Math265 Course Project Part A TJ

Math265 Project Part A Name: Taylor Johnson In this part of the project, you will be modelling the behavior of a resistor-capacitor (RC) circuit that is being charged by a DC voltage source. In this circuit, initially, the capacitor is uncharged. When switch ‘A’ closes, the voltage source (ε) begins charging the capacitor (C) through the resistor (R). Applying Kirchhoff’s Voltage Law results in the equation ε − V R − V C = 0 Using the definition of capacitance, C = q V C , where ‘q’ is the charge on the capacitor and ‘V C ’ is the voltage across the capacitor, we can solve for the voltage as V C = qC . Next, using Ohm’s law, the voltage drop across the resistor is V R = IR . The current through any portion of the circuit is defined as the rate of change of the charge: I = dq dt . Inserting these values into our equation above, we have ε − R dq dt − qC = 0 This differential equation can be solved to determine an expression for the charge on the capacitor as a function of time: Q ( t ) = Q 0 [ 1 − e − t τ ] where Q 0 = Cε is the final charge on the capacitor and τ = RC is known as the time constant I. Graphing Charge vs Time (25 points) Consider an RC circuit that is being charged with a voltage source ε = 5 V . The charge on the capacitor as a function of time is shown in the table below, with time in seconds (s) and charge in Coulombs (C). 1 | P a g e

Time (s) Charge (C) 0 0 0.0005 1.11×10 -6 0.001 1.97×10 -6 0.0015 2.64×10 -6 0.002 3.16×10 -6 0.0025 3.57×10 -6 0.003 3.88×10 -6 0.0035 4.13×10 -6 0.004 4.32×10 -6 0.0045 4.47×10 -6 0.005 4.59×10 -6 Open Desmos by copying and pasting the following link into your web browser or use control+click ( https://www.desmos.com/calculator/a3woq4autu ). Enter the data from the table in the column ‘Q’. To enter numbers in the correct format, type 1.11*10^-6 to enter 1.11×10 -6 Adjust the fitting parameters ‘a’ and ‘b’ using the sliders so that the curve matches the points and record your fitting parameters below: a= 0.000005 b= 500 Noting that a = Q 0 = εC , where ε = 5 V , determine the capacitance of the capacitor. Show your work. C= 0.000001 a(0.000005)= Q 0 =5(C) 0.000005/5 = 5(C)/5 0.000001=C Noting that b = 1 τ = 1 RC , determine the resistance of the resistor. Show your work. R=2000 2 | P a g e

Using Desmos and the quantities R, C and ε from section I., plot the current as a function of time, using the equation above. In Desmos, type the values for ε, R and τ in the gray areas in the equation provided. Note that ‘x’ is our time variable and ‘y’ is the current. The current can also be approximated using the data points from part a) using i = ∆q ∆t As a sample calculation, for time t=0s, Current = q 2 − q 1 ∆t = 1.11 × 10 − 6 − 0 0.0005 = 0.00221 Continue this process and fill in the empty cells in the table below maintaining three significant figures. Time (s) Charge (C) Current (A) 0 0 0.00221 0.0005 1.11×10 -6 0.000860 0.001 1.97×10 -6 0.000447 0.0015 2.64×10 -6 0.000260 0.002 3.16×10 -6 0.000164 0.0025 3.57×10 -6 0.000103 0.003 3.88×10 -6 0.000071 0.0035 4.13×10 -6 0.000048 0.004 4.32×10 -6 0.000033 0.0045 4.47×10 -6 0.000024 0.005 4.59×10 -6 Type the data points into Desmos under column I 1 and plot. Copy and paste your graph with the theoretical and approximated current versus time below. Screenshot of Desmos: 4 | P a g e

III. Summary of Section I. (10 points) Write a two paragraph summary of your findings from Section I. Explain the setup and the results. Section I. Summary Section one was, as it was titled, graphing charge versus time using the Desmos Graphing Calculator. We had to use the calculator to figure out the capacitance of the capacitor and the resistance of the resistor in order to finish the calculations of the circuit. Using the measurements of the charge on the capacitor over a period of time with measurements taken at different time intervals as points on the graph, I was able to adjust the sliding bars of the variables to match the graph and use those variables to calculate the capacitors total capacitance and the resistors total resistance. IV. Summary of Section II. (10 points) Write a two paragraph summary of your findings from Section II. Explain the setup and the results. Section II. Summary I was to graph the current versus time in order to calculate the theoretical and approximated current versus time points. I used the setup of the Desmos Calculator provided and filled in the quantities of total capacitance of the capacitor, given source voltage, and resistance of the resistor into the equation to show the graph. 5 | P a g e