Posted: September 20th, 2022

# Question

1 cover sheet

2 Abstracts

3 objectives

4 background theory

5 Experimental procedures

6 Data

7 Graphs (if asked)

8 Results

9. Conclusion

1

Laboratory Report on RC Time Constants

Student’s Name

Professor’s Name

Course Name

Date

Abstract:

A resistor-capacitor (RC) circuit or RC line or RC organization is an electrical circuit consisting of a resistor and a capacitor. These can be driven very well by voltage or current sources, and they will produce different responses. This experiment seeks to examine how RC circuits can be used to transmit signals by jamming at certain frequencies and across other frequencies. RC circuits can be used to transmit signals by jamming at certain frequencies and across other frequencies. The two most normal RC channels are high-frequency and low-frequency channels; Band channels and band termination channels usually require RLC channels, but unrefined channels can be created with RC channels.

Objectives:

The experiment focused on the RC circuit and measure the time constant for the main RC circuit.

Background Theory:

When showing the possible electrical contrast V between the two terminals of a capacitor of capacitance C, the two terminals accumulate charges of opposite sign but equal magnitude, q = V/C. In an idealistic situation for a fully isolated capacitor, this changes q as soon as V changes. One can charge or discharge the capacitor immediately! Practically speaking, you can never test on a single capacitor. Also, isolated capacitors are not very useful.

Now consider a large RC circuit (Figure 1) consisting of a capacitor with capacitance C and a resistor with resistance R. Due to resistance, the current (I) in the circuit cannot be infinitely large. Thus, the capacitor discharge/charge is not instantaneous (Wilson & Hernández-Hall, 2014). At such a voltage, a larger R results in a simpler I and a longer discharge/charge time in that direction. To change the voltage across the capacitor by the same degree, a larger C causes a larger difference in q and therefore a longer discharge/charge time. Therefore, a larger R or C requires more series time for the RC chain. Such a connetion makes stable weather a controlled and useful frontier.

Some of the useful equation for RC time constants are:

𝑉𝑐 =𝑄/C

𝐼(𝑡)𝑅 = 𝑉𝑐(𝑡)

𝑄(𝑡)= 𝑄0𝑒^−𝑡/RC

𝐼(𝑡)= 𝐼0𝑒^−t/ RC

(𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝑉𝑐(𝑡)= 𝑉o𝑒^−𝑡/RC

(𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝑉𝑐(𝑡)= 𝑉𝑆(1−𝑒^−𝑡/𝑅𝐶)

(𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝐼(𝑡)= 𝑉𝑆/R e^-t/RC

Experimental Procedure:

The first step that I took was set up the RC circuit with R=100 Ω
(Figure 2(a)) then I connected the shift lead S1 (charge) S1 to the positive battery lead (approximately 3.0V) charging as DC power supply, and connect the switching lead S2 (discharge) to the battery ground wire. The warning considered for this step was that volts above 5.00 V would make the capacitor to dissolves. Secondly, the center wire SO of the double throw change was connected to the positive pole of the ammeter before connecting the unfavorable pole of the ammeter to the positive pole of the 1.0 F capacitor and the negative pole of the capacitor to the open wire of one of the two 100ω resistors (Wilson & Hernández-Hall, 2014).

Another resistor wire was then connected to S2 and through this wire goes to battery ground.

Finally, I connected the voltmeter and capacitor according to the correct polarity. The specialist was then asked to thoroughly check the circuit. While undertaking this process, the charge capacitor was connected to the 100Ω resistor

I then set correct ammeter and voltmeter awareness while one support checks the clock and the other at the ammeter and voltmeter and records the readings in Table below. When finished, the clock was started at t=0 and the double change was reversed from S0 to S1. This starts the loading system.

TABLE 1: Charging capacitor connected to R = 100ω

t (s
)

I (A
)

V

t (s
)

I (A
)

V

(V
)

(V
)

5

25.2

0.29

210

3.0

2.70

30

19.6

0.98

250

2.3

2.78

60

14

1.55

300

1.4

2.87

90

10

1.96

350

0.9

2.92

120

7.2

2.25

400

0.6

2.95

150

5.3

2.46

500

0.3

2.98

180

4.0

2.59

600

.2

3

For the third step, the capacitor was disconnected to the 100Ω resistor while holding for another 2 minutes. Again, one support would check the clock and the other reviews the voltmeter and record the readings in Table 2. When we were ready, the clock was started at t=0 and the doubled change was simultaneously reversed in S2. This step aided restarting of the release system.

