Magnetic Fields and Electromagnetism
Core concepts
A moving charge experiences (\vec F = q\vec v \times \vec B), and a current-carrying wire experiences (\vec F = I\vec L \times \vec B). The force is always perpendicular to both velocity and field, so it does no work and changes direction, not speed. Circular motion in a uniform field has radius (r = mv/(|q|B)).
Currents create fields via the Biot–Savart law and Ampère's law, (\oint \vec B\cdot d\vec s = \mu_0 I_{\text{enc}}). Useful results: long straight wire (B = \mu_0 I/(2\pi r)); center of a loop (B = \mu_0 I/(2R)); solenoid (B = \mu_0 nI). The right-hand rule gives field direction around a wire and force direction on a moving charge.
Traps
- Sign of charge flips the force direction; a negative charge curves the opposite way.
- A charge moving parallel to (\vec B) feels no force; perpendicular motion gives circular paths, angled motion gives helices.
- Forgetting that Ampère's law needs symmetry, just like Gauss's law.
- Mixing up the right-hand rule for force (fingers along v, curl to B, thumb along F) versus field around a wire.
- Assuming magnetic forces do work; they don't, so kinetic energy is constant in a pure magnetic field.
Worked example
A proton moves at (2\times10^6,\text{m/s}) perpendicular to a (0.5,\text{T}) field. Force magnitude: (F = qvB = (1.6\times10^{-19})(2\times10^6)(0.5) \approx 1.6\times10^{-13},\text{N}). Radius: (r = mv/(qB)). Rule of thumb: compute the force magnitude with (|q|vB\sin\theta), then use the right-hand rule for direction, reversing it for negative charges.