Physics: Charging and Discharging a capacitor Lab

Use details from Part A and Part B to answer Questions in Part C

Part A = References (Protocol + Instructions + Procedure + Formulas ) and Reference Graphs

It includes all information needed, so questions cannot be splitted since every one of them are interlinked together

Purpose The purpose of this lab is to investigate the concept of capacitance. You will build RC circuits with capacitors and resistors and will collect and graph charging and discharging data as well as predict the charging and discharging time of specific capacitors. Equipment Voltage source, multimeter/voltmeter, breadboard, 2 resistances, 1 capacitor, stopwatch, wires, switch. INTRODUCTION A parallel-plate capacitor is the arrangement of two electrodes (conductors) closely spaced and charged equally but oppositely. (a) Parallel-plate capacitor The plates are wide compared to the distance between them. Plates have area A. (b) Cross section This cross-section view shows a small region near the center of the plates, far from the edges. Because the opposite charges attract, the charges are on the + + + + + inner surfaces of the plates. Fig 1 Capacitors are useful because they store electric potential energy, since energy was used to separate the charges on the two conductors. In general, the electric potential energy Ueler, measured in joules, is the interaction energy of a charge with the source of an electric field. A measure of the amount of energy it takes to place a charge q at the specific place in an electric field is the electric potential V = Uelee/q or AV, potential difference or voltage. The unit of potential is the joule per coulomb, or volt V: 1 volt = 1 V = 1J/C. For a capacitor, V=2vQ=CV. The capacitance C of a capacitor is then Capacitance= charge -vc=2 potential difference The unit for capacitance is the farad F: 1 F = 1 C/1 V A larger capacitance C allows the storage of a larger charge Q at a specific voltage V. The capacitors you will use have 1000 pF or 680 pF. Charging and Discharging a Capacitor If you connect the two plates of a parallel-plate capacitor with a metal wire (a resistor), the capacitor discharges because the charges that had been separated on the two plates flow, producing a variable electric current I (measured in amperes, A) IAQ At The values of the resistance and the capacitance in an RC circuit determine the time it takes the capacitor to charge or discharge. (a) Before the switch closes (b) Immediately after the switch closes (c) At a later time The charge separation on the capacitor The current has reduced the charge The switch will close at 7 = 0. produces a potential difference, which on the capacitor. This reduces the causes a current. potential difference. R R R Charge Co Current is the flow of charge, so the (AV) = Qo/C The reduced potential difference current discharges the capacitor. leads to a reduced current. Fig 2 The switch closes at this instant. When a capacitor is discharged across a fixed resistance (using a resistor R), the current and the capacitor voltage decreases exponentially to zero after the switch closes. I=lenc 1=0 As time goes on, the current and the capacitor This function is used for both charging and discharging - only voltage decrease. the direction of the current is different. 2AVEAVAC with Io the initial current, I the voltage at a moment t (both in amperes) AVc the voltage on the capacitor at a moment t AVo is the initial voltage (both in volts), time t (in seconds), R is the resistance (in ohms), and C is the capacitance. Fig 3 If you take the logarithm of the voltage formula, you get a linear relationship between InA Vc and c: In AV =InAV.- RC =RC is defined as the time constant: a characteristic time for a circuit. A long time constant implies a slow discharge; a short time constant, a rapid discharge. The slope of the line representing the graph InA Ve vs t is the opposite of the inverse of the time constant. In AV=InAV.--, slope=1 Inv [a] (b) Fig 4 from Charging and Discharging Capacitors https://www.d.umn.edu/-djohns30/phys 1002- labs/Lab%204%20Capacitors.pdf Capacitors can charge or discharge in milliseconds or hours, depending on the time constant, or the resistance and capacitance of the equipment. In a circuit that charges a capacitor, once the switch is closed, the electric potential of the battery causes a current in the circuit, and the capacitor begins to charge. The capacitor charges until its voltage V becomes equal to that of the voltage supply, c. A capacitor needs a time interval of 5 times t to charge up to 99.3% of maximum value. 