### The Determination of the Charge to Mass Ratio of the Electron

A magnetic field produced by Helmholz coils is used to deflect electrons into circular paths whose radii are known. By knowing the energy of the electrons and the magnetic field strength, the ratio of the charge to mass (e/m) of the electron is determined.

## Theory

Electrons are thermally emitted from a surface and accelerated through a potential difference V. The kinetic energy of the accelerated electrons equals the energy they gain as a result of being accelerated through the potential difference. In other words:

½ m v2 = eV

and solving for velocity,

v = (2eV/m)1/2 .

In this equation m is the mass of the electron and e is the charge of the electron.

The beam of electrons enters the region where a magnetic field B is set up by the Helmholz coils. The beam is deflected into a circular path of radius R by the magnetic force and undergoes a centripetal acceleration. This can be expressed as

evB = mv2/r

When the velocity is eliminated between the above two equations, then the charge to mass ratio can be written as

e/m = 2V/(B2r2)

The magnetic field due to the Helmholz coils can be expressed as

B = 8moNI/(125)1/2a

where N is the number of turns of wire on each coil, I is the current through the coils, a is the mean radius of the coils, and mo is the permeability of free space. (mo = 4p x 10-7 Tm/A)

## Experimental Procedure

1. Set the apparatus on a level table. The room light should not be too bright, because the electron beam will be hard to see.

2. In order to minimize the influence of the earth’s magnetic field, use a compass to locate magnetic north and align the Helmholz coils so they are parallel to the needle.

3. With the power switch off, connect the line cord to the line voltage.

4. Turn on the power switch. The unit will perform a 30 second self-test, indicated by the digital display changing values rapidly. During the test the controls are locked out allowing the cathode to heat to the proper operating temperature. When the self-test is complete, the display will stabilize and show "000". Although the unit is now ready for operation, a 5 to 10 minute warm up time is recommended before taking measurements.

5. Turn the Voltage Adjust control up to 200 volts and observe the bottom of the electron gun. The bluish beam will be traveling straight down toward the bottom of the tube.

6. Turn the Current Adjust control up and observe the circular deflection of the beam. When the current is high enough, the beam will form a complete circle within the tube. The diameter of the beam can be measured using the internal centimeter scale within the tube. The scale numbers fluoresce when struck by the electron beam. Find the smallest value of the current that causes the scale numbers to fluoresce.

7. For three different voltages (200V, 250V, 300V) record the current readings for 5 different beam diameters and complete the table below. When all the data is collected, turn down the voltage and current and turn off the apparatus.

 Number of turns of wire on each coil:  N = 130 Measured average radius of coils:      a ± Da =

## Accelerating Voltage = 200 V

Diameter (m) Radius (m) Coil Current (A) Magnetic Field (T) rB
0.10
0.09
0.08
0.07
0.06
 Average rB ± error =

## Accelerating Voltage = 250 V

Diameter (m) Radius (m) Coil Current (A) Magnetic Field (T) rB
0.10
0.09
0.08
0.07
0.06
 Average rB ± error =

## Accelerating Voltage = 300 V

Diameter (m) Radius (m) Coil Current (A) Magnetic Field (T) rB
0.10
0.09
0.08
0.07
0.06
 Average rB ± error =

Now calculate the e/m value for each of the three accelerating voltages using the average rB value for each voltage. Also calculate the uncertainty in the e/m value using the fractional error rule: D(e/m) = (e/m) [ 2D(rB)/(rB) + 2Da/a ]. (Note here we have tried to highlight that we may have used an incorrect value for a in our formula for B which could bias the experimental value of rB in a way not taken into account by just looking at the spread of rB values.

 True value of e/m = Voltage (V) e/m ± D(e/m) (C/kg) % difference from true 200 250 300

Show your experimental values and their range relative to the true value on a number line below.

Explain any differences between the true and experimental values.