" = R # C. Create your sketch so that Q(t=τ) is sketched above the delineated tic mark. Your sketch. 1" e " t & (t) = Q max

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1 Physics 241 Lab: Circuits DC Source Name: Section 1: 1.1. Today you will investigate two similar circuits. The first circuit is the charging up the capacitor circuit. In this circuit (shown below) the capacitor begins without any charge on it and is wired in series with a resistor and a constant voltage source. The voltage source begins charging the capacitor until the capacitor is fully charged. The charging up equation that describes the time # dependence of the charge on the capacitor is Q Cap (t) = Q max 1" & % (. The final charge on the $ ' capacitor, Q max is determined by the internal structure of the capacitor (i.e. its capacitance): Q max = C "V source. Use a graphing calculator (or mad graphing skills) and make a quick sketch of Q Cap (t) vs. t on the axes. Assume that the source voltage is 9 V, the resistance is 1.0x10 3 Ω and the capacitance is 1.0x10-3 F. The amount of time that equals the resistance times the capacitance is called the time constant: " = R # C. Create your sketch so that Q(t=τ) is sketched above the delineated tic mark. Your sketch below: 1.2. The second circuit is the discharging the capacitor circuit. In this circuit (shown below) the capacitor begins with some initial charge and is wired in series with a resistor. The capacitor begins discharging through the resistor until no charge remains on the capacitor plates. The discharging equation that describes the time dependence of the charge on the capacitor is Q Cap (t) = Q o. (Also think of a switch:

2 Use a graphing calculator (or mad graphing skills) and sketch a graph of Q Cap (t) vs. t on the axes below. Assume that the resistance is 1.0x10 3 Ω and the capacitance is 1.0x10-3 F. Find the initial charge on the capacitor by assuming the capacitor had been charged to 9 volts by a battery before being discharged through the resistor. Create your sketch so that Q(τ) is sketched above the delineated tic mark. Be sure to include charge values along the y-axis. Your sketch below: What is the decimal value of e -1 to 3 decimal places? Engineers usually approximate this number as 1/3 (.333) in order to think quickly about exponential decay. For example, if you plug in t=τ (one decay time constant), the amount of charge left on the capacitor has decayed to approximately 1/3 of its initial value. Approximately how much of the initial charge is left on the capacitor after the circuit has operated for t=3τ seconds? Your work and answer: 1.3. Now examine the time dependence of the voltage across the capacitor for the same discharging capacitor in part c. As the charge on the capacitor changes, the voltage difference across the capacitor plates also changes. In fact, the definition of capacitance easily relates V Cap (t) and Q Cap (t) by a constant: V Cap (t) = Q (t) Cap. Therefore, the equation describing the time dependent decay of the voltage C across the capacitor is simply V Cap (t) = V o, where V o = Q o. You will experimentally test this C equation later in this lab. Sketch a graph of V Cap (t) vs. t on the axes below using your answer to the previous question (graph of Q CAP ). Be sure to include voltage values along the y-axis. Your sketch below:

3 As the capacitor discharges, it causes a current to flow through the resistor. Because energy must be conserved, the magnitude of the voltage across the resistor is the same as the voltage across the capacitor (they are the only circuit components). Because the resistor is Ohmic, the current through the resistor can be related to its voltage and resistance. This gives a time dependent equation for the current through the resistor of charge on the capacitor, the voltage across the capacitor, and the current through the resistor all exhibit the same exponential decay function, and are simply related to each other using properties you already know. Relate this equation for resistor current to the others by using Ohm s law to determine I o in terms of R, C and Q o. Your work and answer: I Res (t) = I o. You should notice that the time dependence of the Section 2: 2.1. A differential equation is an equation that involves derivatives. Most all equations designed to model reality in the physical sciences make use of differential equations so a good working knowledge of this type of mathematics is essential to any working physical scientist or engineer. The following table compares an algebraic equations to a differential equations using two examples: Examine the differential equation y(t) = 4sin 3t Your work and answer: d 2 y(t) = "9y(t). One solution to this differential equation is dt 2 ( ). Check the solution by plugging it into the differential equation to see if it works.