3R BIR 0.63 8 Current. i Current AVC 0.376/R S 1=0 1= RC 2RC 3RC RC 2RC 3RC Time Time (a) (b) (c) Fig 5 from Castano, Diego and Castro, Victor, "Experiment 2.06: Series RC de-Circuit" (a) Simple RC-circuit. (b) The potential difference increases with time until it reaches a value of c. (c) The current decreases with time until it reaches a value of zero. I = le-URG AVe = =(1-9 (RC ) Current and voltage while charging a capacitor For (= T, AV =1-e =(1-ezz(1-0.37)=0.63 1=be =0.37- With some algebraic manipulation, the exponential term in the voltage versus time formula above can be isolated. AVC=1-PT E AVC_E-AVE E E . If you take the logarithm of this expression, you get: =L= In E-AVE In E-AV= In( z)- The last expression represents a linear relationship between In e-AV) and time t written in the slope intercept formula y = mx + b, with the slope m=1 4PRE-LAB You must complete this part before attending the on campus lab. . On the PHY359SAC lecture course, click/hit Physics II in Course Menu on the left. Open Physics II page, then scroll down to find the RC/RL/LC Circuits module and watch the video. Focus on O Principles Behind the RC/RL/LC Circuits o RC Circuit Watch How a Breadboard Works Tutorial at https:/ /www.youtube.com/watch? EWCIKMFGEmul Pre-lab question 1 What is a breadboard used for in this lab? PROCEDURE A: Charging a capacitor 1. Write down the values for the voltage source & the capacitance of the capacitor and the two resistors provided. 2. Estimate the corresponding time constants based on these values. 3. A new capacitor has to be charged and recharged a couple of times before you could perform the experiment. Use the smaller resistance provided, the capacitor, and the voltage supply to charge the capacitor, then remove the battery to discharge it. 4. Set up the voltage supply & at 12 V and the voltmeter/multimeter be on the 20 V scale. 5. Connect the switch, the voltage supply, the larger resistor, and a capacitor in series but do not close the circuit until you are ready to monitor the voltage across the capacitor and the time, then connect the voltmeter across the ends of the capacitor in the circuit shown in Figure 6. Fig 6 from Castano, Diego and Castro, Victor, "Experiment 2.06: Series RC de-Circuit breadboard 6. Start the stopwatch and close the circuit simultaneously. For every 10 seconds ( more often or less often, so that you have approximately 20-30 data), read the voltage as precisely as possible - it will rise quite fast initially. o Example: for T = 60 s, the capacitor needs 5 x 60 s = 300 s to fully charge, so recordings at every 10 s are reasonable; you would end with 30 readings. 5o For larger time constants, your recordings should be less frequent. 7. Continue for 4 to 5 minutes, or until the voltage on the capacitor is approximately 10 V. PROCEDURE B Discharging a capacitor 8. Once the capacitor is charged, disconnect the battery from the circuit so that the capacitor discharges. Again, record the voltage and time every 10 seconds. Keep recording until the voltage reaches 0.5 V. 9. Repeat steps 5-8 with the second, smaller resistor. DATA ANALYSIS 10. Import in Excel your data for the charging RC circuit with the first resistor Ri, then graph the dependence of AVe (on the y-axis) on time t (on the x-axis). 1 1. Create in Excel a new column for In (8 - AV.) 12. Use Excel to graph a line representing the dependence of In(E - AVc) (on the y-axis) on time t ( on the x-axis) for the charging capacitor with the first resistor Ri; include the equation of the line in your report. 13. Calculate the time constant 7 from the slope m of the line based on the general expression m=- 14. Compare this measured value of the time constant 7 with its theoretical value In = R, Cand suggest possible sources of discrepancies. 15. Import in Excel your data for the charging RC circuit with the second resistor Ra, then repeat step 10 for the charging circuit with the second resistor Ry. 16. Import in Excel your data for the RC discharging circuit with the first resistor Ri, then graph the dependence of AVe (on the y-axis) on time t (on the x-axis). 17. Create a new column for In(AV.) 18. Use Excel to graph a line representing the dependence of In(AVc) (on the y-axis) on time t (on the x-axis) for the charging capacitor with the first resistor Ri; include the equation of the line in your report. 