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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)

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.

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