The Magnetic Force

A charge at rest beside a current-carrying wire feels nothing. Set it moving and it is pushed. That speed-dependent push reveals a new field: the magnetic field.

A field that only acts on moving charges#

A wire carrying current contains equal amounts of positive and negative charge in any small volume, so it is electrically neutral and makes (almost) no electric field. A charge q held at rest near it feels no force. But throw a positive charge parallel to the current and it curves toward the wire. Something there acts on moving charges only: the magnetic field B⃗, and the push it gives is the magnetic force.

Defining the field B#

Fire a charge through a point P in different directions and at different speeds. You find:

1. Along one particular line through P there is no force at all. B⃗ is defined to lie along this line.
2. Moving at angle θ to that line, the force is proportional to v sin θ.
3. The force is perpendicular both to the field and to the velocity.
4. The force is proportional to q and reverses for a negative charge.

All four are captured by one vector product, which also fixes the sense of B⃗:

The SI unit of B is the newton per ampere metre, called the tesla (T), the same as the weber per square metre. The gauss is also used: 1 T = 10⁴ G. A tesla is large: the Earth’s field is about 10⁻⁵ T, and laboratory superconducting magnets reach about 10 T. (B used to be called “magnetic induction”.)

Electric and magnetic fields are one thing#

Observer S sees a charge moving at v beside a wire: no electric field, but a magnetic force pulls the charge toward the wire. Observer S₁ moves along with the charge, so for S₁ the charge is at rest and no magnetic force can act on it. Yet S₁ must see the same attraction (the two observers don’t accelerate relative to each other, so they agree on accelerations). So S₁ must see an electric field.