# Charge To Mass Ratio

#### Purpose

To determine the charge to mass ratio for the electron.

#### Objectives

• To learn how to use statistical average and standard deviation for repeated measurements to improve accuracy in the final result.
• To learn how to calculate the propagation of error when working and algebraically manipulating measured quantities.
• To learn how to derive the formula for the charge-to-mass ratio of a charged particle.
• To learn what sections must be included in a scientific lab report.

#### Equipment

• Q/M Apparatus

#### Theory

When electrons moving with a speed v enter region with magnetic field B they are deflected from their original trajectory in circular path. The radius of curvature is set up by Newton’s Second Law for that electron:

$F_{\text{net}}=qvB = ma = m\frac{v^2}{r}$   (Eq. 1)

The electrons usually are sped up to those velocities v by applying an electric field E created by a high-voltage difference ΔV between two electrodes at a distance d. The relationship between the two then will be:

$|E|=\frac{|\Delta V|}{d}$   (Eq. 2)

The speed with which the electrons emerge  then will depend on that high-voltage difference and can be calculated by using the conservation of energy principle:

$\frac{1}{2}mv^2 = q\Delta V$   (Ed. 3)

Using Eqs. 1  and 3, we can write for the ratio between the charge and the mass of the electron the following formula:

$\frac{q}{m} = \frac {2 \Delta V}{B^2 r^2}$   (Eq. 4)

#### Experimental Setup

The experiment will be completely set up for you. All you need is to turn on the power supply and take measurements.

#### Preliminary Activity

The class will be divided into four groups. There will be only 4 experimental apparatuses set up. This is one lab experiment where you will be permitted to work in large groups.

The magnetic field value depends on the electric current through the apparatus. Look at the bottom front corner of the apparatus and record down that relationship:

$B = 7.8\times 10^{-4} I$   (Eq. 5)

If the current is in Amps, Eq. 5 will give you B in Tesla. You do not need to convert units.

#### Experimental Activity

###### Record the values for the High voltage in Volt and the electric current in Amp.
 ΔV (V) uncertainty ΔV (%) I (A) uncertainty I (%) B (T) uncertainty B (%)

Note.

The uncertainty in (%) are relative uncertainties. Take the absolute uncertainty (smallest possible measurement from your multimeter) of the measured quantity and divide it by the value.

For example: If you measure the volts as 10.1 V, then your measurement is 10.7 ± 0.1 V, and the relative uncertainty in the volts is:

$\frac{0.1}{10.7} \approx 1\%$

Repeat for the current measurements. Since you are not directly measuring the magnetic field, B, use the relative error for the current to be the same as the relative error for B. For example, if you calculate the percent error for I to be 5%, then take the relative error for be to be the same 5%.

###### Measure the radius of the circular trajectory of the electrons.

EACH member of your lab team must measure the radius. The best way to measure it is to mark the locations of the left and right edge of the circle is and to take the average value of those two.

Record your data in a table:

 Student Name xL (m) XR (m) R (m) Average R: St. Deviation: Relative Uncertainty:

You will use the average value for the radius, R, in your calculations.

Use the standard deviation as a measurement if the absolute uncertainty in the measurement for R. Calculate the relative uncertainty of R as the ratio between the st. deviation and the value for R.

#### Results

###### Q/M Ratio

Using your measured values, calculate the q/m ratio.

###### Propagation of error

Using the relative uncertainty of your measurements, calculate the total relative uncertainty of your result for the q/m ratio.

$\frac{q}{m} = \frac {2 \Delta V}{(B)^2 r^2}$

In the formula above, you have used once ΔV, twice the value for the current I (it is squared in the formula), and twice the value for the radius. So your total relative uncertainty should be the sum:

$\text{uncertainty }\Delta V + 2 \times \text{uncertainty } I + 2\times \text{uncertainty }r$

#### Questions and Assignments

1. Does your q/m value agree with the established within the range established by the error calculated by you?
2. Write a lab report on this experiment. You will be graded on the following components:
• Include Title, Purpose of the lab experiment, Theory section, Description of the experimental apparatus, Measurement, and Results
• Your report must be typed and well organized
• Coherent paragraphs (no bulletized lists). No spelling errors.