10−7 N A−2 is the permeability of free space — a constant you’ll find in the data booklet. Because r sits on the bottom, doubling the gap halves the force.
Where the formula comes from
You can build this straight out of last page’s tools. A single straight wire makes a field B = μ0I ÷ 2πr at a distance r. Apply that to wire 2’s field arriving at wire 1, then use F = BIL (with the field square-on, sin θ = 1):
wire 1 sits in wire 2’s field
field there: B = μ0I2 ÷ 2πr
push per metre = B × I1
combine →
F÷L = μ0I1I2 ÷ 2πr
Picturing the Push
The end-on view makes the “each wire is in the other’s field” idea click. Below, both currents come out of the page (dots), so their fields circle anticlockwise. Wire 2 sits right in wire 1’s field, and the force on it points straight back toward wire 1 — the same happens to wire 1, so the pair is pulled together.
Both currents point out of the page (·), so their fields loop anticlockwise. At wire 2, wire 1’s field points up; Fleming’s left-hand rule then sends the force back toward wire 1. Mirror it for wire 1 and you get the equal, opposite partner force — the two wires attract.
Quick recap: two parallel wires each sit in the other’s field, so they push on each other — attracting when the currents run the same way, repelling when they run opposite. The force per metre is F ÷ L = μ0I1I2 ÷ 2πr, and the two forces are always equal and opposite.
🧭 Working out attract-or-repel
Check the current directions — same way or opposite ways along the two wires?
Same → attract, opposite → repel. A quick memory hook: friendly currents (running together) pull in; disagreeing currents push apart
Need the direction on one wire? Find the field the other wire makes at its position (right-hand grip rule), then use Fleming’s left-hand rule with that field and this wire’s current
The partner force is equal and opposite — never draw two forces of different lengths
For a size, use F ÷ L = μ0I1I2 ÷ 2πr — and remember the answer is a force per metre until you multiply by the length
WE 1
Two long, straight, parallel wires sit 2.5 cm apart. One carries a current of 6.0 A and the other 9.0 A, both in the same direction. (a) Calculate the force per unit length between the wires and state whether it is attractive or repulsive. (b) Find the force on a 0.40 m length of one wire.
Part (a) — known values
I1 = 6.0 A, I2 = 9.0 A
r = 2.5 cm = 0.025 m, μ0 = 4π × 10⁻⁷
Put them into F ÷ L = μ₀I₁I₂ ÷ 2πrF ÷ L = (4π × 10⁻⁷ × 6.0 × 9.0) ÷ (2π × 0.025)F ÷ L = 4.32 × 10⁻⁴ N m⁻¹≈ 4.3 × 10⁻⁴ N m⁻¹, attractiveSame-direction currents, so the wires pull together.Part (b) — force on 0.40 mF = (F ÷ L) × L = 4.32 × 10⁻⁴ × 0.40F ≈ 1.7 × 10⁻⁴ N
WE 2
Two long vertical wires, P and Q, hang side by side with Q to the right of P. P carries current upward and Q carries current downward. (a) State whether the wires attract or repel. (b) State the direction of the force on wire Q, and how it compares with the force on wire P.
Part (a) — compare the currents
P is up, Q is down → opposite directions
the wires repelPart (b) — direction on Q
P’s field at Q points into the page;
Q’s current is down → Fleming’s LHR gives…
force on Q points right (away from P)The force on P is equal in size but points left — away from Q. Equal and opposite, by Newton’s third law.
💡 Top tips
Same → attract, opposite → repel. It’s the reverse of magnets, where like poles repel — so state it carefully and don’t run on autopilot
The answer is a force per metre. F ÷ L has units N m⁻¹; multiply by the length to get an actual force in newtons
μ0 = 4π × 10⁻⁷ N A⁻² is in the data booklet — don’t try to remember a decimal for it, and keep the 2π on the bottom
Forces come in an equal-and-opposite pair. Even if the two currents differ, each wire feels the same size force — draw both arrows the same length
⚠ Common mistakes
Swapping the rule round — writing “same direction repel” by copying the like-poles-repel habit from bar magnets
Quoting F ÷ L as if it were the total force, and forgetting to multiply by the wire’s length
Dropping the 2π, or putting r on the top instead of the bottom, so the force grows with distance instead of shrinking
Drawing the two force arrows unequal because the currents are unequal — the pair is always equal and opposite
Up next: we shrink the picture right down from wires full of charge to a single charge on its own. A lone particle flying through a magnetic field feels a force F = Bqv — the same motor effect, now acting on one moving charge at a time.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
