Electrons in a Magnetic Field

Charges moving in a magnetic field experience a force.  This force is given by the equation

\(\vec{F_B} = q \vec{v} \times \vec{B}.\)

For electrons in a magnetic field, the charge q is actually –e, so the equation becomes

\(\vec{F_B} = -e \vec{v} \times \vec{B}.\)

The force is found using the cross product (or vector product) of the velocity vector and the field vector.  By the nature of the cross product, the force will always be perpendicular to the velocity.  If the magnetic field is constant, and perpendicular to the velocity, the electrons will move in a circular path.  If the field has a component not perpendicular to the field, the electrons will follow a helical path along the field.  (Note that a helix has a constant radius, unlike a spiral.)

The classic e/m experiment uses this fact and relates the centripetal force caused by the magnetic force and the velocity caused by the accelerating potential that gives the electrons their initial speed.

Picture of electrons in a magnetic field

Electrons in a magnetic field moving in a helix.

Electrons in a magnetic field moving in a helix.

The bluish glow in the picture is caused by electrons interacting with a low pressure mercury vapor in a glass ball.  The orange glow in the foreground is caused by a glowing filament that provides the electrons.  The filament is surrounded by a metal cylinder held at a potential about 25 Volts higher than the filament.  This potential difference accelerates the electrons.  Most of them hit the cylinder, but some escape to interact with the mercury vapor.

The green rods in the back are coated with a material that fluoresces when hit by electrons.  In the experiment, one adjusts the the magnetic field so that the main electron beam hits each of the individual posts.  The posts are glowing because some of the electrons interacting with the vapor have lost enough energy that they follow a larger circular path.  The electron beam becomes fuzzy because of the interactions with the vapor.