Magnetic flux density
Magnetic flux density B — The force per unit current per unit length on a conductor placed at right angles to the field. Measured in tesla, where 1 T = 1 N A⁻¹ m⁻¹.
At earlier levels a magnetic field is drawn with lines and left there. To calculate anything you need a number for how strong the field is, and that number is the magnetic flux density, B.
It is defined through the force it produces. Put a wire of length L carrying current I at right angles to a field, and the force on it is F = BIL. Rearranged, B = F/(IL) — which is what the unit N A⁻¹ m⁻¹ is saying.
The tesla is a large unit. The Earth's field is about 50 microtesla, a fridge magnet around 5 millitesla, and a hospital MRI scanner between 1.5 and 3 T. If a calculation gives you a field of hundreds of tesla, something has gone wrong.
When the wire is not perpendicular to the field, only the perpendicular component counts, so the force becomes F = BIL sin θ. A wire lying along the field feels no force at all — a fact worth checking your working against.
- F
- forceN
- B
- flux densityT
- I
- currentA
- L
- length in the fieldm
Flux density is really a statement about how crowded these lines are. Bring the poles closer and the lines bunch — the field between them strengthens, and so would the force on any current placed there.
The direction of the force
The force is perpendicular to both the current and the field, which is why it needs a three-dimensional rule rather than a diagram in the plane of the page.
Fleming's left-hand rule gives it. Hold the thumb and first two fingers of the left hand mutually at right angles: First finger along the Field (north to south), seCond finger along the Current (conventional), and the thuMb then points along the Motion.
Reversing either the current or the field reverses the force. Reversing both leaves it unchanged — a common exam question, and one you can answer by applying the rule twice rather than memorising the result.
Left hand for the force on a current; right hand for the current induced by motion. Using the wrong hand reverses every answer in the question, so it is worth saying "left for motor" under your breath each time.
Left hand, right hand
Left hand — the motor effect: a current in a field produces motion. Right hand — the generator effect: motion in a field produces a current. Fleming named them so the pair could be told apart; using one for both is the single most common error in this half of the syllabus.
The force on a moving charge
A current is charge in motion, so a single moving charge in a magnetic field feels a force too. Following the definition through gives F = qvB sin θ.
This force has an unusual property: it is always perpendicular to the velocity. A force at right angles to the motion changes direction but never speed — so a magnetic field can bend a charged particle's path but can never do work on it or speed it up.
A charge entering a uniform field at right angles therefore travels in a circle, with the magnetic force acting as the centripetal force. Setting qvB = mv²/r and rearranging gives the radius, and that relation is what mass spectrometers and particle accelerators are built on.
It also explains the aurora. Charged particles from the Sun spiral along the Earth's field lines and are funnelled towards the poles, which is why the lights appear there and not over the equator.
- q
- chargeC
- v
- speedm s⁻¹
- r
- radius of the circular pathm
- m
- masskg
An electron of mass 9.11 × 10⁻³¹ kg and charge 1.60 × 10⁻¹⁹ C enters a uniform field of 0.012 T at right angles, moving at 3.0 × 10⁶ m s⁻¹. Find the force on it and the radius of its path.
- The velocity is perpendicular to the field, so
sin θ = 1.Always check the angle before dropping the sine. F = qvB = 1.60 × 10⁻¹⁹ × 3.0 × 10⁶ × 0.012.F = 5.8 × 10⁻¹⁵ N.Tiny in newtons, but the electron is tiny too.- This force is centripetal, so
qvB = mv²/r.The magnetic force does no work — it only turns the particle. r = mv/(qB) = (9.11 × 10⁻³¹ × 3.0 × 10⁶) / (1.60 × 10⁻¹⁹ × 0.012).r = 1.4 × 10⁻³ m, about 1.4 mm.A tight circle — which is why bubble-chamber tracks curl so sharply.
F = 5.8 × 10⁻¹⁵ N, r = 1.4 mm
Fields made by currents
Every current makes a magnetic field of its own, and three arrangements matter.
Around a long straight wire the field lines are concentric circles, weakening with distance. The right-hand grip rule gives the direction: thumb along the conventional current, curled fingers along the field.
A flat coil concentrates that field through its centre, and more turns give a stronger field.
A solenoid — a long coil — produces a field very like a bar magnet's: nearly uniform inside, spreading out at the ends. Which end is north again comes from a grip rule: curl the right fingers the way the current goes round, and the thumb points to north.
Two parallel wires therefore exert forces on each other, because each sits in the other's field. Currents in the same direction attract; opposite currents repel — the reverse of what most people guess.
Key points
F = BIL sin θ— zero when the wire lies along the field.- Left hand for the motor effect, right hand for induction.
F = qvBis always perpendicular to v, so it turns a charge without speeding it up.r = mv/(qB)— faster or heavier means a wider circle.- Parallel currents in the same direction attract.