Magnetism and Electromagnetism
Core concepts
Magnetic forces act only on moving charges and on current-carrying wires: $\vec F=q\vec v\times\vec B$ and $\vec F=I\vec L\times\vec B$, with magnitude $F=qvB\sin\theta$. The right-hand rule gives direction (thumb = $v$ or $I$, fingers = $B$, palm = force for positive charge). A charge moving parallel to $\vec B$ feels no force; perpendicular motion gives circular motion with radius $r=mv/(qB)$. Long straight wires create $B=\mu_0 I/(2\pi r)$; solenoids create $B=\mu_0 nI$. Faraday's law says induced emf equals the negative rate of change of magnetic flux: $\varepsilon=-N,d\Phi_B/dt$, with $\Phi_B=BA\cos\theta$. Lenz's law: induced current opposes the change in flux.
Traps to watch
- Magnetic force does no work because it is always perpendicular to velocity-it changes direction, not speed.
- Use the left hand for negative charges, or reverse the force direction.
- Flux depends on the angle between $\vec B$ and the area normal; maximum flux when $\vec B$ is perpendicular to the loop's plane.
- Lenz's law opposes the change, not the flux itself.
- A stationary charge in a magnetic field feels zero force.
Worked example
A proton ($m=1.67\times10^{-27},\mathrm{kg}$, $q=1.6\times10^{-19},\mathrm{C}$) moves at $2\times10^6,\mathrm{m/s}$ perpendicular to $B=0.5,\mathrm{T}$. Radius: $r=mv/(qB)=(1.67\times10^{-27})(2\times10^6)/[(1.6\times10^{-19})(0.5)]\approx 4.2\times10^{-2},\mathrm{m}$.
Rule of thumb: if velocity and field are parallel, force is zero; only the perpendicular component of $v$ bends the path.