The pre-lab and the instructions for the pre-lab are located at http://www2.hawaii.edu/~jmcfatri/labs/capsprelab.html . The pre-lab is due at the beginning of the class. One pre-lab is due per group.

The objectives of this lab are:

1) to create a capacitance meter
2) to determine the capacitance of capacitors in series and parallel
3) to measure the capacitance of a homemade capacitor

A capacitor is a circuit element which stores charge. Anything that stores charge is a capacitor. You are a capacitor! (If you weren't, you wouldn't be able to scuff your feet on carpet and zap your friends!)  You have a capacitance of about 100 pF (100 x 10 ^-12 F). For all capacitors, there is a value which describes how much charge can be put on a capacitor, called capacitance.  Capacitance is measured in Farads (F = C / V).

The simplest type of capacitor is the parallel plate capacitor. A parallel plate capacitor is comprised of two metal plates separated by a distance, d, and which has a voltage difference, V, between them. We can describe this capacitor analytically.(Other capacitors are much more complex!)

We can determine quite easily with the equations that we've learned so far that the capacitance of a parallel plate capacitor is:

C = k ₀A / d

where A is the area of the plate, ₀ is the permittivity of free space = 8.85 x 10 ^-12 F / m, and k is the dielectric constant. If there is no material between the plates of a capacitor (a vacuum) then k is 1. You will learn about dielectrics in your lecture class; in the lab all you need to know is that a dielectric increases the capacitance of a capacitor (i.e. k > 1).

3. Capacitors in Alternating Current circuits:

In an AC circuit, a capacitor acts like a resistor. Energy is lost as the electric fields of the capacitors are created and destroyed.  We can treat the capacitors as resistors with a resistance of: X = 1/ C, where  is the angular frequency of the AC voltage source. Instead of resistance, we call this quantity the reactance of the capacitor .

You can add resistances in series and parallel; the same rules apply for reactances:

series: X_eff = X₁ + X₂
parallel:1/X_eff = 1/X₁ + 1/X₂

You can substitute in the definition of the reactance of a capacitor into the above equation to get:

series:

1	=	1	+	1
Ceff		C1		C2

parallel: C_eff = C₁ + C₂

We will verify these two equations in our experiment.

The Bridge Circuit we are using in this experiment is one way to find the capacitance of an unknown capacitor. The unknown capacitor in the circuit is C₂, as labeled in the circuit diagram in the procedure and in your pre-lab. You know now that a capacitor acts like a resistor in an AC Circuit. So we will first analyze the behavior of resistors in the bridge circuit, then later replace the resistors with capacitors. Then our circuit will look like the picture below.

Now let's use Kirchhoff's Voltage Rule to find R₄, which replaces the unknown capacitor. Let's go through two loops. The first loop, which I will call "top" starts at the battery, goes through R₃ , then through R4 then back to the battery. The second loop, which I will call "bottom" starts at the battery, goes through R1 and then through R ₂ and back to the battery. I will also make an important assumption: I assume that the resistance of the oscilloscope, like all voltmeters, is very high and therefore no current goes through it.

"top" equation: V₀ - I_topR₃ - I _top R₄ = 0
"bottom" equation V₀ - I_bottomR₁ -I_bottom R₂ = 0

Rearranging gives:

I_top = V₀/ (R₃+R₄)
I_bottom = V₀/ (R₁ + R₂) .

Now let's find the voltage difference that the oscilloscope sees. We will call V_A the voltage at the top (where the oscilloscope is connected) and V_B is the voltage at the bottom.

