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Magnetic Fields

20. Magnetic Fields

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20. Magnetic Fields — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Describe magnetic fields and the forces they exert on currents and moving charges, explain the Hall effect and velocity selectors, and apply Faraday's and Lenz'…

  • Magnetic field (B-field)
  • Detecting fields
  • Field lines
  • Earth's field
  • Magnetic flux density B
  • Force on a conductor

Part 1 of 10Magnetic fields and field lines

  • Magnetic field (B-field) = a field of force created by moving electric charges or permanent magnets; a stationary charge produces no magnetic field.
  • Detecting fields — they pull on magnetic materials such as iron and deflect a plotting compass, and iron filings map the pattern.
  • Field lines — arrows leave a north pole and enter a south pole, showing the direction of the force on a free north pole.
    • Lines closer together mean a stronger field, strongest at the poles, and lines never cross.
    • Like poles repel and unlike poles attract.
    • A uniform field has equally spaced parallel lines, for example between two opposite poles held close.
  • Earth's field — it acts like a bar magnet, and a compass needle points to geographic north because that is near the Earth's magnetic south pole.

Part 2 of 10Flux density and force on a current-carrying conductor

1Magnetic flux density B3 points
= the force exerted per unit current per unit length on a straight conductor perpendicular to the field.
  • The unit is the tesla (T): 1 T gives a force of 1 N on a 1 m wire carrying 1 A at right angles to the field.
  • The flux density is higher where the field lines are closer together.
  • The Earth's field is about 0.032 mT and a fridge magnet about 5 mT.
2Force on a conductor3 points
— F = BIL sin θ, where θ is the angle between the conductor and the field and L the length in the field.
  • The maximum force is F = BIL when the conductor is perpendicular to the field (θ = 90°), with F, B and I mutually perpendicular.
  • The force is zero when the conductor is parallel to the field (θ = 0°).
  • The force increases with larger B, I or length in the field.
3Conventional current
— it runs from positive to negative, opposite to electron flow.

Part 3 of 10Fleming's left-hand rule

  • Handhold the thumb, first finger and second finger of the left hand at right angles to each other.
  • First fingerthe direction of the magnetic Field B.
  • Second fingerthe direction of conventional Current I (flow of positive charge).
  • Thumbthe direction of Motion, the force F on the conductor.
  • Page symbolsa dot (⊙) means a field out of the page, like the tip of an arrow approaching, and a cross (⊗) means into the page, like the feathers receding.
  • Generatorsthe right-hand rule is for generators and is different.

Part 4 of 10Force on a moving charge and circular motion

  • Equation — F = BQv sin θ, with θ the angle between the velocity and the field.
    • The maximum force is BQv when the particle moves perpendicular to the field.
    • There is no force when it moves parallel to the field.
  • Direction — use Fleming's left-hand rule with the second finger along the direction of conventional current.
    • For a positive charge that is the direction of motion, and for a negative charge (electrons) it is opposite to the motion.
  • Distinction — BQv is for an isolated moving charge and BIL is for a current in a wire.
  • Why a circle — the force is perpendicular to the velocity, so it changes direction but not speed, and acts as the centripetal force.
  • Radius — equating BQv with mv2/r gives r = mv/(BQ).
ChangeEffect on radius r
Larger speed vLarger circle (r ∝ v)
Larger mass mLarger circle (r ∝ m)
Larger charge QSmaller circle (r ∝ 1/Q)
Stronger field BSmaller circle (r ∝ 1/B)
  • Acceleration — the magnetic force causes the centripetal acceleration a = F/m.

Part 5 of 10The Hall effect

1Hall effect
— when a field is applied perpendicular to the current in a conductor, the moving charge carriers are pushed to one side, so one side becomes negative and the other positive.
2Hall voltage VH3 points
= the p.d. across the conductor produced by the perpendicular field, VH = BI/(ntq).
  • n is the number density of carriers (m−3), t the thickness and q the charge of each carrier.
  • A smaller n gives a larger VH, so semiconductors are used for Hall probes.
  • With positive carriers in place of electrons (same current direction), the Hall voltage reverses.
3Derivation2 points
— the two sides act as oppositely charged plates with E = VH/d.
  • In equilibrium the electric and magnetic forces on a carrier are equal and opposite, qE = Bqv, so VH/d = Bv.
  • With I = nAvq and A = dt, v = I/(nAq) gives VH = BI/(ntq).
4Hall probe3 points
— a thin slice with a flat end, connected to a voltmeter, used to measure flux density.
  • It is held with the field perpendicular to the flat surface for a maximum reading.
  • Tilting reduces the reading, which is zero when the field is parallel to the surface.
  • VH is directly proportional to B, so after calibration it measures B, even the Earth's field.

