Measuring Electron Charge-to-Mass Ratio: Physics Lab Report

Introduction and Theoretical Background

The ratio of charge to mass in electrons is the focus of this experiment. By observing the deflective effects that magnetic and electric fields have on an electron beam, the researcher can measure this ratio by achieving a balance between the magnetic and electric fields influencing the beam. An electron has kinetic energy equal to its initial potential energy, which creates a definite and known relationship between the velocity of the electron and the charge-to-mass ratio, as described by the standard formula for electron acceleration through a potential difference.

The velocity of an electron can only be determined if the overall force acting upon it is balanced — or essentially non-existent. If the experimental setup is arranged so that the only two forces acting on the electron beam are the controlled electric and magnetic fields, the combined effect of those forces can be brought to zero, allowing the velocity to be calculated directly from the balance condition.

Using this velocity, the electric field E between two infinite parallel planes can be defined in terms of the applied voltage and plate separation. A pair of Helmholtz coils with separation distance equal to their radius R is used to generate the magnetic field B. The permeability of free space is a known constant equal to 4π × 10⁻⁷ H/m. Through further simplification of these expressions and appropriate equipment setup, the charge-to-mass ratio can be obtained by working backwards through the field-balance equations while measuring and controlling the electric and magnetic fields.

The detailed procedure is described in Handout Experiment 2A. Ensuring that the circuits shown in the relevant figures are connected correctly and positioned properly is of primary importance before any measurements are taken.

Experimental Method

The power supply for the deflecting plates is turned on while the power supply for the magnetic coils is turned off. Setting the anode voltage to 3000 V causes the electron beam to curve upward between the two deflecting plates — a direct result of the electric field force. After making this observation, the anode voltage is reduced to 1000 V and the power to the magnetic coils is switched on. The magnetic field then exerts a stronger force on the electron beam, causing it to curve downward. The goal is to adjust the strength of each field until the beam travels as straight as possible.

The current through the coils is measured with an ammeter. Observations are recorded at anode potential differences of 1500, 2000, 2500, 3000, 4000, and 5000 V. For each voltage setting, the corresponding current in the magnetic coils (I) and I² are recorded in a data table. These values are subsequently plotted and used to calculate the charge-to-mass ratio. The Lorentz force principle underpins the beam-balancing technique used throughout this procedure.

The line of best fit is very close to the actual data points in the graph, suggesting that the data are both accurate and precise. At the same time, the lack of a perfect fit indicates that some unmeasured and unwanted influences on the charge-to-mass ratio are present. Two broad categories of error are relevant here: careless error and systematic error.

Results and Discussion

Careless errors include observational mistakes, such as difficulty reading the position of the electron beam due to insufficient brightness. Systematic errors arise from any lack of consistency or improper calibration in the apparatus. Creating the proper environment for accurate observation — for example, darkening the room and using a high enough anode voltage (at least 1500 V) so that the electron beam is clearly visible — reduces the potential for careless error. Proper maintenance and calibration of all equipment help to minimize systematic errors.

Regardless of the strength and intensity of the electric field E, the electron beam remains essentially uninfluenced by field non-uniformities when the fields are well balanced. However, slight curves do appear in the beam because the magnetic field is not perfectly uniform across all points in space. Because the beam interacts with the magnetic field over a volume rather than at a single point, local variations in field strength produce small curvatures and lateral shifts. Similarly, the electric field loses uniformity near the edges of the deflecting plates, which exacerbates these slight curvatures.

The established and accepted value for the charge-to-mass ratio of electrons is 1.76 × 10¹¹ C/kg. The charge-to-mass ratio calculated from the experimental data was obtained by plotting I² against the anode voltage V and applying the relevant equation derived from the field-balance conditions. The percentage difference between the calculated value and the accepted value is 96.5%, computed using the standard formula:

Conclusion

The difference between the charge-to-mass ratio calculated in this experiment and the accepted value of 1.76 × 10¹¹ C/kg is 96.5%. This is a very significant discrepancy; the calculated value is nowhere near the accepted value established through prior experimentation and observation.

Several strategies could improve the result in future trials. Plotting I² against V and recalculating using a larger number of data points would increase statistical confidence. Repeating the experiment multiple times to produce additional data sets would allow a more reliable average to be determined. Using different pieces of equipment in the same setup would help to confirm whether faults in specific instruments are distorting the results. Ensuring thorough calibration of all apparatus before each trial, maintaining a darkened room for clearer beam observation, and carefully minimizing magnetic field non-uniformities are all practical steps toward obtaining an experimental value closer to the accepted charge-to-mass ratio.