Electric Fields I: Field Mapping

If we're still doing the e/m lab this year, we should think about either getting different power supplies or packaging up some big honking resistors to make it easier to adjust the current in the coils.

In mechanics, we learned how to compute forces between different objects in order to determine how objects would move. In many cases, we considered forces between two simple objects at a single point – for example the tension of a wire holding up a block or the force of a hand pushing an object along an inclined plane – but when we learned about gravity, we introduced the concept of a field.

A field allows us to consider how a force will affect an object at all points in space or how an object will move when subject to forces from many different sources (or from a source that is not a single point but instead a large, extended object). We change our view from talking about the force between two objects to instead talking about the force that an object feels as it moves through a field.

We encounter fields a lot in the study of electricity and magnetism, so this first lab will be an exploration of electric fields. Such fields are hard to visualize (especially for complex charge or conductor configurations), so we want to spend some time mapping these fields (in Part 1) and observing how charged particles move through then (in Part 2) in order to get a better intuitive feel.

Along the way, we will also introduce some of the specialized tools that are used in the study of electricity and magnetism such as power supplies, function generators, voltmeters and ammeters, and (in future labs this quarter) oscilloscopes.

Theory


Electric field

An electric field is a region of space in which electric forces act on electric charges in the region. If a force $\mathbf{F}$ acts on a charge $q$ in the field, the field strength $\mathbf{E}$ is, by definition, the force per unit charge. In vector form, the equation is

$\mathbf{E} = \dfrac{\mathbf{F}}{q}$. $(1)$

Force is a vector quantity having direction as well as magnitude. The direction of an electric field at any point is the direction of the force on a positive test charge placed at the point in the field.

Lines of force

Lines of force were introduced by Faraday to visualize the direction and strength of an electric field. The electrostatic force is everywhere tangent to the field lines at each point and the field strength is proportional to the density of lines at each point.  As an illustration, consider the isolated positive charge $+Q$ in Fig. 1.

Figure 1: Electric field around a positive electric charge

A small positive test charge $+q$ at any point in the field experiences a radial force of repulsion directed outward from $+Q$; the lines of force are therefore drawn with arrows to pointing outward. (If $Q$ were a negative charge, these lines would be directed inwards to indicate an attraction of the positive test charge $+q$.) Near $+Q$, the lines are close together as the field strength is strong. Moving outward, the density of the lines decreases (and so does the field strength).

Figure 2 shows a plane section near a pair of equal charges of opposite sign. Each charge exerts a force on a unit test charge placed in the field. The resultant force is the vector sum of these forces. Thus, at the point $\mathrm{b}$, $f_1$ is the repulsive force on the unit test charge due to the positive charge $+Q$ and $f_2$ is the force of attraction due to the negative charge $-Q$. The resultant $R$ is tangent to the line of force at the point $\mathrm{b}$.

Figure 2: Electric field around two equal charges of opposite sign

Potential difference

Two points in an electric field have a difference of potential if work is required to move a charge from the one point to the other. The amount of work turns out to be independent of the path taken between the two points. Consider the simple electric field illustrated in Fig. 3.

Figure 3: Potential difference between points in an electric field

Since the charge $+Q$ produces an electric field, a positive test charge $+q$ at any point in the field will be acted upon by a force due to that field. If no external force is applied, the field will do work on the charge and move the charge to a lower potential. On the other hand, if we wish to move $+q$ to a higher potential, we would have to apply an external force, doing work on the charge.

The potential difference between two points in an electric field is defined as the ratio of the work needed to move a small positive charge between the points, to the magnitude of the charge moved:

$V = \dfrac{W}{q}$ $(2)$

where $V$ is the potential difference, $W$ is the work done on the charge , and $q$ is the charge moved. If the work is measured in joules and the charge in coulombs, then the potential difference $V$ is measured in volts.

Conservation of energy requires that the work done must be independent of path over which the charge is transported. For example, consider moving a charge from point $\mathrm{B}$ to $\mathrm{C}$ and back to $\mathrm{B}$ in Fig. 3. One can calculate the amount of energy required to move along path $a$ and one can independently calculate the amount of energy required to move along path $b$. However, since the full return trip ($\mathrm{B}$ to $\mathrm{C}$ to $\mathrm{B}$) should result in zero net change in energy, we see that the energy along each individual path – $a$ or $b$ – has to be the same.

