Lab 3: Capacitors

Purpose

We explore capacitors by building different configurations, taking data and analyzing it, and drawing conclusions about the relationships of the variables.

Equipment

Each station should have: A set of capacitors, a digital multimeter, a digital caliper.

Background

Generally, capacitors are two conductive plates where charges can accumulate if work is done to move them (remember they will repel each other) onto one of the plates. As charges build up on one plate they induce a charge on the other plate. Capacitors can be used to store energy, but they also have other applications such as filtering noise out of signals as shown below.

Here a signal plus noise comes from the left.

The capacitor (parallel lines) removes the noise

and outputs the clean signal to the right.

Capacitance

A key attribute of a capacitor is the capacitance (C).

(1)

Where k is a unitless constant that depends on the properties of the particular material between the capacitor plates, it is 1 for empty space and has larger values when materials are inserted; = 8.854*10^-12F/m is the permitivity of free space, a constant of our universe, and a unit conversion factor; A is the area of the plates (A=l*w for our rectangular plates), and d is the distance between the plates.

Here you can see how a capacitor with nothing between the plates behaves; whereas, one with a material between the plates results in the polarization of the material (dielectric) and an increase in the capacitance.	In this diagram you can see how randomly oriented polar molecules become aligned when placed in an electric field such as between the plates of a charged capacitor.
As shown, the electric field is quite uniform between the plates, and diverges heavily outside the plates. This allows us to focus on the interior region of the capacitor as the exterior is relatively insignificant for our purposes.	This space left blank for you to draw something fun, if you want.

(2)

(3)

A couple of the different ways we can calculate the energy stored in a capacitor are shown above. Here U is the energy, C is the capacitance, Q is the charge, and E is the electric field. Later you will have additional ways of relating these and other variables.

The units of capacitance is farad (F) which is the same as Coulomb/Volt. Since Farad is a large unit, usually we express capacitance C in any of the following units:

1F (microfarad) = 10^-6F 1nF (nanofarad)= 10^-9F 1pF (picofarad)=10^-12F

The Parallel Plate Capacitor

The parallel plate capacitor consists of two plates separated by a dielectric (insulator).

Here you can see our ready-made capacitor which consists of a fiberglass board as the dielectric and two extremely thin coatings of copper as the conductive parallel plates.

We note that the electric field in the region between the capacitor plates is given by

(4) E=0

Experimental Approach

We wish to discover each of the variable relationships in equation 1. Copied here:

(1)

We can utilize our multimeters to measure the capacitance of each variation of our capacitor that we build. In this way, we can vary one of the variables on the right side of the equation while holding all others constant and see how it affects the capacitance. Is the relationship direct, inverse, squared, etc.? In each trial we will take several data points, make a curve fit, and use the resulting equation to verify the relationship between the variables. Often theorists will come up with equations to describe features of our universe, and then experimentalists will design experiments which can confirm or deny the relationships given by the theory. Here we have equation 1, so first we will vary the distance, d, between the plates.

Equipment Tips:

Be careful and make sure that the probes of the multimeter are only making contact with one side of the capacitor each. If a probe touches both sides you won't be able to get a reading. In the image here, you can see that this type of probe has been positioned very close to the edge so that the back portion of the probe doesn't make contact with the thin edge of the opposite side, and in the background you can see how the second probe is placed reversed.

2. Capacitance vs Area

Measure the dimensions of the plates to get the area of the plates. This should be essentially identical for each plate.

Table 2. Area of a capacitor

Area of plate 1 [m2]	Area of plate 2 [m2]	A= average of plates' area (m2)

1. Make sure the probes of your multimeter connect to COM and the capacitor symbol on your multimeter.
2. Connect the probes to the sides of a parallel plate capacitor and record the measured capacitance.
3. Devise a means to increase the area of your capacitor to additional sizes, and take measurements for those. Hint: Look closely at the images above, and we are giving you copper conductive tape to use.

Table 3. Capacitance vs Area table.

A [m2] (x)	C [nF] (y) (nano is )

What happens to the capacitance when you double/triple the area? Is this consistent with the theory (equation) for the capacitance?

