Measure the ratio of charge to mass of the electron.

Lab Template

In order to do data processing and analysis, we will use Jupyter notebooks (which run on the Python programming language). Download the following notebook file:

The Millikan Oil Drop experiment, performed between 1900 and 1911 at the University of Chicago, was the first precise measurement of e, the charge of the electron. Measurement of the ratio, e/m for the electron thus also yields the mass of the electron. These are important physical quantities!

The force on a charge moving in a magnetic field (known as the Lorentz force) is

{ $\vec F = q \vec v \times \vec B$   (1) where q is the charge, v is the velocity of the moving charge, and B is the magnetic field.
The magnitude of this force is given by the scalar form of eq.(1),

{ $F=qvBsin\phi$   (2) where { $\phi$ is the angle between the direction of the magnetic field and the direction of motion of the moving charge.\\The direction of the force is given by the right-hand rule (if the charge is positive) and is perpendicular to both the velocity and magnetic field. In the special case that { $\vec v$  is perpendicular to [Math Processing Error]B→ { $\vec B$  , eq.(2) becomes { $\mathrm{F=evB}$   (3) where e is the charge of the electron. The electron beam will follow a circular trajectory within the field with a centripetal force

{ $F=evB=\frac{mv^{2}}{R}$   (4) where m is the mass of the electron and R is the radius of the circular path as shown in Fig. 1.

{ ${/download/thumbnails/234359051/Circular%20e%20traj%20%28142%29.png?version=1&modificationDate=1580841363000&api=v2}$ \\Fig. 1 path of electron in a magnetic field pointing into the page

For a non-relativistic electron, accelerated through a potential V, the kinetic energy is

{ $KE=\frac{1}{2}mv^{2}=eV$   (5) Eliminating v between eqs.(4) and (5) and solving for e/m gives

{ $frac{e}{m}=\frac{2V}{R^{2}B^{2}}$   (6) Since one can determine all of these quantities on the right side of eq. (6), it is now possible to arrive at a value of e/m. Note that eq.(6) is derived using the following simplifying assumptions:

• The magnetic field B is assumed to be perfectly uniform over the entire path.
• The electrons are assumed to be moving at constant speed along their entire path.

MAGNETIC FIELD
The apparatus uses a pair of Helmholtz coils to produce a magnetic field. These coils have the special geometry with the separation S equal to the radius as shown in Fig. 2.

{ ${/download/attachments/234359051/worddav27512d074fd174b78c653d4e60c3baab.png?version=1&modificationDate=1580769781000&api=v2}$ Fig 2. Helmholtz coil geometry

Using the law of Biot-Savart, it can be derived that the magnetic field (in Tesla) produced by the coils is

{ $B= \left( \frac{8}{\sqrt{125}} \right) \frac{\mu_{o}NI}{S}$   (7) where

• { $\mu_{o}=4\pi \times 10^{-7}$  Tm/A is the permeability of free space, * N is the number of turns in each coil (N = 130 turns in your apparatus),
• I is the current (in Amperes) through each coil (they are in series!), and
• S is the separation (in meters) between the coils and the radius of the coils.

A B field map in Helmholtz coils is shown in Fig. 3.

{ ${/download/attachments/234359051/worddav44d0f96478e032ecb4f1fe37cabbb35c.png?version=1&modificationDate=1580769781000&api=v2}$ Fig. 3 B field produced by Helmholtz coils (from Wikipedia)

This section will be most meaningful if read in the laboratory, while viewing the apparatus.

{ ${/download/attachments/234359051/em_wiring_%28142%29.png?version=1&modificationDate=1580841621000&api=v2}$\\Fig. 4 e/m wiring diagram

According to eq.(6), we have 3 measured quantities to explore: V, R and B. One approach would be to hold V fixed while varying B and measuring the resulting R. Repeating these measurements for several values of V would complete the exploration. This method suggests rearranging eq.(6) as

{ $\frac{2V}{R^{2}} = \frac{e}{m}B^{2}$   (8) which has the form

y = “slope” * x.

Plotted appropriately, the data would be expected to fall on straight lines having slope e/m.
Before taking data, vary V and B to get a feel for the ranges which can yield useful measurements.

Question 1: Plot your data in the form of a straight line.
Question 2: Perform linear fits to each of the straight lines and obtain the value of e/m.
Question 3: Estimate fractional uncertainties for each of the measured quantities.
Question 4: Referring to eq.(8), which uncertainty most affects the value of e/m?
Question 5: Make a rough estimate of the uncertainty in your value of e/m.
Question 6: Compare your value of e/m with the literature value given inside the back cover of your lab manual.
Question 7: Is your value for e/m consistent with the literature value when your uncertainties are taken into account? Answer using actual numbers and units!

Use this link to submit your report

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