Capacitance and RC Circuits: Understanding Charge, Voltage, and Time Constants, Lab Reports of Experimental Physics

Download Capacitance and RC Circuits: Understanding Charge, Voltage, and Time Constants and more Lab Reports Experimental Physics in PDF only on Docsity! E4.1 Lab E4: Capacitors and the RC Circuit Capacitors are among the most-commonly used circuit elements in modern electronic devices. Your calculator and cell phone contain dozens or hundreds of them. Any two electrical conductors brought near each other form a capacitor. If a charge Q is transferred from one conductor to the other (so that one has a charge +Q and the other has charge -Q), then there will be a voltage difference V, which is proportional to the charge Q. The constant ratio Q V is called the capacitance C. (1) C Q V V Q C = =, . [To see that V must be proportional to Q, recall that the electric field E is proportional to the charge ( ), and the voltage is proportional to E Q∝ E ( V E∝ ), hence, ] V Q∝ . A parallel-plate capacitor, formed by two flat plates of area A, separated by a distance d, has a capacitance given by (2) C A do = ε , where is the permittivity constant, which has the value in SI units. Eq’n (2) applies when the plates are separated by air or vacuum. When there is a dielectric medium such as plastic or glass between the plates, the capacitance is greater than predicted by (2) and the correct formula is εo εo = × −885 10 12. (3) C A do = κε . κ ( Greek letter kappa) is a dimensionless number called the dielectric constant. The value of depends on the dielectric medium, but it is always greater than one ( >1). The dielectric constants of various media are .. κ = κ κ Area A d dielectric medium, κ A parallel-plate capacitor Material κ nylon 3.4 oil 4.5 Pyrex 4.7 silicon 12 titania ceramic 130 strontium titanate 310 The SI unit of capacitance is the farad (F): 1 1farad coulomb volt= / . A farad is an immense capacitance; a typical capacitor in an electronic circuit has a capacitance of Fall 2004 E4.2 micro-farads (1µF = 10-6F ) or nano-farads (1nF= 10-9F ). Even a µF capacitor is regarded as a rather large capacitance and making such a capacitor in a small package is quite a technological feat. Consider trying to a make a 1µF capacitor with two square plates separated by a 1mm (10-3 m) air gap. The area A needed is 6 3 2 11 o C d 10 10A 100 m 10 − − − ⋅ = ≈ = ε . These plates would be about 30 ft on a side! Capacitors with large capacitance and small physical size are produced by making d extremely small ( d m≈ −− −10 106 8 ) and κ large (κ ≈ ). 100 A capacitor can be thought of as a device which stores charge. If the two sides of a capacitor are connected across a battery of voltage Vo and then disconnected from the battery, each plate of the capacitor will carry a charge of magnitude (one side has +Qo , the other has -Qo). The larger the capacitance, the larger the charge carried. If the two sides of this charged capacitor are then connected by a wire, there will be a large, brief current through the wire as the capacitor discharges. If we use a large resistance R, instead of a wire, to connect the sides of the charged capacitor, the capacitor will discharge much more slowly. The charge and the voltage on the capacitor will decrease exponentially in time, with a time constant Q C Vo = o τ = RC ( τ = Greek letter tau). Consider the simple RC circuit shown below. The capacitor C can be connected to the battery Vo with switch S1 or to the resistor R with switch S2. C RVo S1 S2 C R Vo S1 S2(closed) Qo i If switch S1 is closed and then re-opened, the capacitor will be charged to a voltage Vo and will carry a charge Q C Vo o= . If we then close switch S2, a current i will flow through the resistor as the capacitor discharges. The current i, the charge q remaining on the capacitor, and the voltage v across the capacitor and resistor are related by.. Fall 2004 E4.5 Compute the average of , as well as the standard deviation and the standard deviation of the mean. As usual, compare your result with the known value and comment on any discrepancy. εo Part II. Measurement of the time constant of an RC circuit. In this part, you will observe the exponential decay of the voltage on a capacitor in an RC circuit, using a sophisticated digital oscilloscope. You will measure the time constant τ of the decay and compare with the computed τ = RC . In this part, you will use a commercial resistor and capacitor (not the aluminum plate capacitor of the last section). Before assembling the circuit shown below, measure the capacitance C of the capacitor and the resistance R of the resistor using the capacitance meter and the digital multimeter. From your measured values, compute RC; it should be in the range 10 - 50 msec. Now assemble the circuit and set the power supply voltage somewhere between 3.0 and 3.5 volts. Vo S R C to oscilloscope ground side= 3.2V~ red black/green At first glance, the oscilloscope has a formidable-looking front panel. Fortunately, all the correct front-panel settings are stored in memory and can be recalled at any time with following sequence of commands: 1) Press SAVE/RECALL SETUP button on the upper right of the scope. 