| PHY 251 | Measurement of e/m for Electrons |
In this experiment you will measure e/m, the ratio of charge e to mass m of the electron. The set-up consists of an evacuated (except for a low pressure of He gas) cathode ray tube which produces a beam of electrons of known energy. The tube contains an electron gun which emits, accelerates, and focuses electrons. The electron beam becomes visible when some of the electrons in the beam excite Helium atoms in their path, which subsequently de-excite with emission of visible-light photons.
For an electron charge e and an accelerating voltage V the (non-relativistic) kinetic energy of the electrons is given by:
K = 1/2 m v2 = eV .
The tube is supported at the center of a pair of large Helmholtz coils producing a uniform magnetic field B perpendicular to the velocity of the electrons. Thus, the electrons will be deflected to trace a circular path with a radius determined by e/m, the accelerating voltage V, and the magnetic field B (see equation 1). Helmholtz coils, when separated by a distance equal to their radius, produce a very uniform magnetic field in the plane half-way between the coils. The field strength there equals:
| B = m0N I R2 / (R2 + x2)3/2 , | (1) |
where m0 = 4p×10-7 Henry m-1 is the magnetic permeability of free space, N = 130 is the number of turns on each coil, I the coil current, R the mean radius of the coils, and 2x the distance between coils. Equation (1) is exact on axis at the very center between the coils; check this by deriving the formula!
Determine R and x (and their errors!) with a meter stick. Then, for a given coil current I, vary the accelerating voltage V and read off the radius r of the electron orbit. This is done most accurately by tuning the voltage V such that the beam coincides with one of the marks on the glass measurement gauge inside the tube. Perform measurements for a wide range of voltage settings, then repeat the sequence for several different current settings.
Rotate the apparatus around the vertical axis, and see if you can find, and measure, the magnitude and direction of the local Earth magnetic field.
Loosening the bolts that tighten down the tube allows one to rotate the beam direction out of the central (vertical) plane and produce helical beam patterns (Do this only under supervision, and be sure to return the tube to its exact original position afterwards!). Keeping the acceleration voltage and coil current fixed, explain the behavior of the beam path radius as function of the rotation angle.
The ratio e/m is given by:
| e / m = 2V / (B r)2 | (2) |
Derive Equation (2). For each I plot V versus B2r2 on graph paper. You should obtain straight lines. Determine e/m from the slope and average over the different values obtained for different values of I. Perform an error analysis estimating the uncertainties in all measured quantities and the relative importance of the various sources of error. Compare your final result with the accepted literature value and discuss the cause of possible deviations. Discuss how you would improve the precision of the e/m measurement.