Two pieces of equipment that will be very helpful to us in our investigation of capacitance are the function generator and the oscilloscope. The former produces voltages that are periodic in time \((\rm{for\:example},\:V(t)=V_{0}\sin{\omega t})\), and the latter allows you to observe and measure these periodic waveforms. In the first part of this lab you will learn how to use these devices by playing with them extensively; no further introduction is needed to begin. However, a brief introduction to capacitors and \(RC\) circuits is needed for the remainder of the lab.
A capacitor is a device that stores charge. It turns out that the amount of charge stored (\(+q\) on one “plate”, \(-q\) on the other) is proportional to the voltage difference \(\Delta V\) across the capacitor, where the proportionality constant \(C\) depends on the geometry of the device and on the details of what’s between the plates (vacuum? a cell membrane?):$$ q = C \Delta V.$$
We call the proportionality constant \(C\) the capacitance. Don’t confuse it with the abbreviation for coulomb, \(\rm{C}\)! Capacitance is measured in farads, \(\rm{F}\), where \(1\:\rm{F}=1\:\rm{C}/\rm{V}\). As you will see, typical capacitors have capacitance values that are only small fractions of a farad.
It takes time to charge (or discharge) a capacitor. You can show that if you put an uncharged capacitor in series with a battery and a resistor at \(t=0\), the voltage across the capacitor at time \(t≥0\) will be $$\Delta V = V_{\rm{batt}}\left(1-e^{-t/RC}\right).$$
Similarly, if the plates of a charged capacitor are connected by a resistor, the capacitor will discharge according to $$\Delta V = V_{0}e^{-t/RC},$$ where \(V_{0}\) is the initial voltage across the capacitor.
We define the time constant \(\tau=RC\), which characterizes how long it takes to charge or discharge a capacitor by some amount; \(1\) time constant is the time it takes to gain \(63\%\) of the final charge when charging a capacitor, or to lose \(63\%\) of the initial charge when discharging a capacitor. Measuring time constants for \(RC\) circuits provides us with a nice way to measure capacitance if we know resistance, or vice versa.
Note that this is a two-week lab, with the weeks graded separately. We will stop here in the middle of the Data Collection and Analysis section at the end of the first week.
Equipment Data Collection and Analysis Summary Questions
