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# 5.2: Activities

## Things You Will Need

For this lab, you will need a stopwatch timer (like found on a smart phone), and this PhET simulator. Go ahead and run this simulator now in another window for use in the activities below.

## Find the Resistance of the Light Bulb

The setup for this lab is very simple. Start by opening the "Light Bulb" section of the simulator. What you find is a capacitor that can be charged by a battery when the switch is set to the left, held at a fixed charge when set in the middle, and discharged through the resistor when set to the right. Prepare the experiment as follows:

• Set the voltage of the battery to is maximum of $$+1.5V$$.
• Move the capacitor plates to their minimum separation of $$2.0mm$$.
• Increase the capacitor plate area to its maximum of $$400mm^2$$.
• Connect the voltmeter across the two capacitor plates.
• Select the boxes for the "Top Plate Charge" and "Stored Energy" bar graphs.

Go ahead and turn the switch all the way to the right and observe what happens. Then when you have seen enough, throw the switch back to the left to recharge the capacitor.

You will note that there is nothing in the simulator (no label or meter) that provides us with the resistance of the light bulb. We can measure the voltage across the resistor, but without an ammeter, we can't use Ohm's law to measure it, either. Your first task is therefore to use what you know about RC circuits determine what this resistance is.

The main thing that distinguishes RC circuits from what we have done before is the time-dependence, so we need to be able to take data in "real time." It turns out that the simulator provides us with a way of doing this. To see how, drag the switch to the right to engage the light bulb, but don't release the mouse – you'll note that the current doesn't flow while the mouse held down. If you release the mouse, current will flow, and if you click and hold it down again (without moving the switch), the current stops again. This means that with one hand you can tap the "start" button on a stopwatch, while simultaneously releasing the mouse with the other hand. The timer records how long the current flows, and then you can tap the stopwatch again to stop it while clicking-and-holding the mouse to stop the flow. While holding the mouse down, turn the switch back to the middle setting (not to the left – you don't want to recharge the capacitor!), and you can write down the data at your leisure before getting the current and stopwatch running again.

1. Take data in the manner described above, noting the elapsed time (starting with $$t=0$$), the charge on the capacitor, the voltage across the plates, and the stored energy for at least 8 moments in time, spread out fairly evenly to the point where the bulb gets pretty dim (but not all the way dark). Arrange this data into a table.
2. Plot your $$Q$$-vs-$$t$$ data points using the desmos graphing calculator. You plot points by typing them into the left panel as a series of ordered pairs: (1.2, 3.4), (–2.3, 6.0), etc.

We know that the charge on the capacitor is supposed to decay exponentially:

$Q\left(t\right)=q\;e^{-\frac{t}{T}}\;,$

where $$q$$ is the starting charge, and $$T$$ is the time constant. Put this equation into the left panel of the desmos calculator, in a new box below the ordered pairs of your data. [Note: To insert an exponent, type a '^' (shift-6).] When you do, the calculator will give you the option of adding "sliders." Sliders make it convenient to change the constants in the formula. In this case, $$Q$$ and $$t$$ are our variables, so add sliders for $$q$$ and $$T$$.

1. Set the constants $$q$$ and $$T$$ so that the graph matches the data points. You can type numbers into sliders for greater precision, and it's easiest to stay in pico-units like shown in the simulator – don't put factors of $$10^{-12}$$ into your graph! Include a copy of this graph (with data points and function) with your lab report. There are many ways to do this, but perhaps the simplest is to use the scanning app on your phone to scan the graph directly from your computer screen.
2. Use the details of the graph that you have matched to the data to calculate the resistance of the light bulb.

Next we will confirm our result by repeating the experiment from a different starting point. We're using the same light bulb, so the resistance should come out the same. Set up the system as follows:

• Turn the switch back to the left, to recharge the capacitor (with the same minimum separation and maximum area settings as before).
• Disconnect the battery by turning the switch to the middle position.
• Pull the plates apart to their maximum separation of $$10.0mm$$ (leave the plate area where it was).
• You can remove the voltmeter – you won't need voltage measurements for this part.

Noting that the plates start with the same charge at $$t=0$$ as in the previous experiment, start the capacitor discharge and the stopwatch simultaneously, let the capacitor run down for awhile, and then stop the discharge and stopwatch simultaneously. You only need to do this once (you don't need a bunch of data points as before).

1. Note any obvious physical differences you witness in the process from the previous experiment, and give a brief explanation for what you think is the cause of this difference.
2. Rather than use a graph to find the resistance, use the two data points for this run (at $$t=0$$ and at your stopping time) to compute the resistance, showing it comes out the same as before.

## Accounting for Energy

Clearly the potential energy stored in the capacitor is reduced as energy leaves the system in the form of light from the bulb. We will now confirm that these numbers add up. The rate at which the energy leaves the light bulb is the power. Now that we know the light bulb's resistance, we can use it with the voltage values recorded in the first part of the lab to determine the power dissipated at each moment that data was taken.

1. Add a column to the data table you created in part (1) to include the power emitted by the light bulb at each moment in time.
2. Clear whatever data is currently in the desmos calculator, and plot the points for power vs time. Noting that the power curve is also exponential, find the function that fits it, and as before, include a copy of this graph (with data points and function) with your lab report. You can try to use some physics to figure out what this function should look like, but it isn't necessary to do so. Instead, you can just write the function below into the graphing calculator, create sliders for $$p$$ and $$a$$, and adjust them until the curve matches the data (you may have to increase the sensitivity of a slider by reducing its range, or alternatively, type-in values to get the level of precision you need to make the function match the data).

$P\left(t\right) = p\;e^{-at}$

Now that you have the power loss as a function of time, you can determine the energy lost between any two points in time by integrating this function between those two times. To do this with the graphing calculator, go to the panel in the desmos calculator containing the equation for $$P\left(t\right)$$ and follow these steps:

• Change the left-hand side of the equation from '$$P(t)$$' to '$$U$$' (we are adding up the energy by integrating the power).
• Click the cursor just to the right of the equal sign, and type 'int' (without the quotes) – an integral symbol should appear.
• Enter two separate times for which you collected data into the limits of the integral.
• Click the cursor at the end of the equation and type-in the differential 'dt' – the value of the integral should then appear in the panel.
1. Show that this gives the correct value for energy lost by the circuit during that time span. Confirm that it works for another time span as well. At least one of these calculations should not include the data point at $$t=0$$.

## Lab Report

Download, print, and complete this document, then upload your lab report to Canvas. [If you don't have a printer, then two other options are to edit the pdf directly on a computer, or create a facsimile of the lab report format by hand.]