Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose of the Experiments... 2 Principles an Error Analysis... 2 The Instrument an Types of Experiments... 3 Experiments Step by Step... 4 Data Analyses an Result Reporting... 7 Appenix... 7 1
Purpose of the Experiments Through measurements of the capacitance of a parallel plate capacitor uner ifferent configurations (the istance between the two plates an the area the two plates facing each other), one verifies the capacitance formula, which is euce irectly from Gauss s Law in Electricity. These experiments will enforce the unerstaning of Gauss s Law an the assumptions use in eriving the formula for calculating the capacitance of a parallel plate capacitor. Through the comparison of experimental ata with theoretical calculations, an ata analyses with error estimation an correction, stuents obtain better unerstaning of theoretical moeling, formulism an their eparture from realistic configurations. Stuents receive complete training in ata acquisition, analyses, incluing error corrections, an result presentation plus report writing. Also inclue in the experiments are the measurements of ielectric constants of several soli an liqui samples to enhance the unerstaning of the concept of an insulator s ielectric constant. Through these measurements, stuents get training in using instruments such as igital multi-meter, igital capacitance meter, micrometer hea, an unerstan measurement errors that come from instruments. Principles an Error Analysis A parallel plate capacitor is mae of two metal plates that are parallel to each other. The capacitance is etermine by the istance between the two plates, the area A they face each other plus the insulating material an its ielectric constant. The formula for the capacitance is euce irectly from Gauss s Law in Electricity. When an electric voltage V is applie to the two plates, charges introuce to the plates establish an electric fiel E between them, as illustrate in the figure on the left. When the size of the plates is much larger than the istance between them, this electric fiel E approximates to that when the plates are infinitely large an can be calculate through Gauss s Law. In the figure σ is the charge ensity, ε is the ielectric constant of the material between the plates. Accoring to the efinition of capacitance C!, we can erive the formula for the capacitance of! this parallel plate capacitor: C = Q V = ε A Here ε = ε r ε 0 is the ielectric constant,ε 0 = 8.854 10 12 F/m is the permittivity of vacuum,ε r is the relative ielectric constant of the material between the plates. For air ε r 1,for an orinary printing paper ε r 3.85,for water ε r 12.9. 2
Possible sources of errors: (1) from theoretical moeling: the electric fiel along the eges of the plates cannot be calculate using Gauss s Law. When the istance between the two plates is not much smaller than the size, this ege effect is no longer negligible hence the above formula uner the assumption that the plates are infinitely large is no longer correct or at least precise. When the two plates are slightly not parallel with each other, one can fin the correction to the capacitance to be ΔC = C Δ. Here Δ represents the average unparallelism. It is clear that this correction becomes significant when is not much larger than Δ, for example <10 Δ. When the assumption of parallelism is no longer vali, for example when Δ, the charges on the plates are no longer evenly istribute, the electric fiel generate by each plate is no longer parallel to each other hence the fiel between the plates cannot be obtaine by a simple aition of the fiels of the plates. (2) from measuring instruments: thickness of the two metal plates that make up the parallel plate capacitor plus measuring wires an probes provie a boy capacitor that is aitional to the parallel plate capacitor. This boy capacitance is small an is not a function of the istance between the plates. It only becomes significant when is large (hence C is small). (3) from mistakes of the operator: for example, misrea the instruments or typos in recors. By changing the parameters in the above formula (, A an ε = ε r ε 0 ) an measure the capacitance of the parallel plate capacitor, one can verify this formula that is euce from Gauss s Law in Electricity. The Instrument an Types of Experiments The instrument consists of a parallel plate capacitor with variable (the istance between the plates) an A (the area the plates face each other), as shown in the figure below, another parallel plate capacitor with fixe an A, plus a igital capacitance meter an a multi-meter (use to measure resistance). In aition there are three insulating plates that can be use as ielectric material between the two metal plates, an a ruler. In the variable capacitor the parameter is set by the micrometer hea of the stage with a precision of 0.01 mm an a range from 0.01 mm to 10 mm, which is 0.01% - 10% of the size of the metal plate (a 10 cm 10 cm square). The parallelism of the two metal plates is etermine by the stage assembly, an is about 0.03 mm. The parameter A is change by sliing the bottom plate with a precision of 1 mm. The range of A is from 0 to 10 cm 10 cm. There can be two types of experiments. The first type uses the variable capacitor to check the relationship between the capacitance C an the parameters, A an ε r. Setting the value of through the micrometer hea, one measures the corresponing C to verify that 3
C 1. At a chosen, for example = 0.5 mm or 1.0 mm, sliing the bottom metal plate to change A, one measures the corresponing C to verify that C A. To check the effect of ε r in C, one select an insulating plate an place it between the two metal plates, raise the bottom plate so that the istance is the thickness of the insulator. Be careful an avoi over tightening an stripping the precision threas of the micrometer hea. By measuring the capacitance with an without the insulator one can calculate the value of ε r of this material. The secon type of experiments employs the fixe capacitor. Measure the capacitance of this capacitor in air, an then immerse the capacitor in the (nonconucting) liqui of which you want to measure its ielectric constant. As an example of how physics is the founation of instrumentation, we use this fixe capacitor as a meter to measure the liqui level of a known ε r. Place the capacitor with the plates vertical in a container an then a liqui (for example istille water), when the liqui reaches a height h, ignoring the boy capacitance at this moment, one can obtain the value of h through this formula C = ε 0 a "# a + ( ε r 1)h$ %. Calculate for h an then verify that using the ruler. Experiments Step by Step The following experiments shoul be carrie out by a group