Current-generated fields, loops, and the Lorentz force
Moving charges and currents create magnetic fields. Unlike electric fields, magnetic field lines always form closed loops -- there are no magnetic monopoles. The force on a moving charge in a magnetic field is perpendicular to both the velocity and the field: F = qv × B, the Lorentz force.
A current-carrying wire creates a B-field that circles around it, following the right-hand rule. The strength falls off as B = μ₀I/(2πr).
Cross-section of a current-carrying wire. The magnetic field forms concentric circles obeying the right-hand rule. Arrow brightness indicates field strength, which falls off as 1/r from the wire.
Key insight: The circular field pattern around a wire is a direct consequence of Ampere's law. Reversing the current reverses the field direction.
A circular current loop creates a field pattern identical to a magnetic dipole -- just like a bar magnet. The field on the axis has a clean formula, and far away it falls off as 1/r³.
Field lines of a current loop resemble a bar magnet (magnetic dipole). The graph shows B along the axis, peaking at the center and falling off as 1/z³ far away.
Key insight: All magnetism in matter comes from current loops at the atomic scale (orbiting electrons, spin). There are no fundamental magnetic charges.
A charged particle in a uniform magnetic field follows a circular orbit. The Lorentz force F = qv × B is always perpendicular to the velocity, so it changes direction but not speed. The cyclotron radius r = mv/(qB) depends on mass, speed, charge, and field strength.
A charged particle in a perpendicular magnetic field follows a circular orbit (Lorentz force). Enable the E-field to see E×B drift: the particle drifts sideways while circling.
Key insight: The magnetic force does no work (it is always perpendicular to velocity). It steers particles without changing their energy. This is the principle behind cyclotrons, mass spectrometers, and the aurora.