IB Physics SLTopic 4 — Force FieldsPaper 1 & 2F = Bqv sin θ~8 min read
Force on a Moving Charge
A current in a wire is really just charge on the move — so if a wire in a field feels a force, a single flying charge must feel one too. It does. Fire an electron or a proton across a magnetic field and the field grabs it, bending its path. This is the motor effect zoomed all the way in, down to one charge at a time.
📘 What you need to know
A charge moving through a magnetic field feels a force — a stationary charge feels none, and a charge moving along the field feels none either
The size of the force is F = Bqv sin θ, where B is flux density (T), q the charge (C), v the speed (m s−1), and θ the angle between the velocity and the field
The force is biggest when the charge moves perpendicular to the field (θ = 90°), giving F = Bqv
Force, field and velocity are mutually perpendicular — all three at right angles
Direction comes from Fleming’s left-hand rule, but the current is the flow of positive charge: for a positive charge, current runs with the velocity; for a negative charge (an electron), current runs the opposite way
The force always acts at right angles to the motion, so it never speeds the charge up — it only bends its path
Careful: F = Bqv is for a single moving charge; F = BIL is for a whole current-carrying wire — different tools
Why a Moving Charge Gets Pushed
On the wire pages, the force came from the moving charges inside the wire — the metal just carried them along. Strip the wire away and the physics stays: any charge in motion makes its own little magnetic field, and that field can’t sit quietly next to an applied field, so the charge gets shoved sideways.
Two conditions matter. First, the charge must be moving — a charge sitting still makes no magnetic field, so it feels no magnetic force. Second, it must move across the field lines: a charge drifting straight along the field slips between the lines without cutting them, and feels nothing. Everything in between is set by that sin θ.
How Big? F = Bqv sin θ
The equation looks almost identical to the wire’s F = BIL sin θ, which makes sense — we’ve just swapped “current × length” for “charge × speed”.
Force on a moving chargeF = Bqv sin θ
Reading it through: F is the force on the charge (N); B is the flux density (T); q is the size of the charge (C); v is its speed (m s−1); and θ is the angle between the velocity and the field. Line the motion up square-on to the field (θ = 90°, sin θ = 1) and the force hits its maximum, F = Bqv; line it up along the field (θ = 0°) and the force vanishes.
θ is the angle between the charge’s velocity (teal) and the field (blue). The force grows with sin θ — largest when the charge cuts straight across the field, zero when it drifts along it.
Which Way? The Positive-Charge Twist
Direction still comes from Fleming’s left-hand rule — thuMb = force, First finger = field, seCond finger = current. The catch is that the second finger wants the conventional current, the flow of positive charge. So you have to translate the charge’s motion into a current direction first:
A positive charge: current points the same way as the velocity.
A negative charge (an electron): current points the opposite way to the velocity.
Get that translation wrong and your force comes out backwards — it’s the single most common slip on this topic. The picture below shows why it matters: same velocity, same field, but the sign of the charge flips the force.
Same velocity (teal), same field (into the page), opposite charges. The orange current arrow follows the velocity for a positive charge but reverses for a negative one — so Fleming’s left-hand rule flips the force from up to down.
🧭 Finding the force on a moving charge
Check the charge is moving across the field. Still, or moving along the lines? Then the force is zero
Turn the motion into a current. Positive charge → current the same way as v; negative charge → current the opposite way
Apply Fleming’s left-hand rule: First finger = field, seCond finger = that current, thuMb = force
For a size, use F = Bqv sin θ (and sin θ = 1 when the motion is perpendicular to the field)
Remember the force is perpendicular to v — it changes the charge’s direction, never its speed
Quick recap: a charge moving across a field feels F = Bqv sin θ — biggest when perpendicular (F = Bqv), zero when parallel or stationary. Direction is Fleming’s left-hand rule, using the flow of positive charge (so an electron’s current runs opposite to its motion). The force stays perpendicular to the velocity, so it only ever bends the path.
WE 1
A proton (charge 1.60 × 10⁻¹⁹ C) travels at 8.0 × 10⁵ m s⁻¹ through a uniform magnetic field of flux density 0.35 T. (a) Calculate the force on the proton when it moves perpendicular to the field. (b) Calculate the force if it instead moves at 30° to the field, at the same speed.
Part (a) — perpendicular (θ = 90°, sin θ = 1)
B = 0.35 T, q = 1.60 × 10⁻¹⁹ C, v = 8.0 × 10⁵ m s⁻¹
F = Bqv = 0.35 × 1.60 × 10⁻¹⁹ × 8.0 × 10⁵F ≈ 4.5 × 10⁻¹⁴ NPart (b) — at 30° to the fieldF = Bqv sin 30° = 4.48 × 10⁻¹⁴ × 0.5F ≈ 2.2 × 10⁻¹⁴ NHalf the sin, so half the force — tilting the motion toward the field always weakens the push.
WE 2
An electron travels to the right across the page and enters a region where the magnetic field points into the page. (a) State the direction of the magnetic force on the electron. (b) State how the force would differ for a proton moving in exactly the same way.
Part (a) — the electron
Electron moves right, so its (conventional) current points left
Field into the page + current left → Fleming’s LHR gives…
force is downwardsPart (b) — the proton
A proton’s current points the same way as its motion (right), the opposite current to the electron
force is upwards (opposite to the electron)Same speed, same field — but flipping the sign of the charge flips the force.
💡 Top tips
Electrons run the current backwards. In Fleming’s left-hand rule the second finger is the flow of positive charge, so for an electron point it opposite to the motion
F = Bqv, not F = BIL. Use Bqv for a single moving charge and BIL for a whole current-carrying wire — a classic mix-up
sin θ uses the angle between v and B. Perpendicular motion (90°) is the maximum; motion along the field (0°) gives nothing
The magnetic force does no work. It’s always at right angles to the velocity, so it changes direction but never speed — handy as a sanity check
⚠ Common mistakes
Forgetting to reverse the current for a negative charge, so the force comes out upside-down
Using the wire equation F = BIL when the question is about a single particle (use Bqv)
Thinking a stationary charge, or one moving along the field, still feels a magnetic force — both give zero
Claiming the magnetic force speeds the charge up — it only ever bends the path, because it acts perpendicular to the motion
Up next: because that force stays perfectly perpendicular to the velocity, a charge fired into a uniform field doesn’t just bend — it loops all the way round into a circle. Next page we set F = Bqv equal to the centripetal force and pin down the radius of that path.
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