Experimental Measurement of the Charge to Mass Ratio of the Electron Stephen Luzader Physics Department Frostburg State University Frostburg, MD

2 Outline of topics Purpose of the experiment Some theory about the motion of an electron in a magnetic field Helmholtz coils The experimental apparatus How to analyze the data

3 Purpose The electron is a fundamental constituent of all matter in the universe. It is important to know the properties of the electron, including its mass and electric charge. The purpose of this experiment is to measure the ratio of the electron’s charge to its mass, a quantity referred to as “e/m”.

4 If an electron with a velocity pointing right enters a uniform magnetic field pointing out of the page, the resulting force will cause the electron to move in a counterclockwise circle. Some theory on the motion of an electron in a uniform magnetic field 

5 The radius R of the circle can be calculated using the definition of centripetal force: where m is the mass of the electron. Since the velocity is perpendicular to the magnetic field, the magnitude of the centripetal force is Combining these equations gives an expression for e/m:

6 If the electron achieved its speed v as a result of being accelerated through a potential difference V, then its kinetic energy is Solving this equation for v and substituting into the equation for e/m gives If we can devise an experiment to make electrons orbit in a magnetic field, we can obtain a value for e/m.

7 A special arrangement of two coils called Helmholtz coils is used to create a uniform magnetic field over a fairly large region of space a It can be shown that if two circular coils are separated by a distance equal to their radii, the magnetic field will be uniform over a large region between the coils. In the diagram, a is the radius of the coils. a

8 The magnitude of the magnetic field on the axis of a coil at a distance x from the center is where I is the current in the coil and N is the number of turns in the coil. At the midpoint of the Helmholtz coils, which is a distance x = a/2 from the center of each coil, the magnitude of the total field from both coils is

9 Summarize what we have deduced so far: 1.An electron whose velocity is perpendicular to a uniform magnetic field will move in a circular orbit. 2.The value of e/m can be calculated if we can measure the accelerating potential, the radius of the orbit, and the magnetic field. 3.By using a special arrangement of coils called Helmholtz coils, we can calculate the magnetic field if we know the radius of the coils, the number of turns in each coil, and the current through the coils.

10 The Experimental Apparatus A glass bulb containing a small amount of mercury vapor is placed between a pair of Helmholtz coils. An electron gun in the bulb shoots a beam of electrons into the mercury vapor. Some of the electrons strike mercury atoms, causing them to emit light. This light allows us to see the orbit of the electrons so we can measure its radius.

11 The Experimental Procedure 1.Measure the accelerating potential. 2.Measure the radius of the electron orbit for at least 3 different currents in the Helmholtz coils. 3.Calculate the value of B for each value of I. 4.Calculate an average value of e/m (with standard deviation) from the measured values of V, R, and B.

12 The radius of the orbit is determined by observing the circular path of the electrons against a scale. The radius can be calculated from the measured values x 1 and x 2 of the right and left sides of the circle: + x 2 x 1 For example, if x 1 = 3.0 cm and x 2 = -4.1 cm, we conclude that R = 3.55 cm.

13 We also need the following information about the Helmholtz coils, which is provided by the manufacturer: a = 15 cm N = 130 turns

14 For your analysis, you must carry out the following steps. 1.Prepare a table with all important data, including accelerating potential, Helmholtz coil currents, positions of right and left sides of electron orbits, number of turns, and coil radius. 2.Calculate the radius of each orbit. 3.Calculate the magnetic field for each orbit. 4.Calculate a value of e/m for each orbit. 5.Calculate an average value for e/m, including uncertainty.