19. Calculate the time constant To from the slope m of the line based on the general expression m=- Reference Figures 1, 2, 3 from College Physics: A Strategic Approach, 4th Edition. Knight, Jones, Field. 2019, Pearson Education Inc. Castano, Diego and Castro, Victor, "Experiment 2.06: Series RC dc-Circuit" (2022). Physics Lab Experiments with Simulated Data for Remote Delivery. 15. https:/suworks.nova.edu/physics_labs/15C=474 LIF R1 163 k(2 R2 48 k(2 E 8.00V Charging with R1 Charging with R2 Discharging with R1 time (s) Voltage (V) time (s) Voltage (V) time (s) Voltage (V) 0 D.00 0 7.99 10 0.85 1.48 10 7.13 20 1.60 10 2.63 20 6.48 30 2.30 15 3.50 30 5.98 40 2.90 20 4.30 40 5.43 50 3.45 25 5.00 50 4.08 60 3.90 30 5.65 60 4.38 70 4.35 35 6.25 70 3.95 80 4.75 40 6.66 80 3.60 90 5.10 45 7.02 90 3.24 100 5.40 50 7.33 100 2.96 110 5.70 55 7.57 110 2.69 120 5.97 60 7.81 120) 2.44 130 6.21 65 8.00 130 2.22 140 6.43 140 2.02 150 6.64 150 1.83 160 6.82 160 1.67 170 6.99 170 1.51 180 7.14 180 1.38 190 7.28 190 1.26 200 7.42 200 1.14 210 7.53 210 1.04 220 7.64 220 0.95 230 7.74 230 0.86 240 7.83 240 0.79 250 7.91 250) 0.72 260 8.00 260 0.65 270 0.60 280 0.55 320 330 340\fData set is provided, in the protocol, it says 1000 pF or 680 pF for the capacitors but has been replaced below with a capacitor of 474 uF (Please check whether the values and calculations are correct given the data sheet and calculate and fill in the values in all tables for the remaining) Using RC-data set Capacitance (F) 474 ULF=0.000474 F Theoretical time constant 1 microfarad (HF) = 10^-6 farads (F) Larger resistance R, (k() 163 k0 v= 163 000 0 01th = RIC (S) (163 kn)(0.00047 4 F)= 0.0773 seconds Smaller resistance R, (kn) 48 k0=48000 0 Top = RC (9) (48 ko) (0.000474 F)=0.023 seconds Voltage supply = (V) 8.00 VProcedure A: Charging a capacitor 1. Measurements for time and voltage while the capacitor is charging through R, and, then, R (at least 20 measurements for each.) Add more rows to your table as needed. 2. Calculate In (# - AV) for each V value for R, (kQ) only, rounded to 2 decimal digits, and fill in the new column In (# - AV)\f230 7.74 240 7.83 250 7.91 260 8.00 Procedure B Discharging a capacitor 3. Measurements for time and voltage while the capacitor is discharging through R, and, then, R (at least 10 measurements for each.) Add more rows to your table as needed. column In V. 4. Calculate InV for each V value for R, (KQ) only, rounded to 2 decimal digits, and fill in the new Table 2 Discharging capacitor R2 (k?) Time :(9 Voltage V (V) InV 0 Time t (9) 7.99 Voltage V (V) 0 8.00 10 7.13 5 6.71 20 6.48 10 5.72 30 5.98 15 4.83 40 5.43 20 4.01 50 4.08 25 3.50 60 4.38 30 2.90 70 3.95 35 2.50 80 3.60 40 2.01 90 3.24 45 1.80 100 2.96 50 1.54 110 2.69 55 1.34 120 2.44 60 1.13 130 2.22 65 0.95 140 2.02 70 0.82 150 1.83 75 0.71 160 1.67 80 0.62\fUse Excel to plot a graph representing the linear dependence of of: [on the y-axis] on time t [on the x-axis] with data from Table LEI on1}-'.Lahel and scale the axes and title the graph. Include your graph below. Use Excel to plot a graph ln in [s 513;] [y-axis] vs time t [Jr-axis] with data from Table 1, for Charging R1 [kJ only. Label and scale the axes and title the graph. Include a trendline that best approximates the linear relationship; and its equation. Determine the time constant from the slope of the trendline found for the previous question.Compare the theoretical Tm and measured [from question T} time constant for the RC circuit. suggest possible sources of discrepancies. Use Excel to plot a graph representing the linear dependence ofdlr': [on they-axis] on time t [on the x-axis] for the charging capacitor with the second resistor R2. Label and scale the axes and title the graph. Include your graph below. Compare the graphs you obtained for questions 5 and 9 to comment on the effect of the resistor on the shape of the charging curve.Use Excel to plot a graph InV (v-axis) vs time t (x-axis) with data from Table 2, for Discharging R1 (k$2) only. Label and scale the axes and title the graph. Include a trendline that best approximates the linear relationship, and its equation.Determine the time constant r, from the slope of the trendline found for the previous question.Compare the time constant you determined in questions 8 (for charging the capacitor) and 13 (discharging the capacitor) through Rj. Should it be the same or not? Discuss