4 2.2. When analyzing circuits, you often must write a differential equation describing the behavior of the circuit. This is most easily done by using conservation of energy to write a voltage equation. Then use fundamental concepts to relate voltage to charge on the capacitor to create a differential equation for Q(t). Examine the discharging circuit for today s lab and the construction of the differential equation that describes it: DISCHARGING WITHOUT SOUE 0 = V res (t) + V cap (t) (conservation of energy) 0 = R " I res (t) + Q cap (t) C 0 = R dq cap (t) dt dq cap (t) dt + Q cap (t) C = # Q cap (t) R " C SOLUTION: (Ohm's law and definition of capacitance) (current through resistor is equal to rate of charging on capacitor) (rearrange to get final differential equation) Q Cap (t) = Q o Check the solution function by substituting it into both sides of the differential equation for Q cap (t). Differentiate where appropriate to prove the left-hand side of the equation equals the right-hand side with the solution substituted in for Q cap (t). Your work and answer: What trivially happens to the initial charge Q o in this checking process? Your answer to this should allow you to see that the initial amount of charge on the capacitor Q o is not determined by the differential equation. Basically you choose the initial amount of charge to put on the capacitor plates and the differential equations determines how quickly that charge discharges through the resistor. Your answer:

5 2.3. The other circuit for today s lab, charging the capacitor, also has a differential equation to describe its behavior in time: CHARGING WITH CONSTANT SOUE V source = V res (t) + V cap (t) V source = R " I res (t) + Q cap(t) C dq cap (t) = V source dt R # Q cap(t) R " C (conservation of energy) (Ohm's law and definition of capacitance) (current through resistor is equal to rate of charging on capacitor then rearrange) $ SOLUTION: Q Cap (t) = C "V source 1# e # t ' & ) % ( (Not a question)

6 Section 3: 3.1. A large capacitance and large resistance translate into a slow time constant so that you may easily measure the rate of decay with a stopwatch. You are supplied with a 1000 µf electrolytic capacitor. Electrolytic capacitors are one-way capacitors. Be careful to only apply voltage correctly to the electrolytic capacitor or you will damage it (the negative terminal is marked on the capacitor). You will discharge your capacitor in an circuit with approximately 10 kω. Remember V Cap (t) = V o. What time constant should you expect with R = 10 kω and C = 1000 µf? Your work and answer: Since approximately 4 time constants allows the circuit to discharge to 2% of its initial value, how long should you measure the decay of the capacitor s charge? Your answer: 3.2. Charge an electrolytic capacitor without resistance in the correct direction using the 9-Volt battery (this happens quickly since there is very little resistance). Then switch to discharge the capacitor through a ~10 kω resistor (if the resistance is too small, the capacitor will discharge too rapidly to measure). Collect (voltage, time) data by having the DMM measure voltage across the capacitor while it discharges through the resistor using a stopwatch. You should collect more data at the beginning when there is rapid voltage change. [Collect your data now.] Make a raw graph of your data by plotting V cap (t) vs. t. [Make your graph now.] Next you will linearize your data by taking the natural logarithm of your voltages. Since V Cap (t) = V o, taking the natural logarithm of the function cancels the exponential: # ln V o & # % ( = ln( V o ) + ln & % ( = " 1 $ ' $ ' t + ln ( V o). The function y(t) = " 1 t + ln ( V o) is the equation of a line with slope -1/() and y-intercept ln(v o ). Thus, if you make the graph of ln( V Cap (t)) vs. t on regular (Cartesian) graph paper, you will obtain a line with a slope equal to -1/ if your data is exponentially related. Graph your linearized data by taking the natural logarithm of your measured voltages ln(v) and plot these vs. t on regular graph paper. This should give you a line with slope equal to -1/. Make your graph now. Then find C from the slope and R:

7 Section 4: The following picture shows the digital oscilloscope and labels its most common features. You now need to practice using the digital oscilloscope so that you are prepared to make measurements with it. Keep in mind that the oscilloscope is simply a tool that allows you to analyze the details of a rapidly changing voltage. With that in mind, you will now practice the more common measurements that are made as well as their uses. Create a sinusoidal wave with your function generator with a very small voltage (i.e. use a special feature of the function generator and a frequency in the khz range. Also add a small negative DC offset. Use the autoset button to quickly get your signal on the screen so you can adjust your function generator DC offset correctly. Be sure that your channel is on 1x probe and that your trigger is set to the correct source. Do this now.

8 Input the sine wave voltage source on channel 1 and determine the average voltage of your sine wave. Use the measurement feature set to measure the mean Your observations: Get the digital oscilloscope to tell you on its screen the wave s period and frequency. Use the measurement feature set to measure the period. Your observations: Get the digital oscilloscope to tell you on its screen the wave s amplitude. Use the measurement feature set to measure the peak-to-peak voltage. Your observations: Adjust the amplitude of the wave on the function generator until you see that the wave spends more of its time being negative than positive (with some positive). This will change your answer to the last question (see this). Use a two cursor measurement of time and get the oscilloscope to tell you on its screen how much time the sine wave spends being positive. Then do the same thing to find out how much time the wave spends being negative. Your observations and answers: Now use a two-cursor measurement in voltage and get the oscilloscope to tell you on its screen the voltage drop of the wave from its maximum positive value to zero. Your observations: Change your sine wave to a triangle wave of 500,000 Hz and use the DC offset so that the minimum of the triangle wave is zero volts. Examine a part of the triangle wave that is decreasing. Use a two cursor measurement in time and space to find how long it takes for the triangle wave to decrease from its highest value to one half of that value. Your observations and answer:

