Electricity and magnetism

Magnetic field and magnetic force

A magnetic field describes magnetic action on magnets and currents; a current-carrying wire experiences F = BIl sin θ.

F=BIl\sin\theta,\qquad \vec F=I\vec l\times\vec B

Definition: Magnetic flux density B

Magnetic flux density $B$ (tesla, T) specifies field strength and direction. A straight wire segment of length $l$ carrying current $I$ at angle $\theta$ to $\vec B$ feels $F=BIl\sin\theta$ . The force is perpendicular to both wire and field; use the right-hand cross-product rule for conventional current.

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Reading the model

Example: Worked example

A 0.20 m wire carries 3.0 A perpendicular to a 0.50 T field. Find the force.

Solution

Substitution gives $F=0.50\times3.0\times0.20=0.30$ N.

The force direction on a current-carrying wire follows the left-hand rule or the cross product of current and magnetic field. Reversing either the current or field reverses the force; reversing both leaves it unchanged. Two parallel current-carrying wires each lie in the field produced by the other, so they attract when their currents run in the same direction and repel when the currents run oppositely.

The force law can be used to measure $B$ : for a perpendicular wire, $B=F/(Il)$ , with tesla equivalent to $\mathrm{N/(A\,m)}$ . The magnetic force acts only on the portion of wire inside the field; do not include wire length outside that region.

Magnetic field lines visualize the field: their tangent gives the direction of $\vec B$ , while line density indicates relative strength. They form closed loops with no isolated beginning or end.

Quick check

What is the magnetic force on a wire parallel to $\vec B$ ?

If $B$ doubles while current, wire length and angle stay fixed, the force:

References

Young, Freedman (2019). University Physics