The Charge to Mass Ratio of an Electron

By

Kate Perry

and

Scott Hesser

Abstract:

We measured the charge to mass ratio of an electron using an apparatus developed by K.T. Bainbridge. Bainbridge’s device fires electrons into a uniform magnetic field produced by Helmholtz coils. The particles follow a circular path and from its radius, the strength of the magnetic field and the energy of the moving particles, one can deduce the charge to mass ratio. We found the ratio of e/m to be 1.760605´10¹¹ coulombs/kg. This is with in 0.1% of the accepted value.

Introduction:

In 1897 J.J. Thomson made the first measurements of the charge to mass ratio of an electron (e/m), using cathode ray tub. Thomson accelerated electrons through a potential difference and down a tube. Part way through the tube the electrons passed through a magnetic field and were deflected from their original path. Thomson used the measurements of the deflection to determine e/m. Bainbridge’s apparatus is very similar. However, rather than applying a magnetic field at one location, the electrons are constantly in the presence of a magnetic field and therefore are continuously being deflected into a circular path. Regardless of the differences, both used a magnetic field to determine a property of the electron. In the early 20^th century, magnetic fields were the only thing available to examine the electron, and it is important to understand how this was done.

Theory:

A charged particle will experience a force when moving through a magnetic field. The force is determined by the charge of the particle (q), its velocity (v), the strength of the magnetic field (B), and the angle (q) of the path to the magnetic field. The equation for this relationship is:

F=qvB sinq eq.1.

In Bainbridge’s experiment an electron is propelled in a direction that is perpendicular to the magnetic field. The resulting path of the electron is a circle. This can be modeled by the formula for centripetal force:

F=mv²/r eq.2.

Combining these two equations provides the formula for the electron’s charge to mass ratio:

q/m=2V/ B² r²

Bainbridge used a pair of Helmholtz coils to form a uniform magnetic field. In the figure below the field is directed out of the plane of the paper and is represented by the uniformly spaced dots. When an electron is released in this field the direction of the force exerted on it can be found with the right hand rule. For a positive charged particle your outstretched fingers of your right hand will point along in the direction of the particle. Your fingers must be able to bend in the direction of B. Your thumb will point up in the direction of the force. This is for a positive charge, the direction is opposite for a negative charge. The electron in the diagram has an initial velocity to the right and is perpendicular to the magnetic field. The right hand rule shows that the force acting on the electron will push it up. Because this is a uniform field, and the electron’s path always remains perpendicular to the field, the electron is moved around in a circle.

The electron is traveling in a circular path with a radius r. If we increase the magnetic field the force gets stronger and the radius decreases. If we increase the velocity of the electrons the radius increases. Because the force is always directed in to the center it is a centripetal force which is describe as:

F=mv²/r eq.2.

This can be combined with the equation 1 to get:

mv²/r= qvB sinq eq.3.

The electrons are accelerated by being passed through a potential difference. The expression of kinetic energy for a particle passed through a potential difference V is:

Vq= mv²/2 eq.4.

This is solved for v²:

v²=Vq2/m eq.5.

By squaring eq.3 and changing sinq to 1, because q equals 90, we arrive at:

m²v⁴/ r²= q² v²B² eq.6.

Substituting in equation 5 provides:

m²V²q²4/r²m²= q³VB²2/m eq.7.

This is solved for q/m:

q/m=2V/ B² r² eq.8.

And that’s it!!! If the potential difference powering up the electrons is known, as well as the strength of the magnetic field, measuring the radius of the path will allow you to find q/m, or for the electron e/m. To make the path of the electron visible for measurement the experiment is done in a sealed chamber with mercury gas. The electrons excite the mercury gas and cause it to glow around the path.

Another important part of this lab is the uniform magnetic field of the Helmholtz coils. A current passing through a wire produces a magnetic field. If the wire is coiled the magnetic field is intensified proportionaly to the number of times the wire is looped around. Helmholtz coils are two identical coils of wire which are mounted parallel to each other along the same axis with the distance between them equal to their radius. The direction of the magnetic field is parallel to the axis of the coils and is relatively equal in magnitude everywhere between the coils.

The strength of the field can be determined from the Biot-Savart Law. This law says that at a point x distance from a coil with radius R, along the coil’s axis, the strength is:

B=m_o IR2/(2(R²+x²)^3/2).

For the Helmholtz coils there are two identical coils so the strength is doubled. Also, the distance x is equal to one half the radius so we substitute in R/2. The resulting equation is:

B=8m_o I/(5^3/2 R).

Because the field strength is proportional to the number of turns in the coil, N, the final equation is:

B=8m_o NI/(5^3/2 R).

Experimental procedure:

The procedure for this is relatively easy. The Bainbridge’s apparatus consists of a mercury filled bulb with an electron gun inside it. The bulb is located between a pair of Helmholtz coils to produce a uniform magnetic field. There is a ruler across the back of the apparatus to measure the radius of the electron path. The current to the Helmholtz coils is measured with a multi-meter and the magnetic field strength is determined from the equation:

B=8m_o NI/(5^3/2 R).

For this particular pair of coils B can be determined as I´7.8´10^-4. The voltage for the electron gun is also measured with a multi-meter. When measuring the radius it is important to measure at the outside of the stream. For several reasons some of the electrons lose energy and fall toward the center. We made a total of twenty measurements, each time setting the voltage and current at random settings and then averaged our value of e/m.

Results and Discussion:

This experiment was rather easy to perform and our results were very impressive. Our data is presented in the chart below.

We found our average value of e/m to be 1.760605´10¹¹ coulombs/kg. This is within 0.1% of error to the accepted value of 1.7588028´10¹¹ coulombs/kg. That should prove that Bainbridge had a pretty good idea. Now Bainbridge also suggested that one should measure the earth’s magnetic field in the lab to compensate for the added B. We did not do that, but we believe that if you are concerned about this, a solution might be to position the apparatus so that the produced magnetic field is perpendicular to the earth’s field. This would make the electron path parallel to the earth’s magnetic field, and would assure that the earth’s magnetic field caused no force on the electrons.

Conclusion:

Our results were damn good and this method of measurement is extremely reliable. This design can be used for other particles and in principle it is very similar to other worthy devices such as the mass spectrometer. The Helmholtz coils are a significant part of this design’s success.

References:

K.T Bainbridge, The American Physics Teacher, 6, 35. (1938).

Giancoli, Douglas C. Physics, Principles with Applications. New Jersey:

Englwood Cliffs. 1991.

Halliday and Resnick, Fundamentals of Physics. New York: John Wiley and

Son’s Inc. 1970.