of 2 to 3 stuents. Each stuent must take the measurement while the other(s) help recor ata. The recore 2 to 3 sets of ata of the same measurement settings will be use to fin an correct operator mistakes. Experiment 1, verify that C is inversely proportional to, that is C 1, in the formula of a parallel plate capacitor C = ε A. Steps: Rea the appenix about how to use the micrometer hea. Slie the bottom plate until the two metal plates completely overlap. Connect the igital multimeter to the two plates an measure the resistance between them. As the plates o not touch each other, the multi-meter reas infinite resistance. Now raise the bottom plate until the multi-meter just start to rea 0 ohm. Recor the reaing of micrometer hea an set that to be = 0 mm. Now you can procee to set the an use the capacitor meter to measure the corresponing C. Recor ata in the Excel file provie, or in any format you choose. That ll be your raw ata. The Excel file has formulas entere for ata analyses such as estimating the errors an plotting the results. One measurement result, as an example, is shown in Figure 1. In Figure 2 we compare the measurements with those calculate from the formula C = ε A, incluing those after the correction of unparallelism an boy capacitance. The estimate of Δ is from minimizing the ifference between measurement an calculation in the range of = 0.20 0.35 mm. Of course this is 4
a coarse estimate. Stuents are encourage to propose an carry out their own error estimate. Figure 1, the measure capacitance (raw ata, no correction ae) as a function of 1/. By eye this is a straight line. Figure 2, comparison of the measure capacitance (raw an correcte) with calculation, as a function of. After correction an > 2 mm, the ifference between measure an calculate is below 5%, an ecreases systematically, inicating the ege effect that is more significant as increases. Stuents shoul follow their instructor to perform more thorough ata analyses, for example to inclue the errors from the capacitance meter, an set confience level in checking the assumption that C 1 Experiment 2, verify that C is proportional to A, that is C A, in the formula of C = ε A. Steps: slie the bottom plate until the two metal plates completely overlap. Set the esire, for example use = 1 mm. This value shoul be etermine taking into account the unparallelism, the ege effect an the meter error. Stuents will nee to justify their choices, an compare results from ifferent settings. The area A 5
that the two plates overlap can be change by sliing the bottom plate. Measure C as A changes an plot the result. An example is shown in Fig. 3. A question: when the two plates o not overlap at all, what o you rea from the capacitance meter an what that reaing means? Is this a way to measure the boy capacitance? If yes, compare the number with what you ve estimate in experiment 1, an explain the ifference if there are any. Figure 3, the measure capacitance (raw ata, no correction ae) as a function of A. By eye this is a straight line. Experiment 3, verify the relationship of C with the ielectric constant in the formula C = ε A. The steps are similar to those in Experiment 1. Place one sample of the soli insulating plate between the two metal plates, raise the bottom plate so that there is no air gap between the sanwich, an measure the capacitance. After that remove the insulating plate while keeping the istance between the two metal plates (the stage can be locke if neee), an measure the capacitance again. Compare the two numbers of C, one can calculate the ielectric constant of the insulating material. Check what you get with what you can fin from reference or internet an explain possible ifferences. Do you nee to correct of factors such as the boy capacitance in your calculation? Experiment 4, measure ielectric constants of non-conucting liqui samples. Use a container for the liqui, compare the capacitance of the fixe capacitor in air an in liqui to calculate for the ielectric constant, as in Experiment 3, pay attention to corrections such as the boy capacitance. Now compare your results with what from references an see if the ifference is smaller or larger as in the case of Experiment 3, an why? 6
Experiment 5, measure liqui level using the fixe capacitor The fixe capacitor is mae of two square metal plates of the size of 10 cm 10 cm, with the istance = 1.00 mm. Set this capacitor vertically in the container of Experiment 4, an fill the container with liqui (for example water). When the liqui reaches a height h, measure the capacitance. The height h can be measure with a ruler. Repeat this measurement at several values of h, an recor ata. Use ε this formula C = 0 a "#a + (εr 1) h$% to calculate h from C, an compare your results with the numbers obtaine using the ruler. If they o not agree, what kin of corrections you nee to make in orer to get a better result? Do you know other liqui level measuring instruments an the physics behin them? Data Analyses an Result Reporting Experimental ata submitte must be in the format of the raw ata. Data analyses can be performe using Excel but stuents are encourage to use more powerful tools such as root (https://root.cern.ch/rupal/). In the lab report on each of the 5 experiments, one nees to explain the physics, provies the iagram to show how instruments are connecte an which range of the meter was use (an why). The methos in ata analyses an error estimation must be clearly state. The final results can be presente using graphs (as in this manual) or tables. Questions raise in this manual also nee to be answere. If you have questions about the experiments, you can write them in the iscussion part of the your report. Appenix How to rea a micrometer hea: the scale on the micrometer is 1 mm. The ticks on either sie the longituinal line is offset by 0.5 mm. The knob has 50 markers with each corresponing to 0.01 mm, or 10 microns. This is the positioning precision of the stage an the istance between the two plates. When the two plates are very close to each other, be careful when turning the knob so as not to strip the fine threas. How to use the capacitance meter: in the experiment we only use the 2000 pf or 200 pf range. Before any measurements, the meter nees to be zeroe. The probes can be plugge into the two holes marke with Cx. The other sie of the probes (with a clamp) is connecte to the metal plate. Measurement precision an other information about this meter is in the following table: 7
Range Relative precision Resolution Measurement frequency 200 pf 0.5% 0.1 pf 800 Hz 2000 pf 0.5% 1 pf 800 Hz 20 nf 0.5% 10 pf 800 Hz 200 nf 0.5% 100 pf 800 Hz 2000 nf 0.5% 1 nf 800 Hz 20 µf 0.5% 10 nf 80 Hz 200 µf 0.5% 100 nf 8 Hz 2000 µf 1% 1 µf 8 Hz 20000 µf 2% 10 µf 8 Hz 8