Experiment 5: RC Circuits
The purpose of this lab is to learn and understand RC Circuits. An RC circuit is composed of at least one resistor and at least one capacitor. A capacitor is composed of two plates with either air or an insulator also known as a dielectric between the plates. We do not want the plates to be touching, because then we would only have a conductor. The insulator between the plates is also known as the dialectic, which affect how the capacitor will store charge. In an RC circuit, voltage will flow from the battery to the capacitor and through the resistor. When the capacitor is charging, the voltage across the battery is decreasing until the capacitor is fully charged. When the capacitor is fully charged, then the voltage through the battery is zero. That would also mean that the voltage of the circuit would drop until it is also zero. Now when the capacitor completely discharges, the voltage through the battery increases.
We were able to measure the half-time of the charging and discharging of the capacitor by connecting the circuit to the oscilloscope with the signal generator providing the potential for the circuit. The time constant was calculated from the half-time of charging and discharging. The time constant is a measure of the length of time a capacitor took to charge and discharge. We used the average of the charging and discharging time constants to calculate the capacitance by using the equation τ = RC. Since we know the resistance and the time constant, we are able to solve for the capacitance and compared the observed and theoretical values in order to verify the capacitance. For the RC circuit with one capacitor, we compared the theoretical and observed time constant and obtained a percent difference of 9.5%. For the RC circuit with two capacitor in series and the RC circuit with two capacitors in parallel, percent differences between the observed and theoretical values were all 9.5%. So we were able to verify the capacitance for all the parts of experiment one. In part two of the experiment, we made a capacitor by placing a sheet of wax paper between two aluminum foil plates.
First, we measured the thickness of the wax paper by using the micrometer. Through this method, we measured the thickness to be 35±05 micrometers. Then we found the average time constant and then calculated for the capacitance. Then from the capacitance, we were able to solve for the distance between the plates. The distance between the plates through this method was solved to be 580 micrometers. Our percent difference for the theoretical and measured thickness was 180%. The calculated thickness should be equal to the measured thickness, but this was not the case.
½ timecharge = ½ time??discharge ? = 0.3 division x 250 μs/division = 75 μs
τchrage = τdischrage =
τaverage = = 110 μs
τtheoretical = RC = 100Ω x 1μF = 100μs
Percent Difference =
= = 9.5%
Uncertainty capacitance = 20% x 1μF = 0.2 μF
½ timeuncertainty = 0.1 division x 250 μs/division = 25 μs
½ timeaverage = = 35 μs
τuncertainty = = 36μs
τuncertainty average = 0 μs
Two Capacitors in Parallel
Ctotal = C1 + C2 = 1μF + 1μF = 2μF
Percent Difference =
= = 9.5%
Two Capacitors in Series
Ctotal = = = 0.5 μF
Percent Difference =
= = 9.5%
Thickness of Wax Paper
d = 65μm – 30μm = 35μm
d = = m2 / C2= 580μm
Percent Difference =
= = 180%
To find the time constant, we first need to determine the peak and trough of the graph on the oscilloscope. We then adjusted the graph by centering it and translating it along the x-axis so that the peak is align with the y-axis. Then we read the time between a peak and a trough and divided that value by 2 to obtain the half time, which is the time required for the voltage to get 50% of the way to the final value. With the half time, we can obtain the time constant by using the formula, .
For the RC circuit with one capacitor and one resistor, we calculated the theoretical time constant to be 100 microseconds, because we know the values of resistance and capacitance. So, we used the formula . We estimated the half time to charge and discharge of the graph on the oscilloscope and then use the formula, τchrage = τdischrage = to find the time constant for charging and discharging. Then we used the average of those two time constants to compare with the theoretical time constant. The percent difference between the theoretical and observed time constant is 9.5%. For the RC circuit with two capacitors in parallel, we expected a greater equivalent resistance, because the capacitances are additive when they were arranged in a parallel manner. So the theoretical equivalent capacitance is equal to 2.0±0.3 microfarads. The half time for this RC circuit to charge and discharge is 150 microseconds. So the time constant for this RC circuit is 110 microseconds.
Then using , the capacitance is calculated to 2.20micro-Farad. Comparing the theoretical value to the observed value, we calculated the percent difference to be 9.5%. The percent difference was not significant, so using the oscilloscope to find the time constant is a fairly accurate way to calculate the observed capacitance. For the RC circuit with two capacitors in series, we expected the equivalent capacitance to be lower than the equivalent capacitance of two capacitors in parallel. By using the formula , we obtained 0.50.1microfarads for the theoretical equivalent capacitance. The half time for charging and discharging is both 3825 microseconds. We used the average half time to calculate the time constant to be 5525 microseconds. Then we use the average time constant to calculate the capacitance to be 0.55 microfarads. The percent difference is also 9.5%, which implied that finding the capacitance through this method is valid. For the second part of the experiment, we created a capacitor by using two sheet of aluminum foil with the same area and a wax paper.
We inserted the wax paper, which is the dialectic, between the aluminum foils to create a capacitor. Then we connected the capacitor to the RC circuit. In this part, we need to find the half time, the time constant, and capacitance in order to calculate for the thickness of the wax paper, or distance between the plates. In order to solve for the distance, we used the formula C=, which k is the dialectic constant, is the permittivity of free space, A is the area, and d is the distance between the plates.
We rearranged the formula do solve for distance, so we used the formula, d=We measured the length and width of the plate, so the area is the multiplication of the length and width of the area plate. After the calculations, we obtain 580 micrometers for the thickness of the paper while the theoretical thickness is 35 micrometers. The percent difference between the theoretical and calculated thickness of the wax paper is 180%, which is significant to indicate that this method does not validate the space between the plates. The high percent difference can be attributed to the gaps between the foils, which we could not eliminate by putting pressure on the plates. A direct relationship between the distance and dielectric constant exists. So if the dielectric constant is too big, then the distance between the plates would be too large, which would make the distance between the plate too large to measure experimentally. Now if the dielectric constant was too small, then the distance would also decrease and it would be negligible to affect the capacitance. The purpose of the signal generator was to produce a frequency that allocated enough time for the capacitor to fully charge and then fully discharge.
As we increased the frequency, we observed that the period lower, which mean that the capacitor cycled between the charged and discharged stage rapidly or at a higher rate. This prevents the capacitor from fully charging and discharging. A higher period would be observed if we decrease the frequency. The errors in the lab that contribute to the percent differences can be attributed to the gaps between the dielectric and the plates, parallax, and internal resistance of the wire. The gaps in between the dielectric and the plates attributed to the increase in the distance between the plates. With gaps between the plates and wax paper, our dielectric includes air and the wax paper. We never accounted for the electric constant for air in our calculations for distance. For the parallax, we sometimes have trouble discerning the exact time for the voltage to decrease from maximum voltage to half of that voltage on the screen. This could increase or decrease the half time, which would then increase or decrease the time constant and the capacitance calculations. The internal resistance of the wire was not included in our calculations, because we only used the labeled resistance of the resistor in our calculations. The internal resistance of the wire would increase the equivalent resistance. And increased equivalent resistance would implied a lower capacitance because an inverse relationship exist between resistance and capacitance exist from the formula τ = RC. Then a lower capacitance would mean an increase in distance because an inverse relationship exist between capacitance and distance from the formula d=.