Lab10

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Jan 9, 2024

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Lab #10: Inductance PI: Natalie Cupples DA: Marissa Harris Researcher: Luke Katalinic INTRODUCTION: Marissa Harris (DA) The purpose of this lab was to use inductance far aday’s law. This version allows us to measure the current and that L is constant. First, we found the relationship between when the circuit and the current when everything was constant. This allowed L to be removed and to make sure that everything was calculated. Then we measured the circuit as it got faster and faster than allowed us to isolate L. PROCEDURE: Luke Katalinic (R) The equipment required for this lab included a function generator, an oscilloscope, a decade resistance box, an induction coil, jumper wires, and a multimeter. We began by setting up the equipment in the fashion of the circuit shown in Figure 1 below. Before turning on the power from the generator, we recorded the resistance of each individual component present in the circuit using the multimeter. Figure 1

After turning on the power from the generator, we set it to sine waves at a frequency below 200 Hz as instructed by our TA. We then placed one of the oscilloscope wires between the coil and the resistor to measure the external resistors voltage, V r . Our data was obtained by measuring the sine waves amplitude from peak to peak with cursor 1 and 2, which gave us our V r . After obtaining the current throughout the circuit as well as Z for low frequency, we began taking measurements for a high frequency, which was instructed by our TA to be above 5,000 Hz. We did six trials for both high and low frequency, testing the values following. For low frequency we ranged between 16 Hz and 196 Hz, as for high frequency we tested values ranging from 5,065 Hz to 16,500 Hz. Once we gathered all our V r values for the various frequencies, Ohm’s law (I = 𝑉𝑟 𝑅 ) was then used to calculate the current (I) in the circuit. We are then able to calculate the impedance Z with the equation Z = 𝑉𝑟 𝐼 . The only real uncertainty present in this experiment comes from the Oscilloscope data and the DMM. The Oscilloscope affects V r , which obtains a systematic uncertainty of ± 0.08 V by measuring the thickness of the black line. The DMM affects the various resistances measured, with a systematic uncertainty of ± 0. 001 Ω. ANALYSIS: Marissa Harris (DA)

Graph 1 In this graph, it shows the Frequency vs. Vr graph. What this means is that as the frequency increased there was no change really in Vr but when 3.36 was reached there almost linear increase in the frequency. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 3.68 3.68 3.52 3.52 3.36 3.36 0.284 0.168 0.124 0.118 0.104 0.094 0.082 0.076 Frequency (Hz) Vr (V) Frequency vs Vr y = 0.5113x + 24.924 -2000 0 2000 4000 6000 8000 10000 12000 -5000 0 5000 10000 15000 20000 Z Frequency Z v Frequency Graph 1: In this graph, this shows the frequency vs Z graph. This is the inductance of the solenoid. As you can see in the graph, as the frequency increased there was a direct increase with Z.

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CONCLUSION: Natalie Cupples (PI) Based on the results in Graph 1 (Frequency vs. Vr), it is clear that both the low and high frequencies tend to increase as voltage increases. This trend aligns with Faraday’s Law which describes the magnitude of the voltage in a conductor due to electromagnetic induction. Graph 2 (Frequency vs. Z) shows an illustration of frequency as a function of Z, the electric impedance of the circuit. The upwards slope of the graph can be used to calculate the inductance, L, which gives us information on the experiment as a whole. Graph 3 (Log-Log Plot of Frequency vs. Vr) illustrates the inverse relationship between frequency and voltage as there is a negative slope. Graph 3: This graph shows the log-log plot of the Frequency vs Vr. In the graph you can see that it has a negative log-log relationship. As you can see with the log-log relationship there is a inverse relationship. y = -3765ln(x) + 4545.4 R ² = 0.9516 1 10 100 1000 10000 100000 0.07 0.7 Frequency (Hz) Vr (V) Log-Log Plot of Frequency vs Vr

