a region of space in which a magnetic pole, a current-carrying conductor or a moving charge experiences a force.
Magnetic field line
a line showing the direction of the force on a north pole placed at that point.
Key points
Magnetic fields are produced by permanent magnets and by moving charges (electric currents).
Field lines run from N to S outside a magnet, never cross, and are closest where the field is strongest.
Evenly spaced parallel lines = a uniform field (e.g. between flat poles, inside a solenoid).
Field into the page is drawn as crosses (×); out of the page as dots (·).
Like poles repel, unlike poles attract; a stationary charge feels no magnetic force.
Bar magnet: lines leave N, curve round and enter S; they are most crowded at the poles.
Uniform field into the page (crosses) and out of the page (dots).
Watch out!
A magnetic force acts only on a moving charge or a current. A charge at rest in a magnetic field feels no magnetic force.
Field lines never cross; if two lines appear to meet, the diagram is wrong.
20.2Force on a current-carrying conductor
Definitions
Magnetic flux density (B)
the force per unit current per unit length on a straight conductor placed at right angles to the field.
Tesla (T)
the flux density when a conductor of length 1 m carrying 1 A at right angles to the field experiences a force of 1 N (1 T = 1 N A-1 m-1).
Fleming's left-hand rule
with thumb, first finger and second finger mutually at right angles: first finger = field, second finger = current, thumb = force (motion).
Key points
A current in a magnetic field feels a force perpendicular to both B and I (the motor effect).
Force is maximum when the wire is perpendicular to B and zero when parallel.
Reversing either the current or the field reverses the force.
Measuring B with a current balance: wire fixed over magnets on a top-pan balance; F = change in reading (kg) × g.
Plot F against I: straight line through origin, gradient = BL.
A Hall probe (see 20.3) also measures B directly once calibrated.
Motor effect: B from N to S, current out of the page, so by Fleming's left-hand rule F is upwards.
Current balance: the force on the fixed wire pushes the magnets down (or up), changing the reading; F = Δm g, then B = F/(IL).
Watch out!
θ is the angle between the wire and the field. Parallel (θ = 0) gives zero force.
Use only the length of wire actually inside the field for L.
20.3Force on a moving charge
Definitions
Hall voltage
the potential difference produced across a current-carrying conductor (or semiconductor slab) in a magnetic field, perpendicular to both the current and the field.
Key points
Force on a moving charge is perpendicular to its velocity, so it does no work: speed and kinetic energy stay constant.
In a uniform field perpendicular to v the charge moves in a circle; BQv provides the centripetal force.
Faster or heavier particles make bigger circles; a stronger B makes smaller ones.
In the left-hand rule, a moving electron counts as a current in the opposite direction.
Hall effect: charges pile up on one face until the electric force balances BQv; VH is much larger in semiconductors (small n).
Velocity selector: crossed E and B fields; only particles with v = E/B pass undeflected (independent of mass and charge).
Charge-to-mass ratio: measure r in a known B at a known speed, then Q/m = v/(Br).
Positive charge in B into the page: the force always points to the centre, giving a circle of radius r = mv/(BQ).
Hall effect: carriers are pushed towards the front or back face, so VH appears between them (thickness t measured along B).
Velocity selector: electric force EQ balances magnetic force BQv only for v = E/B.
Watch out!
The magnetic force never changes the speed; it only changes direction. Kinetic energy is constant.
For electrons, reverse the direction given by the left-hand rule (conventional current is opposite).
In VH = BI/(ntq), t is the slab thickness along B, not the width across which VH is measured.
20.4Magnetic fields due to currents
Definitions
Right-hand grip rule
grip the wire with the thumb along the conventional current; the curled fingers give the direction of the field lines.
Solenoid
a long coil of wire; when it carries a current its field is uniform inside and like that of a bar magnet outside.
Key points
Long straight wire: concentric circles, further apart with distance (field weakens with distance).
Flat circular coil: field lines through the centre are nearly straight, like a short bar magnet.
Solenoid: uniform, parallel field inside; bar-magnet pattern outside. Grip rule on the coil: thumb points to N.
A ferrous (soft iron) core greatly increases the flux density without changing the field shape.
Field strength increases with current and turns per unit length.
Two parallel wires: each sits in the other's field, so they exert equal and opposite forces (Newton's third law).
Currents in the same direction attract; opposite directions repel.
Long straight wire, current out of the page: anticlockwise concentric circles, spacing grows with distance.
Solenoid: parallel, uniform field inside; N at the end the lines leave.
Parallel currents in the same direction: the field between them partly cancels, so the wires attract.
Watch out!
Parallel currents in the same direction attract: the opposite of like charges repelling.
The forces on the two wires are equal and opposite even if the currents are different.
20.5Electromagnetic induction
Definitions
Magnetic flux (Φ)
the product of the magnetic flux density and the area normal to the field: Φ = BA.
Weber (Wb)
the flux when a flux density of 1 T passes normally through an area of 1 m2 (1 Wb = 1 T m2).
Magnetic flux linkage
the product of the magnetic flux and the number of turns: NΦ.
Faraday's law
the magnitude of the induced e.m.f. is proportional to (equal to) the rate of change of magnetic flux linkage.
Lenz's law
the direction of the induced e.m.f. is such that it tends to produce effects that oppose the change causing it.
Key points
An e.m.f. is induced whenever the flux linkage changes: moving a magnet or coil, rotating a coil, or changing a current.
Bigger e.m.f.: faster movement, stronger magnet, more turns, larger area.
Straight rod moving at speed v across B: rate of cutting flux = BLv, so E = BLv.
Lenz's law is conservation of energy: work must be done against the opposing force to generate electrical energy.
Demonstrations: magnet into coil with a galvanometer; reverse direction reverses deflection; no motion, no reading.
The minus sign in Faraday's law expresses Lenz's law.
An e.m.f. is induced even in an open circuit; a current flows only if the circuit is complete.
Lenz's law: as N approaches, the coil's near face becomes N to repel it, so work is done against the motion.
Rod moving along rails across B: it sweeps area Lv per second, so E = BLv; the induced current feels a force opposing the motion.
Flux linkage against time and the induced e.m.f.: E is minus the gradient of the NΦ-t graph.
Watch out!
It is the rate of change of flux linkage that induces an e.m.f. A steady large flux induces nothing.
Flux uses the area normal to B: a coil turned edge-on to the field has zero flux through it.
Do not forget N: flux linkage NΦ, not Φ, goes into Faraday's law for a coil.