In this experiment you will measure the electrostatic force between plates of a parallel plate capacitor and compare your results with predictions derived from Coulomb's Law which can be written as,

$$F = \frac{\epsilon_{\circ} A V^{2}}{2 d^{2}}.$$

The constant $\epsilon_{o}$ is called the permittivity of free space.

This is a challenging measurement to make because the forces involved are small and fall off quickly as the plate separation increases. Making accurate measurements with small enough uncertainties to meaningfully test the model will require not only careful measurements, but also a detailed study of instrumental effects which may possibly bias the data. This is an important point to consider when doing experimental physics. You can make careful measurements, and still end up with data that point to an incorrect conclusion if you have not properly accounted for biases which are inherent in your experimental technique. Note that virtually all experiments have to deal with these sorts of biases, which are commonly referred to as systematic uncertainties.

The experiment will be done over two consecutive lab periods.

• The first part will focus on determining what Coulomb's Law predicts about how the force between the capacitor plates should depend on factors such as plate separation and voltage, making measurements to test those predictions and then evaluating how well the data compare to the model.
• The second part will focus on making additional measurements to assess and quantify the effect of instrumental biases which may be present in your data from the previous week, determining how to account for these biases in your measurement and then seeing how this changes your final conclusion.

The experimental apparatus is pictured above and consists of the following.

• A scale capable of measuring up to 200g, on which the bottom plate of the capacitor sits.
• An assembly which holds the top plate of the capacitor in a fixed position.
• A base on which the scale sits. The height of the base relative to the fixed position of the top capacitor plate can be precisely controlled using three micrometer screws.
• A variable 1500V power supply to apply a voltage difference between the top and bottom plates of the capacitor.
• A set of 5 20g masses.
• A single 200g mass.
• A precision ball bearing attached to the end of a wand.
• A ruler.

Taken together these components can be used to calibrate the scale, accurately set and adjust the spacing between the capacitor plates while maintaining a fixed high voltage and measuring the force between the plates.

We will charge the plates with a power supply, adjustable to about 1500 volts. You will use the output labeled 20 MΩ. The high voltage cable should be plugged into the small box on the left side of the apparatus. This box contains a 100 MΩ resistor, which serves to limit the current to a safe value to avoid shock hazard.

Use the link below to get a copy of the file for this lab, and add your group members' names to the document. As soon as your new document opens, be sure to share it with all members of the group by selecting “Share” in the upper right corner and adding their UChicago email addresses.

Your goal is to measure the force between the charged plates of the capacitor as a function of the distance between the two plates. The apparatus allows you to apply up to a 1500V potential difference between the plates of the capacitor. The bottom plate of the capacitor sits on a scale which can be used to measure the force between the plates. The scale with the bottom plate can be raised and lowered to vary the distance between the plates using the three micrometers.

You will need to bring the plates of the capacitor as close together as possible, within a fraction of a mm. This requires precise adjustment of the separation between the two plates using the micrometers. The procedure for setting the distance between the plates is straight forward, however it must be done carefully. Your TA will demonstrate how to do this, but the basic procedure is as follows:

• Establish a known plate separation and record the readings of the three micrometers.
• Use the micrometers to adjust the distance between the plates in known increments.
• Use the scale to measure the force between the plates.

With a little bit of practice setting and adjusting the distance between the plates is quick and easy. Keep in mind the following points while taking data.

• What are your measured quantities and what are their uncertainties.
• The plates must remain parallel, be sure you adjust each micrometer by the same amount and in the same direction.
• Use a voltage close to 1500V. Make sure the voltage between the plates constant and stable.
• At some point the plates will be so close together that they begin sparking. The smallest distance between the plates for which you can collect data is point just before sparking occurs.

Understanding the impact that your apparatus have on your data is a crucial part of experimental investigation. The tools and methods which you use can affect your measurements, potentially leading you to draw incorrect conclusions. In this experiment the scale that you are using to measure the force between the plates of the capacitor will change the distance between the plates in a manner that varies with the applied force. As illustrated in Figure 2 below, the bottom plate rests on the mass pan of the scale, which is attached to one arm of a U shaped piece of metal with a strain gauge attached. When a force is applied to the scale the top arm of the U is pushed downwards, creating strain in the U which is measured by the gauge. You can easily observe this effect by pressing down on the scale with your finger. This means that the instrument you are using the measure the force will change the distance between the plates as a function of the applied force.

Figure 2. Diagram illustrating how the mass pan attaches to the strain sensor.

The actual distance between the plates are the separation which you set using the micrometers plus the deflection caused by the force. This effect will be most apparent when the forces are largest, which corresponds to when the plate separation is smallest. In order to make a precise measurement you will need to assess and correct for this additional displacement. Doing so is the subject of next weeks lab.

The questions of how many data points to collect and what distances to use are ones that you, as the experimenter, need to answer for yourselves. The only way to make informed decisions about things like do you have enough data or are your error bars small enough, is to actually look at your data as you are collecting it. Someone in the group should open the Jupyter Notebook and be entering the data as it is collected. By examining the plot of force vs. distance you can assess the quality of your data.

In the first part of this lab you hopefully were able to collect data which showed an increasing deviation of your data from the form of a straight line as the force between the plates increased. This was to be expected because of how the scale used to measure the force was designed. In today's lab you will determine how much the mass pan of the scale deflects as a function of the applied force and use this data to correct for this effect in the data you took last week. In principle this should improve the agreement between the model and the data.

You will use the same apparatus as last week, with the bar holding the top plate of the capacitor replaced with one that holds a micrometer. You can now use this micrometer to measure the deflection of the mass pan as you add known masses to it.

For your individual summary discuss the following.