12.2 Magnetism and Moving Charges

Any moving charge creates a magnetic field around it. The field gets weaker with distance and stronger with the charge's speed. Its direction is perpendicular to both the charge's velocity and the line from the charge to the point you're looking at, so it circles around the line of motion. It's strongest at points off to the side of the charge's path, and zero straight ahead or behind.

Use the right-hand rule: point your right thumb along the velocity of a positive charge, and your curled fingers show the way the field circles. For a negative charge, the field circles the other way. You only need to describe this field, not calculate it.

A charge q moving with speed v through a magnetic field B feels a force of size:

$$F_B = \lvert q \rvert vB\sin\theta$$

θ is the angle between the velocity and the field. The force is largest when the charge moves at right angles to the field (θ = 90°, so F = |q|vB) and zero when it moves along the field (θ = 0° or 180°). A charge at rest feels no magnetic force at all. On the exam, you'll calculate the force only for angles of 0°, 90° and 180°; other angles are described in words.

The force is perpendicular to both v and B. For a positive charge, use the right-hand rule: point your fingers along v, curl them toward B, and your thumb points along the force. (An equivalent version: flat right hand, fingers along B, thumb along v, and the force comes out of your palm.) For a negative charge, the force is opposite to what the rule gives.

Diagrams often show fields pointing into the page as rows of × marks (like the tail feathers of an arrow flying away from you) and out of the page as dots (the arrow's tip coming toward you).

Because the magnetic force is always perpendicular to the velocity, it does no work. The charge's speed and kinetic energy never change; only its direction does. If the charge moves at right angles to a uniform field, the force acts as a centripetal force and the charge moves in a circle. Setting |q|vB = mv²/r gives the radius:

$$r = \frac{mv}{\lvert q \rvert B}$$

Faster or heavier particles make bigger circles; a stronger field or a bigger charge makes tighter ones.

If a region has both an electric field and a magnetic field, the charge feels both forces independently, and you add them as vectors. When they point opposite ways and balance, qE = qvB, so only particles with speed v = E/B pass straight through. This setup is called a velocity selector.

When a current flows along a flat conductor in a magnetic field that's perpendicular to the current, the moving charges get pushed sideways toward one edge. Charge builds up there, and the opposite edge is left with the opposite charge. That creates a small potential difference across the width of the conductor, called the Hall voltage. Its sign shows whether the moving charges are positive or negative, and its size grows with the field, so Hall sensors are used to measure magnetic fields.