Magnetic fields and induction

Revision notes

Magnetic flux density B quantifies magnetic-field strength in force relationships. For a straight conductor of length L carrying current I perpendicular to a uniform field, F = BIL. A moving charge q with velocity v perpendicular to a field experiences magnetic force F = Bqv. The magnetic force is always perpendicular to the instantaneous velocity, so it can change the direction of motion without doing work on the particle. In a uniform perpendicular field this can produce circular motion, with magnetic force providing the centripetal force.

Magnetic flux through a flat area can be written Φ = BA cos θ when θ is the angle between B and the normal to the area. For a coil of N turns, flux linkage is NΦ. An induced emf is produced when flux linkage changes. Faraday's law relates emf magnitude to the rate of change of flux linkage, while the minus sign in ε = -d\(\dfrac{N\Phi }{dt}\) represents Lenz's law: the induced effect opposes the change that produced it.

Induction can be produced by changing field strength, changing coil area or orientation, or relative motion between magnet and conductor. Faster change gives larger emf. Generators exploit changing flux linkage to transfer mechanical energy to electrical energy. The opposing nature of induced currents is consistent with energy conservation; induction does not provide free energy.

Force on currents and charges

A current-carrying conductor in a magnetic field can experience a force. For a straight conductor perpendicular to a uniform field, F = BIL. More generally the perpendicular component matters. The direction is perpendicular to both current and magnetic field.

A moving charge in a magnetic field experiences a force qvB sinθ. Because the magnetic force is perpendicular to velocity, it changes direction of motion rather than doing work on the particle in the ideal case. This can produce circular motion when velocity is perpendicular to the field.

Magnetic flux

Magnetic flux through area A is Φ = BA cosθ when θ is the angle between the field and the area normal. A coil of N turns has flux linkage NΦ. Be careful with angle definitions: using the angle to the plane instead of the normal changes the trigonometric function.

Faraday's law

An emf is induced when flux linkage changes. The magnitude is proportional to the rate of change of flux linkage, |ε| = |d\(\dfrac{N\Phi }{dt}\)|. Flux linkage can change because B changes, area changes, orientation changes or the conductor moves through the field.

A large flux by itself does not imply an emf; the flux linkage must be changing.

Lenz's law and energy conservation

The induced current acts in a direction that opposes the change producing it. This is Lenz's law and reflects energy conservation. For example, when a magnet is pushed into a coil, the induced magnetic effect resists the motion, so mechanical work is required to generate electrical energy.

Motional emf

A conductor moving through a magnetic field can develop an emf because charges experience magnetic force and separate. For a rod of length l moving at speed v perpendicular to a field B, the induced emf is ε = Blv.

Exam technique

Separate magnetic force, flux and induced emf. Draw the area normal before using cosθ. For induction, state what quantity changes and therefore why flux linkage changes. Use Lenz's law to determine direction only after identifying the original change.

Direction of forces and charged-particle paths

Fleming's left-hand rule gives the direction of the force on a current: first finger along the field, second finger along the current, thumb along the force.

For a charged particle moving perpendicular to a uniform field, Bqv = \(\dfrac{mv^{2}}{r}\), so r = \(\dfrac{mv}{Bq}\).

In a cyclotron the orbital period \(\dfrac{2\pi m}{Bq}\) does not depend on speed, so an alternating p.d. at a fixed frequency can keep accelerating the particles.

Generators and alternating current

Electricity is transmitted at high voltage so the current, and the I²R loss in the cables, is small.

Solenoids and the Hall probe

Inside a long solenoid the field is uniform and proportional to the current and the number of turns per unit length.

A Hall probe measures magnetic flux density: a small p.d. across a semiconductor slice is proportional to B when the current through it is fixed.