# Capacitor Lab

Note: This is for me and anyone else that needs remote physics lab data. My goal is to keep this as simple as possible.

What is a capacitor?

The very basic idea of a capacitor can be two parallel conducting plates with an insulator between them. It could literally just be two metal plates with air between them.

When a capacitor is connected in a circuit, negative charges move onto one of the plates, but they can’t jump to the other side because of the insulator gap. However, these charges DO create an electric field that can interact with the negative charges on the opposite plate. This pushes those charges off the plate such that it appears there is a continuous electric current. Also, with negative charges leaving that plate, it now becomes positive.

Remember that I is the direction of positive moving charges —but it’s almost always negative moving charges.

This build up of charge creates an electric potential difference from one plate to the other. The ratio of charge to voltage is defined as the capacitance.

$C = \frac{Q}{\Delta V}$

Where Q is the charge on just one plate (since the total charge is zero) and C is the capacitance. If Q is measured in Coulombs and &Delta;V is in volts, then C is in units of Farads.

So, what happens when we connect this capacitor to a battery with a resistor? Here is a simulation.

I was going to write a bunch more stuff—but I apparently already did. Here is an older post on RC circuits.

https://rhettallain.com/2019/06/21/rc-circuit-as-an-example-of-the-loop-rule/

Now I will just focus on this lab. Here is some experimental data (in the form of a video).

Here’s what to do.

• Look at the part where the capacitor is charging. Collect data for the voltage across the capacitor as a function of time. More data is better—but don’t go crazy. If you can get a data point every 1 or 2 seconds, that should work.
• Now make a plot of voltage (on vertical axis) vs. time (horizontal axis).
• Repeat this for the discharging capacitor.

But what about a linear plot? For a capacitor (C) discharging through a resistor (R), the voltage should be:

$V(t) = V_0 e^{-t/RC}$

This is not a linear equation. Divide both sides by V0 and take the natural log. This is what happens.

$\frac{V}{V_0} = e^{-t/RC}$

$\ln \left(\frac{V}{V_0}\right) = -\frac{t}{RC}$

Now if you plot ln(V/V0) on the vertical axis and t on the horizontal axis—it should be a straight line. EVEN BETTER, the slope of this line means something. Use the graph to find the value of C if R = 150 Ohms.

Now go back to the other post. See if you can create a numerical model for a discharging capacitor. Here, this might help you get started.

This program is not finished.

https://trinket.io/glowscript/f87df8f621

Questions:

• What happens if you change the time step (dt)?
• What happens if you change the capacitance?
• What happens if you change R?
• What happens if you change the initial voltage?
• Can you make a model for a charging capacitor?

Suppose you built your own capacitor and you wanted to discharge it. For this capacitor, you used two sheets of aluminum foil separated by a page in your textbook. The capacitance can be calculated as:

$C = \frac{\epsilon_0 A}{d}$

where

$\epsilon_0=8.854\times 10^{-12} \text{ F/m}$

Use this to estimate the value of the capacitance. How long would it take to get halfway discharged using a 150 Ohm resistor? Could you essentially repeat this experiment with your homemade capacitor? Hint: no.

Bonus here is a capacitor that you CAN make.

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