PHYS272_Lab6

=Physics for Scientists and Engineers- Lab 6 PreLab =

Assignment: Perform the calculations described below. This assignment is due at the start of the laboratory period. These concepts will be used to complete the lab.

Questions:
1. You measure the resistance of a resistor and an inductor and find that they have values of R = 10.0 ± 0.2 Ω and R L = 1.5 ± 0.1 Ω. You then measure the slope of a line which should be equal to L/R L and find that m = 5.53 × 10 -3 ± 0.05 × 10 -3 H/Ω. Calculate the inductance L and the uncertainty in the inductance. For the uncertainty calculation use the fact that, if y=f(x 1, x 2 ) then math \displaystyle \Delta y = \sqrt{(\partial f / \partial x_1)^2 \Delta x_1^2 + (\partial f / \partial x_2)^2 \Delta x_2^2}. math

2. You next measure the slope of a different graph which should obey the relation m = L/(R + R L ) and find that m = 7.28 × 10 -4 ± 0.05 × 10 -4 H/Ω. Calculate the inductance L and the uncertainty in the inductance. Considering the inductance intervals from this question and question 1, are they consistent?

3. Finally you make one more graph whose slope should obey the relation m = RC and find that m = 3.32 × 10 -3 ± 0.05 × 10 -3 FΩ. Calculate the capacitance C and the uncertainty in the capacitance.

=Physics for Scientists and Engineers – Lab 6 =

Objectives:

 * To observe the relationships between the voltage and current across resistors, inductors, and capacitors in series combinations as the frequency of the source is varied.
 * To observe resonance in an R L C circuit.

Equipment:

 * Pasco Voltage Sensors (3)
 * banana leads (2 short) (2 long)
 * Pasco Signal Generator (internal to the ScienceWorkshop 750 interface)
 * Pasco Circuit Board with 100 and 330 microfarad Capacitors, 8 millihenry Inductor having about 5 Ω resistance, and a 10 Ω Resistor
 * Multimeter
 * Capacitance meter with 200 microfarad range
 * Pasco Signal Interface Data Studio and Graphical Analysis software

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Physical Principles:
The voltage, V, across a resistor varies in phase with a current, I, through it and is related to it by

math V_R = V_{R0} \sin(\omega t) math

In this equation V R is the voltage across the resistor, V R0 is the maximum voltage or amplitude, and ω = 2πf with f the frequency of the current. The voltage across an ideal capacitor (one with negligible resistance) reaches a maximum one quarter cycle after the current does while the voltage across an ideal inductor (one with negligible resistance) reaches a maximum one quarter cycle before the current does. The voltage builds up on the capacitor as the current deposits charge on it, and when it is charged the current is zero. Since the current is increasing most rapidly when it is zero the voltage on the inductor is greatest then. Voltages across ideal capacitors and inductors vary with time according to the equations below.

math \displaystyle V_C=\frac{Q}{C}=V_{C0} \sin(\omega t - 90^{\circ}) math

where C is the capacitance in farads and L is the inductance in henries. The amplitudes of the voltages across the three elements discussed are given by

math \displaystyle V_C= L \frac{\Delta I}{\Delta t}= V_{L0} \sin(\omega t + 90^{\circ}) math

where ω = 2π f and f is the frequency. The coefficients X C and X L of I o are called the capacitive and inductive reactances, math \displaystyle X_C = \frac{1}{\omega C} \mbox{ and } X_L = \omega L math

For a series R L C circuit the current and voltage amplitudes are related by

math V_{R0} = R I_0 \ \ V_{C0}=X_CI_0 \ \ V_{I0}=X_LI_0 math

where Z is called the impedance and is determined by

math Z= \sqrt{R^2 + (X_L-X_C)^2} math

The voltage leads the current if X L is larger than X C by an angle given by

math \displaystyle \phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right) math

and lags the current by the angle φ given by (9) (a negative angle) when X L is smaller than X C. When X C = X L the resulting current amplitude is maximum and the circuit is in resonance. The resonance condition is

math \displaystyle \omega L=\frac{1}{\omega C} math

which gives math \displaystyle \omega = \frac{1}{\sqrt{LC}} \mbox{ and } f=\frac{1}{2 \pi \sqrt{LC}} math

The above equations are correct if ideal capacitors and inductors are used, where their resistances are small compared to their inductances. For capacitors this is generally true but for inductors it is true only for higher frequencies. To correct for the effect of the inductor resistance, the value R in equations for Z and φ must be replaced with R + R L.

For an R-L circuit, as shown in figure 1, the following phasor diagram and equations can be written

math \displaystyle \tan \phi_L =\frac{L}{R_L}\omega math

math \displaystyle \tan \phi_S =\frac{L}{R+R_L}\omega math

For an R-C circuit, as shown in figure 2, the following equation and diagram can be written. <span style="background-color: #ffffff; color: #2b73b7; font-family: Arial,sans-serif; font-size: 14px; text-decoration: none;">

math \cot \phi_S =-RC\omega math

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.5em;">Procedure:
<span style="background-color: #ffffff; color: #2b73b7; font-family: Arial,sans-serif; font-size: 14px; text-decoration: none;">

