Experiment 7

The Mass of an Electron

 

Objective:

 

The objective of this experiment is to measure the mass of an electron by using electric and magnetic fields.

 

Equipment:

 

Tuning-eye vacuum tube, high-voltage dc source, two 12 V dc power supplies, two multi-meters, a few cylindrical objects of circular cross-section (of small diameters such as pencils or plastic rods), a solenoid (with an inside diameter greater than the outside diameter of the tuning-eye tube), connecting wires, and a calculator

 

Theory:

 

It is possible to use an electric field perpendicular to a magnetic field in order to measure the mass of an electron knowing that the electronic charge is e = –1.6x10-19 C.

 

When a charge q moving at velocity v crosses a magnetic field B perpendicular to its field lines, the magnetic field exerts a force Fm on the charge perpendicular to the plane of v and B.  The magnitude of Fm is given by

 

Fm = | q v B | .

 

The perpendicularity of Fm and v guarantees a centripetal force

that is indeed the magnetic force Fm itself.  The centripetal force Fc forces the electron to travel in a circular path.  Equating Fm and Fc , yields:

Dividing both sides by v and solving for R, the radius of revolution, results in

 

If an electron of mass M (to be determined), whose charge is the known value  –e,  moving at speed v, crosses a magnetic field of strength B, it will be given a radius of revolution R that can be calculated from the equation:

( 1 )

The difficult variable to measure in equation (1) is v, the (magnitude of) the velocity of the electron.   Velocity v can be determined as described below.

P.E. lost = K.E gained.

or

 

eV = ½ M v 2.

 

From this equation, v 2 may be calculated as

( 2 )

Solving (1) for v and then squaring it yields:

( 3 )

Equating (2) and (3), gives:

 

Dividing both sides by (e/M),

 

( 4 )

This equation will be used to measure M, the mass of an electron.

 

Procedure:

 

A tuning-eye is an electronic device widely used in older non-transistor radios.  It was used as a visual indicator for best tuning on a desired station.  When the tiny filament in a tuning eye is given a low voltage (V1), it warms up and glows red, as does an electric heater.  In this heated state, the filament releases electrons.  Another voltage (V) may be used to create an electric field in which the released electrons can be energized, accelerated, and brought into motion toward a positive dish.   The negative filament, the positive dish, and the two voltage sources are shown below:

 

 

The positive dish-like surface is coated with a metal oxide that glows green/blue as electrons hit it.  A metal cap is placed over the element which is held by three thin legs.  These legs cast a shadow on the dish, making straight dark lines on it when the tuning-eye tube is in use.  Increasing V makes the dish glow brighter.  An instruction comes with each tuning-eye tube that must be followed for proper use. The following steps should be taken:

 

  1. Connect the filament wires as indicated in the instruction to an appropriate voltage V1.
  2. Wait a few seconds for the filament to warm up and observe its reddish color.
  3. Connect the other wires as indicated in the instruction to the second appropriate source and increase the voltage to the appropriate level V and observe that the dish attains a bluish or greenish color.  Pay attention to the shadow cast on the dish by the cap’s legs and note that they are straight lines as viewed from the top of the tuning eye.
  4. Connect a pre-selected solenoid (with known number of turns per meter) to an appropriate power supply that can provide a few amperes.  An ammeter must be placed in the solenoid circuit for a more-accurate current measurement.  Note that the ammeter wire must be put in its 10-Amp setting.
  5. With a current of about 1A passing through the solenoid, place the solenoid over the tuning-eye tube so that the filament is positioned at the center of it, and observe how the legs’ shadows (straight lines to begin with) bend as the magnet is lowered.  The value you later calculate for B, magnetic field strength, actually occurs at the middle of the solenoid; therefore, make sure that the filament is precisely inside and at the center of the solenoid.
  6. Adjust the dish voltage V, and the solenoid current I to get a nice curve for the shadow of each leg in the tuning eye.  The radius of curvature R must be measured.  Any cylindrical non-metallic object that goes into the solenoid may be used.  While holding that object inside the solenoid and looking straight down on to it and the top of the tuning eye, try to match the curvature of the shadow with the curvature of the object by adjusting V and I. 
  7. Try three different round objects and obtain three different sets of V, I, and R.
  8. For each set, calculate B = μon I , where n is the number of turns per meter of the solenoid.

  9. Use B, V, and R along with e to find the value of M for each set by using equation (4).

  10. Find a mean value for M using the three values calculated.  This is the measured value.

 

Data:

 

Given:

           

        Maccepted = 9.11x10-31 kg.

          n = number of turns per meter of the solenoid obtained from the manufacturer’s paper.

          μo = 4 π x 10-7 (T m/A)

 

Measured:

 

         For each set: V, I, and R

 

Calculations:

 

Follow the steps in Procedure.

 

Comparison of the Results:

 

The accepted and measured values of M may be used to obtain a percent error:

 

Conclusion:          To be explained by students

 

Discussion:             To be explained by students