In this lab, we will be measuring the capacitance of a "fixed-capacitance" capacitor using a variable capacitance capacitor both in series and in parallel.^{Hoveroverthese!}

• 2 Capacitors: 1 Fixed, 1 Variable
• 1 Voltmeter
• 1 Power Supply
• 1 Double-pole double-throw (DPDT) Switch
• 9 banana cables, 2 (or more) alligator clips
• Record data in this Google Sheets data table

Capacitor Basics

Suppose one has two conductive surfaces, one with charge \(Q\) and the other with charge \(-Q\). These conductive surfaces will have a voltage \(V\) between them, which is proportional to the charge \(Q\) with a constant of proportionality that depends on the shape and location of the surfaces.

This constant of proportionality, known as the capacitance \(C\), is defined by:

$$Q=CV\label{Cdef}$$

A common simple capacitor setup is a parallel plate capacitor, which consists of two conductive plates of area \(A\) separated by a distance \(d\) with some insulating material in between. With the assumption that \(A\gg d^2\),¹ this is (approximately) directly computable:²

$$C=\frac{\epsilon A}{d}\label{Cplates}$$

The material in between is usually chosen to enhance the capacitance; such a material is known as a dielectric. Then, \(\epsilon\) is a constant known as the permittivity of that material. If there is a vacuum (or, approximately, air) in between, then \(\epsilon=\epsilon_0\) is the permittivity of free space.

Capacitors In Circuits

In general, capacitors in circuits are governed by two main rules:¹

1. Kirchoff's voltage law: Any two "paths" through a circuit have the same sum of voltage changes across each component.
   • Corollary: components wired in parallel have the same voltage drop across them.
   • Corollary: components wired in series to a power supply have their voltages add to the voltage of the power supply.
2. Conservation of charge: If a portion of wire is not connected to a power supply, the sum of charges on plates attached to that wire remains fixed over time.

In this lab, we will first put two capacitors in series, one of variable capacitance and one of fixed capacitance. They will begin totally discharged, and then (with the flip of a switch) we will charge them to some total voltage \(V_0\). In this process, the final charge on both capacitors are equal.

For a variety of total voltages \(V_0\), we will measure the voltage across each capacitor. We can compute the charge across the variable capacitor, and use this to deduce the charge across the fixed capacitor. From the charge and voltage measurements for the fixed capacitor, we can deduce the capacitance (which can be compared with the listed value).

Then, we will wire the capacitors in parallel. We will begin by charging the fixed capacitor up to a known voltage \(V_0\). With a flip of a switch, we will disconnect the power supply and connect the variable capacitor in parallel with the first capacitor. We will then measure this final voltage, \(V_{||}\).

Since the charge across the fixed capacitor initially is the same as the total charge across both at the end, we can relate the capacitance \(C_1\) to the final "parallel capacitance" \(C_{||}\):

$$Q_\text{tot}=V_0C_1=V_{||}C_{||}\label{Qtot}$$

We know on theoretical grounds what we expect for \(C_{||}\) in terms of the individual capacitances \(C_1\) and \(C_2\):

$$C_{||}=C_1+C_2\label{Cpara}$$

RC Circuits

In real circuits, capacitors do not instantly equilibrate - the charges take time to move. The length of time it takes for it to reach equilibrium depends on the resistance of the other components - how much they resist the flow of charges (i.e., current) through them. The larger the resistance, the slower the charges move, so the longer it takes to reach equilibrium.

This behavior can be summed up as follows.² With a resistance (with SI units of ohms, denoted by Ω) of \(R\), the time it takes to get "close" to equilibrium³ (or time constant) is denoted by:

$$\tau=RC$$

An "ideal" wire should have zero resistance - it would be a perfect conductor, so charges move around instantaneously. (Real wires, combined with realistic contact resistances, can add up to ~Ω-scale resistances across the lengths of wire used in this lab.)

On the other hand, an "ideal" voltmeter should have infinite resistance, so that charges cannot "leak" through it. (A real voltmeter will tend to have ~MΩ-scale resistances.)

Part I: Capacitors in Series

First, take the fixed capacitor (i.e., the object without any dials or anything on it). It should have a listed capacitance (of 5 or 6μF) and corresponding uncertainty; find and record this number as your theoretical \(C_1\) [for Part I; the corresponding boxes in Part II should fill automatically].

Then, wire the capacitors in series, according to the following diagrams:

Click here for pictures that include the voltmeter.