What about absolute potentials?

If point $\mathrm{B}$ in Fig. 3 is taken very far from point $\mathrm{C}$, the force on the test charge $q$ at this point would be practically zero (See Eq. (1)). The potential difference between $\mathrm{C}$ and this point at an infinitely large distance away is called the absolute potential of the point $\mathrm{C}$. The absolute potential of a point in an electric field may, therefore, be defined numeri­cally as the work per unit charge required to bring a small positive charge from a point outside the field to the point considered.

Since both work and charge are scalar quantities, it follows that potential is a scalar quantity. The potential near an isolated positive charge is positive, while that near an isolated negative charge is negative.

In this week's lab we can't move our probe out infinitely far away (or close enough for practical purposes), so we'll be working with potential differences.

Equipotential surfaces

It is possible to find a large number of points in an electric field, all of which have the same potential. If a line or a surface is so drawn that it includes all such points, the line or surface is known as an equipotential line or equipotential surface. A test charge may be moved along an equipotential line or over an equipotential surface without doing any work.

Lines of force perpendicular to equipotential surfaces

Since no work is done in moving a charge over an equipotential surface, it follows that there can be no component of the electric field along an equipotential surface. Thus, the electric field or lines of force must be every­where perpendicular to the equipotential surface. Equipotential lines or surfaces in an electric field are more readily located experimentally than lines of force, but if either is known the other may be constructed as shown in Fig. 4. The two sets of lines must everywhere be normal to one another.

Figure 4: (Solid orange) lines of force and (dashed blue) equipotentials, near two rods with equal and opposite charges.

Potential of a conductor

Electrons in a conductor can move under the action of an electric field. Thus, if an electrical conductor is placed in an electric field, this electron flow, which constitutes an electric current, will take place until all points in the conductor reach the same potential. There will then be no electric field inside the conductor – whether solid or hollow – provided it contains no insulated charge. Thus, to screen a region of space from an electric field it need only be enclosed within a conducting container. Since all parts of the con­ductor are at the same potential, the electric lines of force always leave or enter the conductor at right angles to its surface.

Getting started


Lab format and rubrics

If you were a student in the PHYS 12100, 13100, or 14100 labs last quarter, then this quarter's format will the same as you are used to.

If you are a new student in this lab sequence, take a moment to look over the “Lab Format” section on the lab course homepage here (below the course calendar). There you will see information about what the course goals are, how the labs are structured, and how the lab grading works. In particular, pay attention to the grading rubric which breaks down scoring for each experiment.

Lab notebook template

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

All members of the group are expected to contribute to all aspects of the experiment, including making notes in the lab notebook. If you brought a laptop or tablet to lab, you may want to open multiple copies of the notebook so that different group members can contribute simultaneously. If you did not, then it's OK to have just one group member typing in the document at a time (though you should regularly rotate record-keeping duties… both within the lab period and from lab-to-lab.)

Jupyter notebook

Last quarter, we introduced Python as a tool for doing calculations, making plots, and doing other sorts of data analysis. We will continue to use Python again this quarter, but for this first experiment, we are going to run our code using Jupyter Notebook (installed on the lab computers) instead of Google Colaboratory.

Why not use Google Colab again?

Google Colab is a very powerful tool for executing Python code, but it doesn't handle interactive plots well. For this lab, we want you to be able to click on a plot to add points as you collect data and we need Jupyter – a proper python notebook application – to do that.

Create a new (empty) folder on the desktop for your group. Download the two files from the following links to your lab computer and move them into your group folder.

The file “FieldMappingTool.ipynb” is the notebook containing the Python code. The file “FieldMappingSetup.py” contains extra code that is used by your notebook and just needs to be saved in the same directory as the notebook.

To run these files, do one of the following:

METHOD 1: Double-click FieldMappingTool.ipynb. This should open a terminal window (which needs to run in the background, but which you don't need to do anything with) and new tab in your internet browser window showing you the notebook.

If that doesn't work, then try…

METHOD 2: Open Jupyter Notebook on the lab computer first. (There should be a link on the desktop or search for it in the Start Menu.) This will open a terminal window (which needs to run in the background, but which you don't need to do anything with) and a new tab in your internet browser.

