Magnetic fields

Learning objectives

• Calculate forces on currents and moving charges.
• Describe flux linkage and the Hall-probe principle.
• Apply Faraday’s and Lenz’s laws.

The big idea

A magnetic field exerts a force on a moving charge with a velocity component across the field. The force is perpendicular to velocity, so a purely magnetic force changes direction rather than kinetic energy.

Flux through a flat area depends on the angle between field and area normal. Changing flux linkage induces an emf whose direction opposes the change causing it. This is consistent with energy conservation. Hall probes measure a transverse voltage caused by magnetic deflection of charge carriers.

1. Forces act across motion or current

For a moving charge, magnetic force magnitude is |q|vB sin θ. Its direction is perpendicular to both velocity and field, reversing when the charge sign reverses. For a straight conductor in a uniform field, use BIL sin θ. Parallel motion gives no magnetic force. A magnetic force alone changes a point charge’s direction without doing work, because it remains perpendicular to velocity.

2. Circular paths and Hall voltage

For perpendicular motion at constant speed, magnetic force can supply the radial resultant, giving r = mv/(|q|B). This assumes other forces are negligible. In a conductor, sideways deflection of carriers can establish a transverse electric field and Hall voltage. A simple one-carrier model gives Hall-voltage magnitude BI/(n|q|t), with t the thickness parallel to the magnetic field. Its polarity can reveal the carrier sign when geometry is known.

3. Flux is not the same as field strength

Flux through a flat surface in a uniform field is BA cos θ, where θ is measured to the surface normal. Changing field strength, loop area or orientation can therefore change flux. Flux linkage includes the number of turns. A strong unchanged field through a stationary unchanged coil does not by itself produce a continuous induced emf; the relevant quantity is the rate of change of linkage.

4. Induction and energy conservation

Faraday’s law relates induced emf to the negative rate of change of flux linkage. Lenz’s-law direction opposes the change producing it, rather than always opposing the field itself. Moving a magnet towards a coil can require mechanical work against the induced response. That work can become electrical energy and thermal energy. Reversing the motion reverses the induced response; simply naming north and south poles is not enough without specifying the change.

A method you can reuse

1. Identify field, current or charge-velocity directions.
2. Use the angle appropriate to force or to surface flux.
3. For a particle path, combine magnetic force with radial dynamics.
4. Calculate the change of flux linkage over the required time.
5. Use Lenz’s law and energy conservation to check the induced direction.

Equations and relationships

Investigate and explain

Move a magnet into and out of a simulated coil. Compare induced-current direction, motion speed and turn count.

Compare field geometry and induction as related experiments. These tools do not fully model Hall probes or high-energy particle detectors; use explicit carrier and force diagrams for those applications.

Open the interactive investigation

Check your understanding