Magnetic field
A magnetic field acts on moving charge, not on stationary charge simply because the charge exists. Its force is sideways to the motion, which is why a uniform magnetic field can bend a charged particle's path without directly changing its speed.
Force on a moving charge
F = |q|vB sin θ
Direction: perpendicular to v and B; use the right-hand rule for a positive charge and reverse it for a negative charge.
Force magnitude versus angle
For a moving charge, F = |q|vB sin θ.
When v is parallel or antiparallel to B, sin θ = 0 and the magnetic force vanishes. At 90°, it is maximal.
Why magnetic force bends rather than speeds up
Because the magnetic force is perpendicular to instantaneous velocity, it does no work on an isolated point charge. In a uniform field with v perpendicular to B, the particle follows a circular path with r = mv/(|q|B).
If the velocity also has a component parallel to B, the perpendicular part curves while the parallel part continues, producing a helix.
Currents feel magnetic force too
F = ILB sin θ for a straight wire of length L in a uniform field.
The force direction follows the same cross-product geometry as for individual moving charges. This effect underlies motors and loudspeaker coils.
Currents also create magnetic fields
For a long straight wire, B = μ0I/(2πr). The field direction circles the wire. A current loop produces a dipole-like field with a north and south side, and many loops in a solenoid can create a strong, nearly uniform internal field.
Field lines are a visualization: B is tangent to the line at each point and stronger regions are drawn more densely. Unlike electric field lines, magnetic field lines form closed loops; isolated magnetic monopoles have not been observed.
Direction rules without memorizing a picture
For a positive charge, choose the direction of v first, then B, and use the right-hand cross-product rule to obtain the force. For a negative charge, reverse that result. The force is always perpendicular to the plane defined by v and B, so an answer parallel to either vector signals a geometry error.
Magnetic fields can redirect kinetic motion without changing kinetic energy, but magnetic devices can still transfer energy when electric fields, induced emfs or moving conductors are part of the full system. Saying “magnetic fields never do anything energetic” is therefore too broad; the precise statement is that the magnetic part qv × B does no work on an isolated point charge.
Worked examples
1. Force on a moving proton
A proton moves perpendicular to a 0.20 T field at 3.0 × 106 m s−1. Find the force magnitude.
Solution
F = qvB = (1.602 × 10−19)(3.0 × 106)(0.20) ≈ 9.61 × 10−14 N.
2. Particle path radius
An electron of speed 2.0 × 106 m s−1 enters a 0.010 T field perpendicular to B. Estimate its circular radius.
Solution
r = mv/(|q|B) = (9.11 × 10−31)(2.0 × 106)/[(1.602 × 10−19)(0.010)] ≈ 1.14 × 10−3 m.
3. Field of a straight wire
Find B 0.050 m from a long straight wire carrying 8.0 A.
Solution
B = μ0I/(2πr) = (4π × 10−7)(8.0)/(2π × 0.050) = 3.2 × 10−5 T.