From that browser tab, navigate to where your file is located. (It should be in the new folder you created for your group on the Desktop.)

Experimental procedure


Apparatus

A diagram of the apparatus is shown below in Fig. 5.

Figure 5: An experimental setup for measuring electric field

The experiment will be performed with a tank of tap water about 3/8ths of an inch deep. Normal tap water contains sufficient numbers of ions to be a moderately good electrical conductor. A plastic grid is placed in the bottom of the tank to enable you to specify the location of the electrodes and potentials you measure. Reduced photocopies of the grid are provided to record your arrangement and data.

Several metal shapes are provided, including rectangular bars, cylinders, and rings. You will use these shapes, two at at time.

You'll apply a signal (a sine wave at about $1\text{ kHz}$) to two test electrodes. The resistor shown on the right clip is to prevent excessive current being drawn from the generator.

In Fig. 5, $V_{\text{ac}}$ represents a digital volt meter (DVM) set on an AC (alternating current) voltage scale.

What does that setting look like?

Figure 6: The AC voltage scale is the “V” with a tilde (~) next to it.

You are also provided with a probe consisting of two wires, held a fixed distance apart. When its banana plugs are plugged into an AC voltmeter and its tips are inserted into the water, the probe will measure the electric field in volts/distance.

Why are we using alternating current?

If we were to create a fixed, DC voltage, ions in the water would be attracted to our conductors and would start to react with them. This would corrode our conductors and generally make them unpleasant to work with with the chance of creating noxious chemicals.  However, this effect can be used in other circumstances, such as electroplating metals.

electroplating.jpeg

Field mapping

Attach the black cable from the function generator and the DVM to the same binder clip on the edge of the plastic water tank. Make sure the red cable from the function generator is attached to the resistor on the other binder clip. Set two metal electrodes at convenient locations on the grid in the tank and connect them each to one of the binder clips using the attached wires.

After turning the function generator on, choose one point and click on the plot in the Jupyter notebook to mark it. Be sure to set the voltage in the Jupyter notebook using the slider. Keeping the probe vertical, find several other positions with the same voltage and enter them into the Jupyter notebook by clicking on the corresponding location on the plot.  

Choose several additional voltages and map out equipotential lines for them as well; you will want at least 5 equipotential lines if possible.

After you have finished, save the figure in the notebook; the file will be in the same folder with a filename corresponding to the current date and time.

Other shapes

Clear your previous data and repeat the process of mapping equipotential lines, testing at least 2 more configurations using the different electrodes provided. Group members should cycle through roles here so that everyone gets some practice using the probes, mapping the field, and so forth.

One of your configurations must be two parallel conducting plates with a gap in between. This will be the electric field configuration which we use in Part 2 the for acceleration and deflection of electrons, so we want you to get a feel for how the field behaves this week.

Your other configurations can be anything you like.

Leave your last set of electrodes in place for the final part of the lab.

Extracting field information

Continue to the last section in your Jupyter notebook. Run the cell that generates an approximation of the electric field from your data. (This may take a minute or two). The program will highlight three field vectors for you to test.

Before using the field probe, measure the inside distance between its two wires. This distance (in $\text{cm}$) should be entered as the Correction_factor in your notebook, to account for the fact that the edges of the wires are not precisely $1\text{ cm}$ apart.

Use the two-prong electric field probe to measure the electric field for these points. To do so, hold the probe vertically with the middle at the indicated point in the water, with the prongs parallel to the direction of the field vector. See, for example, Fig. 7.

Figure 7: Configuration for using the electric field probe. Note that the DVM should be connected to only the probe (disconnect the other wires). The point you are probing should be between the ends of the wires; in this instance the probe is measuring the field in the horizontal direction at the point $(7, 0)$.

Remember to disconnect the wire connecting the meter (the “common” jack) to the function generator (the black jack) before connecting the two-prong field probe.

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

To help the next group, delete your group data folder from the desktop (to prevent clutter) and restart the computer (to clear the memory of your old Jupyter session).

Post-lab assignment


Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

Conclusions

The conclusion is your interpretation and discussion of your data.

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.