OBJECT:

1. To study the discharging of a capacitor and determine the time constant for a simple circuit.

2. To use the dual trace mode of an oscilloscope to determine the phase difference between a resistor and capacitor in an AC series circuit; to learn the phasor method of determining the impedance of the circuit. 

APPARATUS

Capacitor, resistor, DC source, double pole double throw switch, computer, calibrated dual trace oscilloscope with 2 probe cables, sine wave generator (Vpp less than or equal to 25 V; f between 1 Hz and 100 kHz). 

THEORY:

Please, read carefully Tipler's PHYSICS pp.690-692 and 760-763.

A capacitor is a device for storing charge and energy. It consists of two conductors, closely spaced, but insulated from each other, that carry equal and opposite charges.
Figure 1.

The charge stored in a capacitor is proportional to the voltage across the capacitor. (See Figure 1.):

q = C V,

A circuit containing a resistor and capacitor is called an RC circuit. In such a circuit, current varies with time.

Figure 2.

When the capacitor is charged to the initial charge q_o (Figure 2) and switch S is closed at time t=0, the current starts to flow through the circuit. The initial current is

, where 

is the initial voltage across the resistor. After a time the charge in the capacitor is reduced, i.e. the capacitor is discharging. The current through the circuit is equal to the rate of discharge of the capacitor.

At any moment of time Kirchoff's loop rule gives us for the circuit

Substituting for I, we have

The solution of this differential equation is

where t = RC is called the time constant of the circuit.

The current through the circuit is

and the voltage across the resistor

So we can see that the charge on the capacitor, the current through the circuit, and the voltage across the resistor decrease exponentially with time, and fall to 1/e of the initial value after time t = t = RC. 

EXERCISE 1:

1. Set up the computer as instructed in Instructions for Computerized Experiments.

2.  Double click of the "Vernier" folder.  Double click on the "Phys 231 labs" folder.  Double click on the "RCTime" experiment.

3.  Insert the voltage meter cable in Din 1 on the ULI Box..

4. Connect the circuit shown in Figure 3.

Figure 3.

5. Turn on the power source, set V_DC to the 10 V scale, and adjust the voltage to 4V.

6. Turn the double throw switch to side "A" to charge the capacitor to 4 V and click "Start".

7. After the capacitor is fully charged (you see a horizontal red line on the graph on the computer), turn the double throw switch to side "B" to discharge the capacitor. Wait until the computer registers 0.1 F, then throw the switch to side "A" again. Data are taken. Refer to Figure 4.

Figure 4.

8. On the "Analyze" menu, choose "Analyze Data A" (so that a check is placed beside it).  Shade the discharge part of the graph by clicking the mouse at the beginning of the discharge curve and at the lowest point of the curve. Refer to Figure 4.

9. On the "Analyze" menu, choose "Fit...".  Make sure that an exponential fit (y=b_o*exp(b₁*t)) is selected.

10. Click on "Try Fit".  After the computer fits the data, click "Results". Record the coefficient b₁.

11. Repeat steps 6-10 five times. You will get five slightly different results for b₁. 

DATA ANALYSIS FOR EXERCISE 1.

1. Find the average and standard deviation of the coefficient b₁.  Record your result in the form b₁=b_1avg ± Db₁.

2.  Calculate the time constant t = 1/b₁.  Find the experimental error for t:

1. Connect a RC series circuit (see Fig.5) with the AC generator.

Figure 5.

 2. Turn the power of the AC generator on, and adjust the peak-to-peak voltage of the generator (V_S) to 8.0 V (full scale on the 1 V setting of Volts/Div).

3. Carefully measure and record the peak-to-peak voltages across the capacitor (V_C) and across the resistor (V_R).  Answer the question: Does V_S = V_R + V_C ?

To discover the reason for this apparent failure (!) of Ohm's Law, make a measurement of the phase difference between V_R and V_C, as follows:

4.  Connect the second probe cable to the 'scope (but DO NOT yet hook it into the circuit), and depress the "Dual" push-button on the 'scope. There should now be a second trace visible on the screen. Use the vertical position control for Ch. II to place the new independent sweep beneath the first voltage trace (Ch. I).

Before you connect the second probe into the circuit, READ THIS FIRST: when two 'scope cables are used simultaneously, the grounds for each cable must be located at the same electrical position (see Fig.6). Failure to do this will cause an internal current flow inside the 'scope which may result in the destruction of a very expensive piece of apparatus!
 

Figure 6.
5. Once the Ch. II connection has been made, observe the waveforms on the screen. Make a sketch of what you find: show the relative amplitudes and positions of the voltages. Use vertical control(s) to find where the peak of one voltage occurs relative to the other. Note that you can vary the amplitude controls for each channel independently, but there is only one time base for the display.

6. For the RC series circuit record: (all voltages are peak-to-peak)

V_R = _____ ± _____ V

V_C = _____ ± _____ V

V_S = _____ ± _____ V

V_C + V_R = _____ ± _____ V

1. Find V_calc = = _____ ± _____ V and its experimental error.

2. Answer the questions :

• What portion of a cycle appears to be between the two voltage waveforms (e.g., half cycle, one eighth cycle)?

 
• How does this relation change if you increase the frequency:
• by a factor of 10
• by a factor of 10²;
• by a factor of 10^-1?
 
• Is the observed phase difference frequency dependent in a RC series circuit?

 
• What (if anything) about V_R and V_C changed when the frequency of the signal changed over a few orders of magnitude?

 
• Can you explain why the "vector method of addition" was useful in showing how V_R and V_C combined to yield V_S? Why can't the voltages simply add (as in DC circuits) to equal the source voltage?