IB Physics HLCharges Moving in FieldsPaper 1 & 2F = BIL sin θ~16 min read
Force on a Current-Carrying Wire
Last section we mapped the magnetic field in loving detail and never once asked what it does. Now we find out. A current-carrying wire makes a magnetic field of its own. Drop it into somebody else’s field and the two disagree — and the wire gets shoved. Not along the current. Not along the field. Sideways, at right angles to both. It is the strangest force in the syllabus, and it is the reason every electric motor on Earth turns.
📘 What you need to know
A current-carrying conductor produces its own magnetic field. In an external field, it feels a force
F = BIL sin θ, where θ is the angle between the conductor and the field
B in teslas, I in amps, L in metres, F in newtons
L is the length of the conductor that lies inside the field — not the whole wire
Maximum force when θ = 90° (wire perpendicular to B), giving F = BIL
Zero force when θ = 0° (wire parallel to B). No exceptions
F, B and I are mutually perpendicular
Directions come from Fleming’s left-hand rule: thuMb = Motion (force), First finger = Field, seCond finger = Current
I is conventional current (+ to −), opposite to electron flow
In 3D: a dot means out of the page, a cross means into the page
Where the force comes from
Two magnetic fields, sharing the same bit of space. The wire’s own circular field wraps around it; the external field runs straight past. On one side of the wire the two reinforce, on the other they cancel. The field lines pile up on one side and thin out on the other — and the wire is pushed towards the thin side.
Current in a wire
makes its own magnetic field
Two fields interact
and the wire is pushed
F = BIL sin θ
Force on a current-carrying conductorF = BIL sin θB = flux density (T) • I = current (A) • L = length in the field (m)θ = angle between the conductor and the field
On the right the wire and the field are at 90°, sin θ = 1, and the force is as big as it can get. The force itself points out of the plane of the page — it is perpendicular to both.
Read that L definition twice: it is the length of the conductor inside the field. Hand a student a 30 cm wire dangling through a 12 cm magnet gap and most of them will cheerfully use 0.30 m. Only the 12 cm is being pushed. The rest of the wire is sitting in no field at all, and no field means no force.
How the force depends on the angle
Everything about this equation lives in the sine. Plot F against θ and you get one clean arch.
Parallel to the field (θ = 0° or 180°) and the force is zero. Perpendicular (θ = 90°) and it is maximum. Everything in between is just the sine.
Change
Effect on the force
Stronger field B
Force increases, F ∝ B
Bigger current I
Force increases, F ∝ I
More wire inside the field
Force increases, F ∝ L
Wire perpendicular (θ = 90°)
Maximum: F = BIL
Wire parallel (θ = 0°)
Zero. No force at all
Seeing it happen
Lay a loose copper rod across two rails inside a uniform field, and wire the rails into a circuit. Switch on the current and the rod rolls — it accelerates in the direction of the force. That is the whole of the electric motor, stripped to one moving part.
Which way does it push?
The force is perpendicular to both the current and the field. That still leaves two possibilities — and Fleming’s left-hand rule picks between them.
The three letters are hidden in the words: thuMb → Motion, First → Field, seCond → Current. Left hand for motors; there is a right-hand version, but that is for generators.
Use the rule in the exam. Actually put your left hand on the desk and twist it about. Nobody has ever lost a mark for looking slightly ridiculous, and thousands have lost marks for guessing. The one thing to be careful about: your second finger points along the conventional current. If the question gives you a beam of electrons, point that finger the other way.
Drawing in three dimensions
The force sticks out of the plane of the page, which means the page has to show three directions at once. Two symbols do the job.
Think of an arrow in flight. Coming towards you, all you see is the point — a dot. Flying away, all you see is the crossed feathers — a cross.
🧲 Working a force-on-a-wire question
Convert everything. mT → T, cm → m. The prefixes are where the marks go.
Which L? Only the length of wire inside the field. Read the question twice.
Find θ. It is between the wire and the field — not between the wire and anything else.
Perpendicular? sin 90° = 1, so F = BIL. Parallel?F = 0. Stop there.
Direction? Fleming’s left hand: thumb F, first finger B, second finger I.
Electron beam? The conventional current points the opposite way to the electrons.
WE 1
A straight wire of length 8.0 cm carries a current of 2.5 A in a uniform magnetic field of flux density 45 mT. Calculate the force on the wire when it is placed (a) at 90° to the field, (b) at 25° to the field, and (c) parallel to the field.