9 Section 5: 5.1. Most digital electronics make extensive use of capacitors. However, the decay rates are typically much too rapid to measure with a DMM. In this part of the lab you will create an circuit using a 0.1 µf capacitor and a 1 kω resistor and you will rapidly charge and discharge the capacitor with an oscillating square wave. What time constant τ will this produce? Your answer: You should choose a frequency of 1/(20τ) Hz so that there is plenty of time for the capacitor to discharge fully. What is this frequency? Your answer: Use your function generator to create a square wave with a voltage alternating between V MIN = 0 Volts and V MAX = 3 volts and the correct frequency. Do this by 1 st setting the frequency. Then set the wave to oscillate between +1.5 V and -1.5V and use the DC offset to shift your signal to have V MIN = 0 volts. The voltage across the capacitor should look like shark fins on your oscilloscope as the capacitor exponentially charges and then discharges. Do this now Now set up an circuit with R=1 kω and C=0.1 µf in series with the same square wave from the previous question. Then answer the following questions: During the time interval that the square wave is at +3 volts, is the capacitor being charged or discharged? Your answer: During the time interval that the square wave is at +0 Volts, is the capacitor being charged or discharged? Your answer: The following is an important reminder that you won t need in today s lab. Many times you may need to find the current in a circuit. Which component must you measure if you want to determine the current of the circuit and why? Your answer and explanation:

10 Observe the voltage across the capacitor and the total circuit voltage simultaneously using a bottom ground configuration. You should see the shark fin pattern that is modulated by the alternating square wave source voltage (turning on then off). Use a double cursor measurement to find the time it takes for your charged capacitor to decrease by half. Your observation: When a physical quantity decays exponentially, the time it takes for it to decay to ½ its original value is called the half-life t ½. Solve the half-life equation for t ½ to find what t ½ should be in this circuit in 1 terms of R and C: 2 V o = V o e " 1 t half. Your work and answer: Combine the results of the previous questions and calculate the experimentally determined capacitance C of your capacitor using your half-life measurement. Your work and answer: Now use the double cursor method to find the time it takes for your capacitor to discharge from ½ of its initial value to ¼ of its initial value. Your observation: The decaying exponential function has the unique property that each consecutive halving of its value occurs in the same amount of time. Using this knowledge, predict how long it should take for your capacitor to discharge to 1/128 of its initial value. Your work and answer: Now use the cursors to collect voltage vs. time data for your decaying capacitor. Then linearize your data, graph it on regular graph paper, and compute to the capacitance C from the slope. Be sure your value is close to the labeled value. Collect your data and make your graph now. Then show your work and answer for C below:

11 Report Guidelines: Write a separate section using the labels and instructions provided below. You may add diagrams and equations by hand to your final printout. However, images, text or equations plagiarized from the internet are not allowed Title A catchy title worth zero points so make it fun. Goals Write a 3-4 sentence paragraph stating the experimental goals of the lab (the big picture). Do NOT state the learning goals (keep it scientific). [~1-point] Concepts & Equations [~5-points] Be sure to write a separate paragraph to explain each of the following concepts. o Describe the operation and features of the digital oscilloscope. o Discuss what differential equations are and how to check their solutions. You should give an example (keep it simple). o Discuss the construction process of the two differential equations that model circuits with DC source. Discuss the solutions to these differential equations. o Discuss all you know about capacitors (should be a lot from lecture). o Discuss how to find the capacitance of a capacitor using an circuit with a DC source. o Discuss how to determine if data has an exponential relationship. o Discuss what the half-life is of an exponential relationship is and how it works. Procedure & Results Write a 2-4 sentence paragraph for each section of the lab describing what you did and what you found. Save any interpretation of your results for the conclusion. [~4-points] Conclusion Write at least three paragraphs where you analyze and interpret the results you observed or measured based upon your previous discussion of concepts and equations. It is all right to sound repetitive since it is important to get your scientific points across to your reader. Write a separate paragraph analyzing and interpreting your results from your open-ended experiment. Do NOT write personal statements or feeling about the learning process (keep it scientific). [~5-points] Graphs All graphs must be neatly hand-drawn during class, fill an entire sheet of graph paper, include a title, labeled axes, units on the axes, and the calculated line of best fit if applicable. [~5-points] o The two graphs from section 3. o The graph from section 5. Worksheet thoroughly completed in class and signed by your TA. [~5-points.]

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