Click here for wire-by-wire instructions.

For simplicity, for the rest of this part, we use the convention that "left" on the switch means "in the state where the capacitors are charged", and "right" means "in the state where the capacitors are discharged." (This is consistent with the orientation in the above pictures and wiring instructions.)

Discharge your capacitors (i.e., put the switch in the right position), set your voltmeter to a 20V maximum, and set your power supply to output 3V. Then, using the voltmeter, measure and record the actual voltage across your power supply.¹ This should remain constant throughout this experiment (although it won't actually change your capacitance results if you vary it).

Now: set the variable capacitor \(C_2\) to 2μF, and record this value. Wire your voltmeter so it is measuring across the variable capacitor \(C_2\) (although it should be reading zero at present, since the capacitor is discharged).

Now, let's check that we have hooked it up correctly. Flip your switch to the left and look at your voltmeter: it should jump up to some (positive) voltage, then slowly start to drop. Then, flip it back to the right; your number should drop to zero. (If it doesn't change as you expect, check your wiring, then consult your TA.)

Alright, now to actually take the measurement. We want to record the number the voltmeter measures after the "jump up," but before the "slow drop"; in this part, this should be the maximum value the voltmeter displays. Flip the switch and record this as the voltage across the variable capacitor, \(V_2\). Then, discharge the capacitors by flipping the switch right again.

Now, move the voltmeter so that now it measures across the fixed capacitor \(C_1\) instead of the variable capacitor \(C_2\). As you did when measuring \(V_2\), flip the switch left and record the "end-of-jump" voltage that appears, this time as \(V_1\). Then, flip the switch right and let it discharge again.

Repeat this process (measuring \(V_1\) and \(V_2\)) for different values of \(C_2\), up to 10μF in increments of 2μF.

Part II: Capacitors in Parallel

Now, we want to wire our circuit so that one direction on our switch connects \(C_1\) to the power supply, and the other direction connects it to \(C_2\).

We also want to be able to discharge capacitor \(C_2\) between trials, so we'll set up our switch to do that, too, for convenience. (We could also "short out" the capacitor by touching a wire to each terminal of it, but there is no need to do so, since our switch can do that job.)

Accordingly, wire up your circuit according to the following diagrams:

Click here for wire-by-wire instructions.

For this part, we use the convention that "left" on the switch means "in the state where we charge the capacitor \(C_1\) and discharge \(C_2\)", and "right" means "disconnected from the power supply, with the capacitors connected together." (Again, this is consistent with the above pictures/instructions.)

Flip your switch to the left position, and set your power supply to output 3V. Turn on your voltmeter with a 20V max; it should be reading approximately 3V. Record this voltage as \(V_0\).

Now, set \(C_2\) to 10μF and flip your switch to the right position. Now, you should record a sudden drop in voltage (as \(C_1\) dischrages into \(C_2\)), followed by a slow decline to zero. (Again, if you don't see this, consult your TA.) Flip it back left to re-charge \(C_1\) and discharge \(C_2\).

The number we are going to want to record (as \(V_{||}\)) is the number that appears at the end of the sudden drop, but before it changes from the slow decline. Unfortunately, unlike the previous part, there isn't an easy criteria here (like the "maximum value"), but it should be pretty easy to judge by eye.²

Now: flip the switch back right and record \(V_{||}\) as described above.

Part III: RC Circuits

You will find an RC circuit, driven by the function generator, wired as follows:

With this setup, the oscilloscope displays both the "input" voltage (from the function generator) and the "resulting" voltage on the capacitor. The variable resistor is set to 100Ω and the capacitance is set to 1μF. The function generator is set to output a ~300Hz square wave.³

What you observe should highlight how the capacitor repeatedly charges and discharges and the function generator changes its input voltage.⁴

On the provided sketch paper, make a sketch of the oscilloscope screen, noting that Voltage is on the Y-axis and Time is on the X-axis.

Now: vary both the capacitance and resistance, and record your observations. In particular: what happens to the "rate of charging" as you vary the resistance? The capacitance?

Part I: Capacitors in Series

For each data point, calculate \(Q_2\), the voltage on the variable capacitor, and propagate uncertainty. Note this is also \(Q_1\).

Make a plot of \(Q_1\) vs. \(V_1\). If you orient your axes appropriately, the slope of this plot should be \(C_1\).

Based on your results, answer the question about compatibility with expectation.