Step 1 — convert everything firstB = 45 mT = 45 × 10⁻³ T | L = 8.0 cm = 0.080 m | I = 2.5 A(a) perpendicular, so sin 90° = 1F = BIL = (45 × 10⁻³)(2.5)(0.080)F = 9.0 × 10⁻³ N = 9.0 mN(b) at 25°, multiply by sin 25°F = (9.0 × 10⁻³) × sin 25° = (9.0 × 10⁻³) × 0.4226F = 3.8 × 10⁻³ N = 3.8 mN(c) parallel, so θ = 0° and sin 0° = 0F = 0 NNotice you never needed to redo the whole calculation in (b) — the perpendicular answer is BIL, and everything else is just that number times sin θ. Work out BIL once, then scale it.
WE 2
A 30 cm length of wire carries a current of 4.0 A. It passes at right angles through the gap between two magnets, where the field is 0.15 T. Only 12 cm of the wire lies inside the field. (a) Calculate the force on the wire. (b) Determine the current needed to double the force.
(a) Step 1 — which length is L?
Only the part inside the field is pushed.
L = 12 cm = 0.12 m, not 0.30 mStep 2 — perpendicular, so F = BILF = (0.15)(4.0)(0.12)F = 0.072 N(b) Step 3 — F is proportional to Idouble F → double II = 8.0 AUse L = 0.30 m and you get 0.18 N — two and a half times too big, and no method marks left to save you. The wire outside the field has current flowing through it, but no field to push against.
WE 3
(a) A horizontal wire lies in a magnetic field directed into the page. The force on the wire is vertically downwards. State the direction of the current. (b) A different horizontal wire carries a conventional current from west to east, in a magnetic field directed vertically downwards. Determine the direction of the force. (c) State the direction of the force in (b) if the current is reversed.
(a) apply Fleming’s left-hand rule backwards
First finger (field) points into the page.
Thumb (force) points downwards.
The second finger is then forced to point to the left.
the current flows from right to left(b) set the hand up in three dimensions
Second finger (current) points east.
First finger (field) points vertically down.
the thumb points due north — the force is horizontal, to the north(c) reverse the current
Reversing I reverses F, and B is unchanged.
the force is now due southIn (b) the wire is horizontal and the field is vertical, so they are always at 90° and the force has its full value BIL. It is also horizontal — the force can never point along B, and it can never point along the wire.
💡 Top tips
L is the length in the field. Underline that phrase in the question.
Work out BIL once, then multiply by sin θ. Don’t restart the sum for each angle.
θ is measured between the wire and the field, and nothing else.
Perpendicular → maximum. Parallel → zero. Say which one applies before calculating.
Convert mT to T and cm to m before you touch the calculator.
Fleming’s left hand for the force on a current. The right-hand rule is for generators.
Second finger = conventional current. For electrons, point it backwards.
⚠ Common mistakes
Using the whole length of the wire instead of the part inside the field
Leaving B in mT or L in cm
Using cos θ. The force is maximum when perpendicular, so it must be a sine
Measuring θ from the wrong thing — it is wire to field
Using the right hand, or using the electron direction as I
Thinking the force points along the field, or along the wire. It is perpendicular to both
Swapping the symbols: a dot is out of the page, a cross is into it
Forgetting that a wire parallel to the field feels no force whatsoever
Quick recap: A current-carrying wire in an external magnetic field feels a force F = BIL sin θ, where L is the length of wire inside the field and θ is the angle between the wire and the field. It is maximum (F = BIL) when they are perpendicular, and zero when they are parallel. F, B and I are mutually perpendicular, and their directions come from Fleming’s left-hand rule: thuMb = Motion, First = Field, seCond = Current, using conventional current. In 3D, a dot is out of the page and a cross is into it.
Now put two current-carrying wires side by side. Each one makes a magnetic field. Each one sits in the other’s field. So each one feels a force — and by Newton’s third law those two forces must be equal and opposite. Run the currents the same way and the wires pull together; run them opposite and they push apart. It is such a clean effect that for over a century it was used to define the ampere. Next page: Magnetic Force between Two Parallel Conductors.
Fleming’s left hand tying your fingers in knots?
Book a free meeting and we’ll drill the angle, the length-in-the-field trap and the direction rule until they’re automatic.