Part 6 of 10Velocity selector

  • Velocity selector perpendicular electric and magnetic fields in a vacuum, between oppositely charged plates, that pass only charged particles of one speed undeflected; it is used in mass spectrometers.
  • Balancethe electric force EQ and magnetic force BQv are equal and opposite at the chosen speed, so EQ = BQv.
  • Selected speedv = E/B, independent of the charge.
  • Other speedsfaster or slower particles are deflected by the unbalanced force and hit a plate.
  • Gravityit is negligible compared with the electric and magnetic forces.

Part 7 of 10Fields due to currents

1Straight wire2 points
— concentric circular field lines centred on the wire, closer near the wire where the field is strongest.
  • Reversing the current reverses the field.
  • Right-hand grip rule: thumb along the conventional current, curled fingers give the field direction.
2Solenoid2 points
— a long coil gives a field like a bar magnet, with lines leaving the north pole and re-entering at the south pole.
  • For a coil, the curled fingers follow the current and the thumb gives the field inside, towards the north pole.
  • The end where the current appears anticlockwise is the north pole.
3Strengthening
— add more turns and a soft iron core, which can increase the field several hundred times.
4Parallel currents2 points
— each wire is in the field of the other, so each feels a force.
  • Currents in the same direction attract and in opposite directions repel.
  • The two forces are equal and opposite, as Newton's third law requires.

Part 8 of 10Magnetic flux and flux linkage

  • Magnetic flux Φ = the product of flux density and the area perpendicular to the field, Φ = BA, in webers (Wb).
    What the symbols mean: Φ = magnetic flux (Wb) · B = magnetic flux density (T) · A = area at right angles to the field (m2)
  • Angled area — Φ = BA cos θ, where θ is the angle between the field lines and the normal to the area.
    • The flux is maximum (BA) when the field is perpendicular to the plane (θ = 0°) and zero when it is parallel (θ = 90°).
  • Flux linkage = the flux multiplied by the number of turns, NΦ = BAN cos θ, in Wb turns.
  • Changing flux — an e.m.f. is induced when B, A or the angle changes with time.

Part 9 of 10Electromagnetic induction

1Induction1 point
— an e.m.f. is induced when a conductor cuts magnetic field lines or the field around it changes.
  • Work is done on the charges in the conductor, separating them.
2Magnet and coil3 points
— a stationary magnet gives a zero reading.
  • Pushing the magnet in gives a reading and pulling it out gives the opposite sign.
  • Faster movement, more turns and a stronger magnet give a larger e.m.f.
  • The e.m.f. exists only while the magnet moves, because only then does the flux change.
3Wire between magnets1 point
— moving a wire across the field induces an e.m.f., opposite on the return.
  • It is larger with a longer wire, faster movement and stronger magnets.

Part 10 of 10Faraday's and Lenz's laws

  • Faraday's law = the magnitude of the induced e.m.f. is directly proportional to the rate of change of flux linkage, ε = NΔΦ/Δt.
    What the symbols mean: ε = induced e.m.f. (V) · N = number of turns · ΔΦ = change of flux (Wb) · Δt = time taken (s); the minus sign is Lenz’s law
  • Lenz's law = the induced e.m.f. acts in a direction that opposes the change causing it, so ε = −NΔΦ/Δt.
  • Verifying Lenz's law — push a known pole of a bar magnet into a coil connected to a sensitive ammeter, with no cell.
    • The ammeter shows a current, and the induced field repels the approaching magnet.
    • Reversing the magnet reverses the deflection.
    • Use the right-hand grip rule on the solenoid, with fingers along the current and thumb along the induced field.
  • Alternating supply — with an a.c. supply the induced e.m.f. alternates at the same frequency.

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