Also add together \(V_1+V_2\), propagating uncertainties. Comment on the outcome in your report.

Part II: Capacitors in Parallel

Calculate the theoretical value of \(C_{||}\) using equation \eqref{Cpara}, using the theoretical value of \(C_1\), and propagate uncertainties.

Then, compute the experimental value using equation \eqref{Qtot} (and the same value of \(C_1\)), and propagate uncertainties.

Based on your results, answer the question about compatibility with expectation.

Part III: RC Circuits

Include your oscilloscope sketch, which should have the following:

• A title for the plot, plus labelled axes ³;
• A sketch of the curve that is visible on your screen (of course), drawn to the best of your ability. Indicate which curve is the function generator and which curve is the circuit response.

Your TA will ask you to answer some of the following questions (they will tell you which ones to answer):

Experimental Questions:

1. For each of the first two parts, draw a circuit diagram illustrating the schematic setup of the circuit.⁴ A few notes to help you out:
   • For the first part, just draw the voltmeter measuring across the variable capacitor \(C_2\) [so you don't end up drawing a bunch of extra circuit diagrams with the voltmeter moved for this part.]
   • Note that there are two ways to represent the switch: you can use an appropriate representation of a DPDT switch (i.e., a switch with six inputs set up in the way we did here), or you can draw two circuit diagrams per part (one for each connection setting on the switch, excluding the disconnected "middle position").
   • If you choose to use a DPDT switch in your circuit diagram, here's a short explanation of the symbol (in terms of the diagramming software linked in the above footnote):
     • The DPDT switch symbol has two connections on the left (call the top one A and the bottom one B), and two connections on the right (call them 1234 from top to bottom).
     • A/B are the "middle terminals" on our switch, and 1234 are the "corner terminals."
     • If our switch is in the left position, this corresponds to connecting A to 1 and B to 3 in the diagram. If our switch is in the right position, this corresponds to connecting A to 2 and B to 4 in the diagram. [Or, equivalently, you can swap left and right in that description, since the switch is symmetric.]

Theoretical Questions:

1. Explain/derive formula \eqref{Cpara}, taking \(C_{||}\) to be defined as in equation \eqref{Qtot}.
2. There is a formula somewhat analogous to \eqref{Cpara} for capacitors in series: if you take the setup described in part I, you relate the charge \(Q\) on either capacitor⁵ to the voltage of the power supply \(V\) and find that \(Q=C_\text{series}V\) for some \(C_\text{series}\) that can be expressed in terms of \(C_1\) and \(C_2\). State what this formula is, and prove that it holds.
3. Derive formula \eqref{Cplates}, using electrostatics. You may assume the plates are in vacuum, so \(\epsilon=\epsilon_0\).
4. By answering the following questions, explain why \(\tau=RC\) makes intuitive sense. Do not just say "the formula \(\tau=RC\) says so" for this part; the point is to justify the formula, not to use it (nor to derive it, as properly deriving it uses calculus).
   1. Explain (qualitatively) why it is reasonable for the "time to equilibrium" to increase with resistance.
   2. Explain (qualitatively) why it is reasonable for the "time to equilibrium" to increase with capacitance.
   3. Explain (quantitatively, using dimensional analysis) why \(\tau=RC\) is the "only option" for the timescale in an RC circuit. (You may assume that the only "inputs" are the resistance \(R\), the capacitance \(C\), and the input voltage \(V_0\), in this explanation.)

For Further Thought:

1. In the first two parts of this experiment, we observe that when we flip the switch, the voltage first very quickly changes to one value, then slowly changes after that. Let's examine how to explain this in terms of the RC behavior we observed in Part III.
   1. Suppose the wire has a 10Ω resistance (which is almost certainly an overestimate) and the voltmeter has a 1MΩ resistance (1Ω=1s/F). Given that our capacitances are of order ~10μF, compute the time constant for each resistance: this is essentially the time it takes for current to "go through" that component. From these timescales, consider: of the two processes described above (the quick change in voltage and the slow change in voltage), which process is controlled by each component's resistance?
   2. On very long timescales (minutes-to-hours), the voltmeter can essentially be considered as another wire - current can pass through it given enough time. Based on this, if we were to leave the capacitors in Part I in the left (capacitors charged) position for a very long time, what voltages would you observe on each capacitor? What if we left the capacitors in Part II in the right (capacitors connected together) position for a very long time?

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

PHY121/122 Plotting Tool