2) On the row of button below the screen, press RECALL SAVED SETUP. 3) On the column of buttons just to the right of the screen, press RECALL SETUP 1. Fall 2004 E4.6 on/off clearmenu cursor toggle cursor control knob horizontal position vertical position Channel 1 signal in 1 2 3 For correct settings, press buttons 1, 2, 3. volts/div sec/div Ch1 50mV M 5ms 1 @ : 2.13V : 1.24V∆ : 11.6ms∆ Ch1 2.91V voltage between cursors time between cursors voltage of active cursor zero volts level volts/div time/div trig level active cursor (solid)inactive cursor (dashed) Portion of waveform off screen Recorded signal With the circuit properly assembled, close the switch S and then open it. The digital oscilloscope will record the brief exponential decay of the voltage across the capacitor when the switch S is opened. The recorded signal should appear as in the diagram above. As you can see, there is a lot of information displayed on the screen. The volts/division of the vertical scale and the time/division of the horizontal scale are shown at the bottom of the screen. The zero volts position is shown with a little (1→) symbol (1 refers to Channel 1). Fall 2004 E4.7 You can quickly read the voltage and time of any point on the recorded waveform by using the two cursors which are controlled with the cursor toggle button and the cursor control knob. (If any screen menus are displayed, press CLEAR MENU before using the cursor controls.) One cursor or the other is made “active” with the toggle button and the active cursor is moved with the control knob. The active cursor is shown as a solid vertical line; the inactive cursor is shown as a dashed vertical line. The vertical distance (∆voltage) and the horizontal distance (∆time) between the two cursors is shown at the upper right of the screen, along with the absolute voltage of the active cursor. The oscilloscope screen displays only a portion of the recorded waveform. You can view more of the recorded waveform with the horizontal position knob. You want to measure time from the moment when the switch S was closed and the voltage on the capacitor began to decay. So set one of the cursors directly over the point where the exponential decay begins. Then toggle to the other cursor. By moving the other cursor, you have a readout of the time and voltage of any point on the waveform. Record the voltage and time of 10 or more points along the exponential decay, including at least 5 or 6 points in the first part of the curve between the initial voltage Vo and Vo/3. Using all your data, make plots of V vs. t and ln(V) vs. t. The plot of ln(V) vs. t should be a straight line with slope -1/τ. Using only the points between voltages Vo and Vo/3, determine the slope m and the intercept b of the best fit line using the file linfit.mcd , which computes the best fit line to any x-y data. Do not use the data points at small voltages, because these have large fractional errors and will reduce the precision of the fit. The file linfit.mcd is on your PC. In Mathcad, open linfit.mcd, enter your x-y data (x = t, y = ln(V)) and linfit computes m, b, δm, and δb. You can switch back and forth between linfit.mcd and your original file with the WINDOW menu item. From your computed slope m and intercept b, plot the best fit line on your graph of ln(V) vs. t. From the slope , compute the time constant m m± δ τ δτ± . Compare τ with RC, computed earlier. Prelab Questions. 1. Demonstrate that eq’n (5) is the solution to the differential equation (4). Hint: this is similar to a question in Lab M1, The Simple Pendulum. 2. A parallel-plate capacitor has area A = 500 cm2, separation d = 0.550 mm, and is filled with a dielectric medium with . What is the capacitance of the this capacitor, in pF? 8 40.κ = 3. In part II of this lab, you will graph ln(V) vs. t for an RC circuit. What should this graph look like? No numbers! - just a qualitative sketch. In terms of R and C, what is the slope of this graph? Fall 2004