lab3_manual_W24

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Feb 20, 2024

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Updated by: Claude Cournoyer-Cloutier 1 February 9, 2024 LAB 3: CAPACITORS AND THE RC CIRCUIT PRE-LAB QUESTIONS Please complete the following questions before coming to the lab, and bring your answers with you to the lab: 1. What is the main purpose and function of a capacitor? Provide a practical example of how capacitors are used within circuits. 2. When we double the separation between capacitor plates, what happens to the capacitance, C? 3. What is the time constant of a series RC circuit, and what unit does it have? What about for a parallel RC circuit? Explain. 4. When building an RC circuit, does the polarity of the capacitor with respect to the voltage source need to be considered? You can find the answer to this within the lab manual along with specifics regarding our capacitors. 5. Within Microsoft Excel, if you wanted to take a column of over a thousand rows and apply a natural logarithm function to each row value, how would you go about doing that in an efficient manner? Write out the explicit command for the natural logarithm function in Excel along with the process for doing this quickly. a. Note: Don’t worry about the version, the specific command will be given in the manual for the version you will work with.
2 Set yourself the following objectives for this lab: ● Learn the basic properties of a capacitor experimentally and theoretically ● Analyze circuits which have current/voltage changing in time ● Understand the meaning of a time constant and how to find it for half an initial voltage ● Discharge a capacitor in parallel to a resistor and find the cumulated charge through integration ● Charge a capacitor through a resistor in a series circuit and experimentally analyze the results The following shows the value of all the questions in this lab: Laboratory 3, Question Grading Scheme Totals Part 1 Question 1 Question 2 Question 3 Question 4 Points /2 /1 /2 /1 /6 Part 2 Question 5 Question 6 Question 7 Question 8 Points /1 /1 /2 /1 /5 Total number of points in this lab /11 The following shows the value of all the tables and graphs in this lab: Laboratory 3, Results Grading Scheme Totals Part 1 Results/Table Graph Points /3 /2 /5 Part 2 Results/Table Graph Points /1 /3 /4 Total number of points in this lab /9
3 THEORY ELECTROLYTIC CAPACITOR A capacitor is a device which stores charge. Its capacitance (C) is the charge it will hold per volt of applied potential, so that Q = CV, (1) where Q is the charge, and V is the potential difference. Capacitance has SI units in the form of Farads (F) which are equivalent to Coulombs divided by Volts. The simplest form of a capacitor is a pair of parallel conducting plates with surface area (A) separated by a distance (d). In this case, the capacitance is ࠵? = ! ! " # , (2) for plates in air or vacuum, ࠵? $ = 8.85 × 10 %&’ ࠵? %( ࠵?࠵? %& ࠵? ) ࠵? ’ and it is called the permittivity of free space. It is a fundamental constant in electricity and magnetism as it describes the capability of an electric field to permeate a vacuum. Large values of C are obtained by increasing A and minimizing the separation d. Rolling thin foil plates into a cylinder with a plastic layer in between gives a large area in a small package. But for capacitances much greater than 1 μF, a thinner insulator is needed. The electrolytic capacitor replaces one of the plates with a conducting solution of aluminum hydroxide. An extremely thin layer of oxide formed on the other (aluminum foil) plate insulates it from the electrolyte. By reducing the separation d in this manner, capacitances of thousands of μF are readily obtained. However, some compromises are made. First, if the applied voltage makes the electrolyte positive with respect to the foil, the aluminum oxide will be broken down (chemically reduced), and the capacitor will short-circuit internally and possibly explode. All electrolytic capacitors have the polarity of the leads indicated on the package. Those used in this lab have a white band with minus signs indicating the negative side. Always connect the circuit so that the voltage across the capacitor is negative at this end, positive at the other. It is a good idea to check with a voltmeter before the second capacitor lead is connected. Second, there is inevitably some internal leakage through the imperfect insulation layer. This effect is insignificant in this lab, as the capacitors will hold their charge for a few days. There is a third effect which you may notice in your measurements. Because of chemical reactions between the electrolyte and the oxide layer, the capacitor does not strictly follow equation (1). In some ways it acts as a cross between a rechargeable battery and an ideal capacitor. This behavior is worse if the device has been charged up for several minutes. When no measurements
4 are being taken, you should disconnect the capacitor from the circuit and short (i.e. connect together) the leads with a wire; this will help to give reproducible results. THE RC CIRCUIT The circuit above is what we call an RC circuit (you might have guessed it, a Resistor and Capacitor circuit). If the capacitor has an initial charge (Q) and voltage (V) associated with it, then the charge will begin to decline once it is connected to the resistor (R). The rate of loss of Q through the resistor is the current I, which in turn depends on V through Ohm’s Law. This gives us a differential equation of (3) where the - sign indicates that Q decreases. Substituting for V using (1) (4) which has the solution (verify by substitution) ࠵?(࠵?) = ࠵?(0)࠵? % " #$ . (5) The potential V therefore declines exponentially as well, ࠵?(࠵?) = ࠵?(0)࠵? % " #$ , (6) Or ࠵?࠵?9࠵?(࠵?): = ࠵?࠵?(࠵? $ ) − * +, , where V 0 =V(0). (6a) The product, RC, has units of seconds (check this for yourself, with R is in Ω and C is in Farads) and is called the time constant of the circuit. It is the time at which the voltage or charge declines by a factor of e ≈ 2.718. For simplicity in the experiment, we are going to find a different time constant. Instead of the time at which V or C declines by a factor of e , our time constant will be when the voltage in dQ dt I V R = - = - dQ dt Q RC = -
5 the capacitor drop to half of its initial value. We will call this time constant ࠵? % & (the Greek letter ࠵? is pronounced “tau”). Analytically, we can find this value by: ࠵?(࠵?) = ࠵? $ ࠵? % * +, , ࠵? = ࠵? & ’ ⇒ ࠵?(࠵?) = ࠵? $ 2 . ∴ ࠵? $ 2 = ࠵? $ ࠵? % - % & +, . ln(2) = - % & +, (7) ⇒ ࠵? = - % & +./(’) (8)
6 SETUP There are only two new circuit elements that we are going to introduce in this lab. One of them is the capacitor, while the other is the switch. Figure 1 shows an image with the capacitor and switch we are using in this lab. We will be using an external capacitor for our circuit. The capacitor has a white band which indicates the negative sign. This switch does not have a polarity associated with it and can be oriented in either direction. Figure 1: Image showing the capacitor and switch that its being used in this experiment. The capacitor is shown on the left, while the switch is shown on the right. Note: the white band on the capacitor indicates the negative terminal for the capacitor.
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