In mechanics, we learned how to compute forces between different objects in order to determine how these objects would move. In many cases, we considered the 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. However, when we learned about gravity, we introduced the concept of a field, which revealed itself to be quite a powerful tool.

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 even from a source that consists of a large, extended object. An immediate example of this is projectile motion: given the presence of the gravitational field of the Earth, we know the trajectory that launched objects will follow over the surface of the Earth. Instead of focusing on the force between two objects (e.g., a ball and the Earth), we now focus on the force that an object feels as it moves through a field (e.g., a ball moving over the surface of the Earth).

In the next few labs, you will apply your knowledge of electric fields to study the motion of charges in their presence. More specifically, you will focus on the phenomenon of electrophoresis, which is the motion of suspended charged particles when they are subjected to an electric field.

Before diving into the motion of charges, however, it is helpful to gain an experimental understanding of how different charge configurations give rise to electric fields and potentials. As such, in this first experiment, we will be mapping out the fields that arise when we submerge charged electrodes of varying shapes in water. As you perform the experiment, keep the following learning goals in mind.

• Relate electrical potential measurements to the electric fields that are associated with these potentials.
  • Electric fields are intangible objects that cannot be directly observed. However, they do have measurable effects on objects around them, and by observing these effects we can learn to indirectly characterize them.
• Become acquainted with specialized electrical equipment that is important in the study of electricity and magnetism and which you will encounter in the following months.
  • These include power supplies, function generators, voltmeters, ammeters, and oscilloscopes.
• Start thinking about the motion of charges in the presence of electric fields.
  • How are ions in water moving when we submerge the charged electrodes?

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

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 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

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:

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.

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.

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.

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.

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.

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.)

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.

Create a new (empty) folder on the desktop for your group. Download the two files below 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.)

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.

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.

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.

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.

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

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 $(7, 0)$.

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!

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.

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.

In addition to your conclusion, try to include in your report answers to the following questions. Feel free to discuss among your peers or with the TA for ideas, but make sure the answers are your own. Do not worry too much about a precisely correct answer. Instead, think about what you have learned about electric fields and the questions that would arise when thinking about ions moving in electric fields.

1. In this experiment, we used an AC voltage instead of a DC voltage to generate the electric fields. The main reason for this is that a DC voltage would cause a reaction between the electrodes and the ions in the water, which has the potential to generate unwanted chemicals. With an AC voltage of amplitude $V_{AC} > 0$, the voltage oscillates back and forth between $\pm V_{AC}$, so that, while the reaction occurs, it is constantly reversed so that there is no net reaction.
   1. On average, where are ions in water moving due to the alternating electric field? Recall that the field is always flipping in direction.
2. Based on the field you got for the parallel plates, what motion do you expect a charge to experience between them?
   1. If a particle has charge $q$ and you apply a voltage $V$ across plates separated by a distance $d$, what is the force on the particle?
   2. If you wanted to separate particles based on how much charge they have, would you then use an AC or a DC field? Why?
3. If you have a particle with charge $q$ that is submerged in water, how would it interact with ions in the water?
   1. If the particle has charge $q>0$ and there are ions with charge $Q <0$ what would happen?
   2. Would the ions affect the force that the particle experiences due to the field? Why?