The tube. The electron gun accelerates electrons through the voltage V, so each gains kinetic energy eV. They enter the space between two plates (length 10 cm, gap 4 cm) and travel on to a screen 20 cm farther on, where they make a spot of light. With E and B both off the spot is at the centre mark of the scale.

Forces. The electric force on an electron is F_E = eE, directed opposite to the field, so towards the positive plate. The magnetic force is F_B = evB at right angles to both the velocity and the field. For electrons going to the right in a field into the screen this is downwards. So the electric force and the magnetic force can be made to cancel:

eE = evB, so v = E / B

Thomson’s method for e/m. (1) With B off, the electric field makes the beam follow a parabola between the plates. The deflection on the screen is y = (e/m) E L(D + L/2) / v², where L = 0.10 m is the length of the plates and D = 0.20 m is the distance from the plates to the screen. (2) Turn on B until the beam is straight again. Then v = E/B. (3) Put v = E/B into the first equation:

e/m = y E / [ B² L (D + L/2) ]

The accepted value is 1.759 × 10¹¹ C kg⁻¹. Thomson found about 1.7 × 10¹¹ in 1897 and concluded that cathode rays are particles much lighter than atoms.

Magnetic field alone. A magnetic force at right angles to the velocity makes the electron move in a circle of radius r = p/(eB) = γmv/(eB), so e/m = v/(Br) for slow electrons. The number in the fourth tile is this radius for the present B and v. It is the radius of the path only when E is off.

Choosing the settings. The electrons are fast (3 × 10⁷ m/s, a tenth of the speed of light), so the fields needed are not large: with E alone the beam hits a plate for E above about 20 kV/m, and with B alone for B above about 0.7 mT. But the two together can be large, as long as they are nearly balanced (E = vB). For a good measurement use a small E (about 5 to 10 kV/m) so that the deflection is small but easy to read.

Relativity. The program uses the exact relativistic equation of motion, dp/dt = q(E + v × B) with p = γmv, with no approximations. At 0.1c the factor γ is 1.005, so the formulas above, which assume γ = 1, give a value of e/m that is about 0.5% too small. Raise the accelerating voltage to see the error grow.

What the picture leaves out. The fields have sharp edges and are uniform between the plates and zero outside (in a real tube they leak out a little at the edges), the beam is a single thin line in a perfect vacuum, and the electrons do not repel each other. The dots moving along the beam show its direction but not its real speed.