Both graphs relay the same data and show very similar trends in frequencies. To create a formula that calculates the maximum induced voltage, I started with the given equation, 𝜀 𝑚𝑎𝑥 = |−𝐿 𝑑𝐼 𝑑𝑡 | 𝑚𝑎𝑥 and substituted 𝑑𝐼 𝑑𝑡 = −? 0 sin (2𝜋𝑓𝑡) for a simplified equation of 𝑉 𝑚𝑎𝑥 = ? 0 (2𝜋𝑓𝐿) . Using all of the derived equations, we created a graph which gave us a slope of 0.5113 on the Frequency vs Z graph. This value divided by 2  gives us L to be 0.0814 H which falls between the correct range of values. Overall, this experiment helped to visualize inductance and how it applies to electric currents and magnetic fields. DA 1: First assess the quality of your raw data. Create two plots of your raw data: a. “Frequency vs 𝑉𝑟 ” b. “Log -Log plot of frequency vs 𝑉𝑟 .” (see Lab 2 for Log -Log plots) Consider making two (or three) different series on each of these plots so you can clearly distinguish the “Low Frequency” behavior and the “High Frequency” behavior. After this first pass you should be able to confidently make statements like: “the Low frequency behavior of 𝑉𝑟 is 𝑓? .” (With ? being an integer.) “And the High frequency behavior of 𝑉𝑟 is 𝑓? .” (With { ? being a different integer.) Researcher 1 : Apply Kirchhoff’s voltage law to your circuit on your low frequency data to calculate the Voltage generated by the function generator. Confirm the Voltage generated by the function generator is a constant regardless of frequency. We know that with Kirchoff’s Laws, there is confirmation that the sum of the individual voltages will equal the EMF from the function generator. ε = V r + V L + V G . This can be seen below in Figure 2. V L = IR L V r = Ir

Figure 2 DA 2 : Return to Lab 4, where Ohm’s Law is defined as Δ 𝑉 = ?? . The inductance of our solenoid must somehow be hiding in the reactance, the imaginary part of ? . Because you have measurements for 𝑉? , the voltage of the power supply, and ? , the current through the power supply, you can directly calculate this ? at all these different frequencies. Create a plot of Frequency vs ? which best displays your data. Sanity check/Hint: ? should qualitatively {increase, decrease, or remain constant} as you in crease the frequency based on Faraday’s Law? PI 1: Assume for a moment that all resistances in your circuit are zero. Take advantage of the fact that the current measured in the external resistor was always sinusoidal: ? = ? 0 cos(2 𝜋𝑓𝑡 ) to calculate the maximum induced voltage: 𝜀𝑚?? = |− 𝐿 𝑑? / 𝑑𝑡 | 𝑚?? . Take advantage of the fact that equation (5) can be written in the form of equation (3). The left hand side is a voltage, and the right hand side, after some simplification, is a current times some constants (no 𝑡 should survive if you consider the maximum values). To create a formula that calculates the maximum induced voltage, I started with the given equation, 𝜀 𝑚𝑎𝑥 = |−𝐿 𝑑𝐼 𝑑𝑡 | 𝑚𝑎𝑥 and substituted 𝑑𝐼 𝑑𝑡 = −? 0 sin (2𝜋𝑓𝑡) for a simplified equation of 𝑉 𝑚𝑎𝑥 = ? 0 (2𝜋𝑓𝐿) . Researcher 2: Explicitly connect the work in PI 1, to the high frequency data your group has collected. Explain why “assuming all resistances are zero” is reasonable for your high frequency data. For the high frequency data, it can be seen that as the frequency increases, V r decreases. Because of this decreasing V r , the impedance (Z) values increase exponentially. This means that it is valid to assume that at high frequencies assuming all resistances are zero; The Z values are so high, the resistance is essentially negligible. The data supports this as the current is keeps

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decreasing to low numbers while the impedance is increasing exponentially. PI 2: Using all the analysis above, calculate 𝐿 . Just to make sure your math is in the correct ballpark, you can be assured that .01 ? < 𝐿 < 1 ? . Using all the derived equations, we created a graph which gave us a slope of 0.5113. This value divided by 2  gives us L to be 0.0814 H which falls between the correct range of values.