V_A = V₀ - I_topR₃ = V ₀ - (V₀R₃) / (R₃ + R₄ )
V_B = V₀ - I_bottomR₁ = V ₀ - (V₀R₁) / (R₁ + R₂ )

V_A - V_B is the voltage difference. That doesn't seem to help us get the R₄, which is what we want to find. But remember that R₁ and R₂ are variable resistors. We can make their value anything we want. Maybe we can't solve this problem in general, but we can solve the problem for a specific case. Let's suppose we change R₁ and R₂ until V_A = V_B . Then the voltage difference is zero, and you will see a 0 V reading on the oscilloscope. Then

(V₀R₃) / (R₃ + R₄) = (V ₀R₁) / (R₁ +R₂)

Solving this, I get

R₄ = R₃ (R₂ / R₁)

Now instead of resistance, let's put the reactance of the capacitors. (R₃ = X₁, R₃ = X₂). Solving this new equation for C₂, you get:

C₂ = C₁ (R₁ / R₂)

and now we have a capacitance meter, because we can find C₂ if we know C₁, R₁, and R₂. 

Part I: Measuring capacitance

1) The circuit below is already constructed for you.  You only need to connect the oscilloscope and the frequency generator (as in your pre-lab).  C₁ = 0.22 µF.

2) Attach capacitor A to the circuit.

3) Vary the two resistances R₁ and R₂ by turning the black knobs on the box. Do this until the amplitude of the output on the oscilloscope is minimized (nearly flat line). You need to put your oscilloscopes on a more sensitive scale to see this (in the mV range at least!)

4) Disconnect the scope and the frequency generator, being careful not to change the value of the resistors.  Measure the two resistances with a digital multimeter.

5) Determine the capacitance of A. C₂ = C₁ * (R ₁ / R₂).

6) Repeat for capacitors B - E.

Part II:  Effective Capacitance of Capacitors in Series and Parallel

1)  Using your measured values for A and B, analytically determine the effective capacitance of A and B when they are placed in series. 

2) Connect A and B in series.  Repeat part I, steps 1-5, but instead of attaching capacitor A, attach the "A series B" capacitor. This is the experimentally determined the effective capacitance of A in series with B.

3) Repeat for A and B in parallel.

Part III:  Homemade Capacitor 

The homemade capacitor is a capacitor made out of two sheets of aluminum foil with a transparency between them.  The capacitance of our homemade device is very small (~10 nF).  To determine the capacitance, we will have to make our capacitance meter more sensitive.

1) Pick the smallest capacitor from part I and connect them in series to the orginal C₁. These two capacitors are now the new C ₁. Camke sure your oscilloscope is attached to the end of the new C ₁, not the old one.

2) Using your modified capacitance meter, find the capacitance of the homemade capacitor.

Important! To measure the capacitance of the homemade capacitor, you will have to touch the leads to either side of the homemade capacitor.

• Do not clip the leads to the aluminum foil (else it is not a parallel plate capacitor, and the ends will begin to curl). 
• Do not touch the metal parts of the capacitor or the lead with your bare hands. If you follow point number 3 below this is not dangerous, but doing this will change the results . You have a small capacitance of 0.1 nF and what you are trying to measure is only of order 10 nF.
• Do not turn the voltage higher than 10% on your frequency generator (or just turn it low if you don't have a percentage out). Trust me when I say that accidentally touching the metal parts of the capacitor at 90% output is not pleasant.
  

Part I: Creating and Testing a Capacitance Meter

Capacitor Code	R1 (    )	R2 (    )	C2 (    )	C2 expected (    )
A
B
C
D
E

Part II:Series and Parallel

Capacitor Configuration	R1 (     )	R2 (     )	C2 (     )	C2 expected (     )
A series B
A || B

Part III Homemade Capacitor:

assume d = 0.006 cm
measure length of foil: _____________±___________
measure width of foil: _____________±____________

Capacitor	R1 (     )	R2 (     )	Chomemade C1 not 0.22 µF (     )
homemade

1.  What is the value of k for our homemade capacitor? (Show your work)
 
 
 
 
 

2.  If I add a capacitor (C1) in series to another (C2), will I increase or decrease the capacitance of the overall circuit?  Justify your answer.
 
  

 

3.  If I add a capacitor in parallel to another, will I increase or decrease the capacitance of the overall circuit?  Justify your answer.