Magnetic Force

The two-wire demonstration of Section 10.1 shows that a current-carrying wire experiences a force when placed in the magnetic field produced by a neighbouring current-carrying wire. To understand this force from first principles, it helps to go back to the most basic case: a single moving charge in an external magnetic field. Consider an electron, of charge $-e$, moving with velocity $\vec{v}$ through a region where a magnetic field $\vec{B}$ is present. Careful experiments (of exactly the sort suggested by Fig. 10.1) show that this charge experiences a magnetic force

and, more generally, for any charge $q$ (positive or negative),

If an electric field $\vec{E}$ is ALSO present at the same location, the total force on the charge is the sum of the electric and magnetic contributions,

a combined expression known as the Lorentz force. Here $\vec{F}_e=q\vec{E}$ is the familiar electrostatic force, and $\vec{F}_m=q(\vec{v}\times\vec{B})$ is the new, purely magnetic contribution this chapter is concerned with.

The cross product in $\vec{F}_m=q(\vec{v}\times\vec{B})$ carries several important physical consequences, all of which follow directly from the geometric properties of the cross product itself (Fig. 10.4). First, because a cross product of two vectors is always perpendicular to BOTH of them, $\vec{F}_m$ is always perpendicular to $\vec{v}$ -- which means $\vec{F}_m\cdot\vec{v}=0$ at every instant, for ANY magnetic field. A force that is always perpendicular to the velocity does no work on the particle (since work requires a component of force ALONG the direction of displacement); the magnetic force can change the DIRECTION a charged particle moves in, but it can never change its SPEED, and hence never changes its kinetic energy either. This single fact underlies the circular and helical motion studied in the next two sections, and is also the reason a charged particle's kinetic energy stays fixed as it circulates inside a cyclotron between accelerating pushes from the electric field. Second, if the charge's velocity happens to be exactly PARALLEL to $\vec{B}$, the magnetic force is zero (since the angle between $\vec{v}$ and $\vec{B}$ is then $0^\circ$, and $\sin0^\circ=0$); this is the key fact behind the parallel-velocity component of helical motion (Section 10.4). Third, if the charge itself is stationary ($\vec{v}=0$), the force is zero even in the presence of an arbitrarily strong magnetic field ($B\neq0$) -- a magnetic field, unlike an electric field, exerts NO force at all on a charge at rest. …