AP Physics C: E&M Unit 12 Cheatsheet: Magnetic Fields and Electromagnetism

What is a magnetic field?

• Vector field that exerts force on moving charges, currents, and magnetic materials
• No magnetic monopoles have been observed; magnetic fields are produced by moving charges/currents (including magnetic dipoles) and by changing electric fields.
• Field lines form closed loops (Gauss's law for magnetism: ∮ B · dA = 0)
• External field points away from north pole, returns to south pole

Magnetic dipoles & alignment

• Arise from current loops (charges moving in closed paths) or from intrinsic magnetic moments of particles (e.g., electron spin, which is not literal rotation)
• Same polarity repels; opposite polarity attracts
• Compass needle aligns with external field
• Breaking a magnet creates two new dipoles (no isolated poles)
• Field magnitude decreases with distance

Field line patterns

• Always form closed loops (no beginning or end)
• Density indicates field strength
• Direction: away from north pole, into south pole (external)
• Inside a bar magnet: south to north

Concentric circles around wire

• Field vectors tangent to circles centered on wire
• Perpendicular to wire and to radial direction
• Right-hand rule: thumb along current, fingers curl in field direction

Solenoid field

• Uniform inside (parallel lines)
• Negligible outside (assumed)
• Strength depends on turns per unit length (n) and current (I)

Magnetic field vs. force

• Field B exists independently; force requires moving charge or current
• Stationary charge feels no magnetic force

Biot-Savart vs. Ampère

• Biot-Savart: Integrates over current elements; works for any geometry
• Ampère: Uses symmetry; faster for symmetric cases (wire, solenoid, slab)

Closed loops requirement

• Magnetic field lines never start or end (unlike electric field)
• Consequence: no magnetic monopoles
• Breaking magnet creates two dipoles, not isolated poles

Right-hand rule variants

• Current/velocity → field: thumb along current, fingers curl
• Cross product v × B: fingers point along v, curl toward B, thumb shows result