5. Series Capacitors

The good news: you just dealt with capacitors connected inparallel (connected side-by-side), and that will be a recurring topic which we will view in more detail later. Now, let's look at what happens when we stack capacitors inseries on top of each other.

You will see these equations and relationships between parallel and series capacitors in detail in future labs! For now we just want to use the concept of parallel capacitors where we have two dielectrics side-by-side and series capacitors where they share a common middle plate (not visible) as in the second diagram.

1. Measure the capacitance of two capacitors with your multimeter. Also measure the thickness of each capacitor.
2. Use copper conductive tape to make a good connection between two of your capacitors such that they are stacked directly on top of each other with only the middle plates making electrical contact. This will look like:
   2. Notice how the tape is stuck to one plate and ready to contact the other, then the plates are stacked. Once stacked, the only plates that matter to us are the two outermost plates.
3. Measure the capacitance of the combination of plates in series.

	d (thickness)	C (capacitance)	k (calculated)
Capacitor 1
Capacitor 2
Series Capacitor	(sum, do not measure directly)

What happens to the capacitance when you double the distance? Is this consistent with the theory, , for the capacitance?

Use the calculated values for k to calculate . We expect this to be low since the dielectric is the same material in each of our capacitors. Show the %Error here.

Store-bought, pre-made, capacitors.

Now that you have some insight as to what a capacitor physically is, we will use compact capacitors as you would find in electronics and explore the relationship between the charge they hold and their properties.

Data & Data Analysis:

Use the following 2-part experimental procedure to collect your data and perform the relevant data analysis.

Part 1. Stored Charge in a Capacitor

Purpose:

To evaluate the relationship between the charge held in a capacitor and the potential difference (voltage) across the capacitor's plates.

Materials:

• Variable power supply
• prewired capacitor
• Voltmeter
• charge meter
• capacitance meter.

The power supplies in this lab work just like batteries, except they're adjustable. Notice that the output of the power supply has a "-" connection on the left and a "+" connection on the right. These act like the terminals of a battery. (Don't use the "GND" in the middle.)

Your instructor will show you how to use the lab's "multimeter" to act as a Voltmeter (to measure voltage differences). Later, you will also use this multimeter in other ways.

Your instructor will also show you how to operate the "chargemeter", to measure the charge on a Capacitor.

Procedure:

WARNING: DO NOT EXCEED 2 V on the variable power supply during this part of the experiment.

1. With the variable power supply set at 0(zero) volts, connect your capacitor to the variable power supply, volt meter, and charge meter as shown in the diagram below:

Figure 1.1 Diagram showing the setup for the experiment

2. Apply about 0.1 volts to the system (this value should be read using the voltmeter) and record the charge stored on the capacitor as demonstrated by your lab instructor.

3. Increase the voltage by about another 0.15 volts, and repeat the previous step. Do this until you reach about 1.5 volts. You can organize your data in a table similar to the figure below (Figure 1.2), except make it vertical instead of horizontal. (Useful Suggestion:Use Google Sheets since you'll be asked to graph next anyway, you can select and copy-paste this table into a Google Sheet to speed your setup). Also, notice that this example table is incomplete in that it does not specify the units.

Voltage [x-axis]
Charge [y-axis]

Figure 1.2 Example Table of Voltage vs. Charge

4. Using your data, make a graph, and apply a curve fit. In order to identify what type of curve fit to use, Interpret your data using the equation:

Q=CV

where Q is the charge on the capacitor, C is capacitance, and V is the voltage.

To fit a curve in Sheets, double click your graph Customize Series scroll down check Trendline, and select Trendline options as appropriate. You will also want to click Label Use Equation, and check the R² box as well.

5. Use the result from your curve fit to find a value for your measured capacitance. (Warning: do not pick a single data point and analyze it; the idea is to get a value using all your data at once.)

6. Explain how you identified your value for capacitance from the curve fit. i.e. What do the parts of the equation mean?

7. Remove the capacitor from the circuit. Then use a capacitance meter to measure the actual capacitance of your capacitor.

8. Show a calculation of the percent difference between this value and the one found using your curve fit.