With a multimeter or a RLC meter, measure and record the values of the resistance R (about 10 ohms) of the resistor and R L (about 5 ohms) of the inductor. <span style="background-color: #ffffff; color: #008800; font-family: Arial,sans-serif; font-size: 14px; text-decoration: none;">[|(click here for multimeter reading uncertainties)] Use a RLC meter to measure the inductance of the inductor.
 * Series R-L Circuit**

Set up a series R-L circuit, with channel A of the Pasco interface box connected across the resistor. Be certain that the RED probe wire is on the positive side of the resistor. Connect channel B of the interface box across the inductor again with the RED probe wire on the positive side of the inductor. Connect the negative side of the inductor to the ground port on the science workshop box and the positive side of the resistor to the sinusoidal output port. Ask the lab instructor to check your circuit. Click on the picture of the science workshop box in Data Studio between the two voltage output ports. Set the output voltage to 3.5 Volts. For frequencies of about 30 Hz, 60 Hz, 100 Hz, 150 Hz, and 200 Hz complete the table 1 by following the instructions listed below. <span style="background-color: #ffffff; color: #2b73b7; font-family: Arial,sans-serif; font-size: 14px; text-decoration: none;"> Pasco Capstone Setup

In Pasco Capstone, open two voltage sensors for analogue channels A and B. Set the sampling rate to 10,000 Hz, and record data for 0.1 seconds. Open a graph that includes all three signals. (See figure 6) Click on the cross-hair icon and use it to pinpoint time the resistor curve crosses the time axis. Repeat this to find the crossing times for the inductor and signal generator voltages as shown in figure 4 and record these values in table 1 as t R, t L and t S.

Compute the phase shifts in radians from φ L = ω(t R -t L ) and φ s = ω(t R -t S ), where t R is the time at which the voltage across the resistor is zero, t L is the time at which the voltage across the inductor is zero, and t S is the time at which the voltage of the source is zero. Plot a linear graph of the data of tan φ L vs. ω so that the slope of the graph should be equivalent to L/R L. Determine the value of the inductance from this relation. Make sure that you display the uncertainty in the slope on your graph and calculate the uncertainty in the inductance as you did in the prelab. Plot a second linear graph of tan φ s vs. ω with slope L/(R+R L ). Determine the value of the inductance from this relation and its uncertainty.

The two inductances calculated should be very close to each other. Calculate the % difference between the first and second inductances. Also state whether the uncertainty intervals overlap.

<span style="background-color: #ffffff; color: #333333; font-family: Arial,sans-serif; font-size: 14px;">**Table 1 Series R-L circuit** <span style="background-color: #ffffff; color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> <span style="background-color: #ffffff; color: #333333; font-family: Arial,sans-serif; font-size: 14px;">**Series R-C Circuit** Use your RLC meter to measure the capacitance of the 330 μF capacitor. Use a banana lead to bypass the inductor and move the leads for Channel B to either side of the capacitor. Repeat the measurements performed in the R-L experiment and complete table 2. Do this for frequencies of 20 Hz, 30 Hz, 50 Hz, 80 Hz, and 150 Hz. You need not include phase shifts for the capacitor in table 2 since they are all very nearly 90°. In your journal calculate at least one capacitor phase shift. Click on the cursor icon and place the horizontal cursor at the top of each voltage, then read the value at the left of the voltage axis. Plot a linear graph of cot φ s vs ω to compute the value of the capacitance from the relation slope = -RC. Include the uncertainty in the slope on your graph and use it to calculate the uncertainty in the capacitance. Plot a second linear graph of V R0 /V C0 vs ω to generate a slope of the same value, slope = RC. Determine the value of the capacitance from this relation and its uncertainty. These values should be close to each other. Find the % differences between the first and second and state whether their uncertainty intervals overlap.
 * ~ f ||~ ω ||~ t R ||~ t S ||~ t L ||~ φ L ||~ φ S ||~ tan φ L ||~ tan φ S ||

<span style="background-color: #ffffff; color: #333333; font-family: Arial,sans-serif; font-size: 14px;">**Table 2 Series R-C circuit** <span style="background-color: #ffffff; color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> In this table, V R0 is the amplitude of V R and V C0 is the amplitude of V C. In order to calculate these amplitudes, copy your data from Data Studio into Graphical Analysis and use a sine fit in order find the amplitude.
 * ~ f ||~ ω ||~ t R ||~ t S ||~ φ S ||~ V R0 ||~ V C0 ||~ V R0 /V C0 ||~ tan φ S ||

Add the inductor to the circuit so everything is connected in series. For a frequency of about one fifth of the resonant frequency describe and explain the phase and voltage relationships. Repeat this for a frequency of about 5 times the resonant frequency. Find resonance by adjusting the frequency until the voltage across the resistor is maximum. Compare this to the theoretical resonant frequency. Place the steel rod in the inductor (which will change the inductance) and determine again the resonant frequency of the circuit. Also measure the new inductance with your RLC meter. Compare the experimental and calculated resonant frequencies.
 * Series R-L-C Circuit**

Further Investigations (+3 pts each)

 * Repeat the Series R-L circuit measurements with the steel rod in the inductor to increase the inductance from the first measurement. How does the increased inductance change your results?


 * Repeat the Series R-C circuit measurements with the 100 μF capacitor. How does the decreased capacitance change your results?


 * Find the resonance value in the R-L-C circuit measurement using two inductance values (with and without steel rod) and four different capacitance values (100 μF, 330 μF, two capacitors in parallel, two capacitors in series). How do the measured resonance values compare with calculated values?