THE EXPERIMENT

A. VOLTAGE, RESISTANCE AND CURRENT WITH TWO RESISTORS IN SERIES

1. Select two resistors, measure their resistance values with the multimeter and connect them in series with a battery as shown in Figure 1.
2. Measure the voltage of the battery and the voltages across R₁ and R₂. Record them along with your estimates of the uncertainty of the measurements.
3. Compute the current from the measured values of the battery voltage and the resistances propagating the uncertainties of each quantity. From this current compute a "theoretical" value for the voltage across R2 and compare it to your measured value.

Cautions:

• Use resistors of sufficiently large size to limit the battery current to about 1mA (10^-3Amp). This will assure that the battery will not be discharged and will provide a stable voltage.
• Be careful not to "short" the battery by connecting wires from both terminals of the battery to the same node. The long wire running down one side of the block on which you will connect to the resistors is a node.
• Do not measure a resistance with the multimeter while it is connected to a voltage source. The multimeter uses its own internal voltage source to produce a current through the resistor to be measured. The current is used as a measure of the resistance. That current will be changed if an external voltage source is also connected. Connect and double check your circuits before attaching meters or the oscilloscope.

Figure 8

Hints:

• A sketch of the connections for the meter, the battery and the resistors for measuring the voltage across R₂ is shown in Figure 8.
• When making a measurement with the multimeter, always use the scale which yields the largest number of digits without going off scale.
• Make certain to record the scale setting of the multimeter or oscilloscope when measurements are made.
• For digital readouts, the uncertainty, or error of the measurement, will be equal to or greater than +/- 1 in the least significant digit.
• For oscilloscope readings, the uncertainty is approximately the width of the trace. If the trace is well focused, this is about +/- 0.1 divisions.

B. SETTING UP THE OSCILLOSCOPE

Read and follow the instructions in the Oscilloscope Information section for "Learning the Controls" and the "Calibration Procedure."

C. SETTING UP THE SIGNAL GENERATOR

Read the summary of controls for the signal generator in the manual and set the controls as indicated in the settings summary for a 10kHz square wave signal. Remove the calibration signal from the oscilloscope and connect the output signal of the generator to one of the scope inputs and measure the frequency of the square wave, in the same manner as the calibration operation. Compare your measurement with the frequency indicated by the signal generator display.

D. MEASURE THE TIME CONSTANT FOR THE RC CIRCUIT

1. Record the measured values and uncertainties of a nominally 500 resistor and a 0.1F capacitor. Since a capacitance meter is not available in this lab, the stated value and tolerance of C (on the label) should be used.
2. Connect the components in a RC circuit with the signal generator as shown in Figure 5, and then connect the scope across the capacitor.
3. Compute the theoretical time constant for the circuit (with uncertainty).
4. Measure the time constant using the simple two point method described in the section regarding the discharge of an RC circuit (pages 1.2-1.4)
5. Measure and record the voltage level as a function of time. A straight-forward way to do this is to adjust the controls such that a single exponential decay fills the screen (see hints below), and simply record the levels as the trace crosses each of the (11) major verticle lines on the scope screen. For example, a horizontal scale setting of 5s/division would provide points at 0, 5, 10 ... 55 s. Remember to include the uncertainties in the measurements. Plot a graph of V_C versus t (including error bars) and then "fit" your data to the functional form of an exponential decay (i.e. y = A e^-x/B). From the coefficients of the fit, determine a value (with uncertainy) for the time constant .
6. Compare both measurements to your computed, theoretical values. Include propagated uncertainties.

Hints:

• The signal generator is represented as a voltage source in series with a 50 internal resistance, as in figure 9. The generator offers a choice of 600 or 50 output impedances. Select 50.
• Use a low frequency square wave from the signal generator so as to allow the voltage on the capacitor to decay completely during 1/2 of the square wave cycle.
• Adjust the controls on the signal generator and the scope until you see a trace which appears similar to figure 10.
• Make certain that you know the vertical position on the screen at which the exponential reaches V = 0. If you do not know this value then your measurements of the voltages will be incorrect. In order to be certain, use a low frequency square wave and adjust the scope so that the trace appears similar to figure 11. Once you have determined the zero line, make certain not to change the position settings since all voltages must be measure relative to this line.
• For the measurement to be fit to an exponential it is best to arrange the horizontal sweep (time scale) such that a single exponential decay fills the screen. Then set the left-right position of the trace such that you can measure the early portion of the decay where the voltage is largest. This will provide the most accurate measurements.

Figure 10 Figure 11

E. DETERMINE THE TIME CONSTANT FROM THE FREQUENCY RESPONSE

In this section you will investigate how the RC circuit responds to sine waves at various frequencies. Use the same circuit as in Part D for the following measurements. Select the sine wave output from the Signal Generator and adjust the oscilloscope so as to display several cycles of the sinusoidal voltage across the capacitor. Perform a quick examination of the amplitude of V_C as a function of the frequency to test that the behavior is as expected from eqn(11). If it is not, ask the TA for assistance.

Measure and record the peak-to-peak voltage across C as a function of the frequency (a frequency range of to 16 should be sufficient). Plot a graph of the amplitude (or peak-to-peak value) of V_C versus frequency (including error bars) and then "fit" your data to the functional form given by eqn(11). From the coefficients of the fit, determine a value of (with uncertainty) and once again compare it to the theoretical value. Comment on the differences between the three values of (simple two point, time domain, frequency domain) that you obtained. Which are more accurate, more precise?

Hints:

• The signal generator displays the frequency in Hz, which is the number of cycles per second. Equation 11 is written in terms of angular frequency, which is the number of radians per second. (2 radians = 1 cycle.)
• The peak-to-peak voltage of a sine wave is twice the amplitude as given in the expression V(t) = Asin(t). In order to measure the peak-to-peak voltage you will want to move the trace horizontally so that first the peak, and then the trough of the wave lines up with a vertical scale line.