Magnetism and Electromagnetism

Magnetic fields come from magnetic dipoles, which always have a north and a south pole; break a magnet in half and you get two smaller magnets. Field lines form closed loops, leaving the north end and returning to the south, and a compass lines up with the field. Iron and other ferromagnetic materials can become permanent magnets when their domains line up, paramagnetic materials respond weakly, and all materials are slightly diamagnetic.

A magnetic field exerts a force on a charge only while it moves: the size is $F_B = qvB\sin\theta$, and the right-hand rule gives a direction at right angles to both the velocity and the field. Because the force is always sideways, a charge moving at right angles to a uniform field travels in a circle. A moving charge also makes its own magnetic field, and the Hall effect, a sideways potential difference across a current-carrying conductor in a magnetic field, comes from this force.

A current in a long straight wire makes a magnetic field that circles the wire, with strength $B = \frac{\mu_0 I}{2\pi r}$ and direction from the right-hand rule. A wire carrying current through a magnetic field feels a force $F_B = I\ell B\sin\theta$, so two parallel wires attract when their currents point the same way and repel when they point opposite ways.

Magnetic flux measures how much magnetic field passes straight through an area, $\Phi_B = BA\cos\theta$. Faraday's law says a changing flux induces an emf, $\varepsilon = -\frac{\Delta\Phi_B}{\Delta t}$, and Lenz's law, the minus sign, says the induced current makes a field that opposes the change. A classic example is a rod sliding along rails in a magnetic field, which produces $\varepsilon = B\ell v$.