TABLE 2 Discharge of the capacitor connected to R = 100ω

t (s
)

V

t (s
)

V

(V
)

(V
)

5

210

0.29

30

250

60

300

90

350

120

400

150

500

180

600

2.7

2.06

0.19

1.48

0.12

1.03

0.08

0.74

0.05

0.54

0.02

0.39

0.01

The fourth step involved setting the RC circuit with R=50 (Figure 2(b)), changing step 1 unless R=100Ω was replaced with R=50Ω (by connecting two 100Ω resistors evenly) and removing the ammeter from the circuit. Ask your vet to take a close look at the circuit. Once this step was complete, the capacitor connected to the 50Ω resistor was charged. Like in the previous steps, one support experimenter would check the clock and the other the voltmeter while recording the readings in Table 3. The clock was then started at t=0 and while the change in double step S1 was reversed all the time. This starts the loading system.

TABLE 3 Charging the capacitor connected to R = 50

t (s
)

V

t (s
)

V

(V
)

(V
)

5

0.98

120

30

2.87

150

60

180

2.95

210

2.98

90

250

3.0

 0.42 105 2.75 15 2.82 1.6 135 45 2.02 2.91 2.33 75 2.52 2.64

Data and Graphs

1. Charging the RC circuit to R = 100

Using Table 1, draw VC-versus-t and install the bend according to equation (5) VC(t) A (1 e Bt) with assembly limits A V0 and B 1 / RC.

Figure 1: The graph of VC-against t

Using Table 1, plot I-versus-t and consider bending according to Equation (3′) I(t)

Ae with adjacent boundaries A V0/R and B 1/RC.

Notice assembled V0/R = 26.6/100 ohms = 0.266 01 = RC = 94.46

2. Discharging the RC element with R = 100

Using Table 2, draw a graph of VC – versus t and adjust the deflection according to Equation; VC(t) Ae Bt with installation limitations A V0 and B 1 / RC.

Note assembly 01 = RC = 90.3

Figure 2: The graph of VC against t

3. Loading RC circuit for R = 50

Using Table 3, draw VC-versus-t and assemble the bend according to Equation (5).

VC(t) A(1 e Bt) with adjacent boundaries A V0 and B 1/RC.

Note assembly 02 = RC = 39.2

Results

The information obtained is very broad because it is still within normal limits. When the capacitor is charged, the voltage increases dramatically in 600 seconds and rises to the ends. On release the opposite occurs, there is a dramatic reduction from the start and smoothing towards the end. This level indicates that the battery is fully charged or empty. At the point where the resistance is lower, the battery can be charged faster as shown in Table 3. This shows that if we assume that there is a higher R or C, the steady time of the RC circuit will also be greater. Most likely the error in this lab may be due to human error.

Conclusion

This experiment indicates that RC circuits can be used to transmit signals by ramming at certain frequencies and across other frequencies. The two most normal RC channels based on this experiment are high-frequency and low-frequency channels. Also, the band channels and band termination channels require RLC channels, but unrefined channels can be created with RC channels. The findings from this experiment indicates that the capacitor discharge/charge is not instantaneous. Another conclusion from this experiment is that at such a voltage, a larger R results in a simpler current and a longer charge time in that direction.

Reference

Wilson, J. D., & Hernández-Hall, C. A. (2014).
Physics laboratory experiments. Cengage Learning.

Appendix 1: Data for Charging Time Constant

Appendix 2: Data for Discharging Time Constant

Charging the RC Circuit to R=100

Computed Vc(V) 10 20 30 40 50 60 12.64 17.29 19 19.63 19.86 19.850000000000001

Discharging the RC element with R=100

Computed V 10 20 30 40 50 60 7.36 2.71 0.99 0.37 0.13 4.9000000000000002E-2

image2

image3

image1

TWU Physics Laboratory |

RC Time Constants

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RC Time Constants

Introduction

In most experiments with electric circuits we

observe what happens at an appreciable time after

a switch has been closed or opened. In this

experiment, observations will be made of the

behavior of the circuit immediately after a switch is

opened or closed.

When a DC power source is connected across a

capacitor the capacitor becomes charged. The

charging is not instantaneous, but follows an

exponential behavior characteristic of the

capacitance of the capacitor. When the switch S in

Figure 1 is closed to complete the circuit, the

charging starts. At any instant of time after the star

t

of the charging process, the voltage which causes current to flow in the circuits is called an active

voltage and it is equal to the source voltage Vs minus the voltage Vc across the capacitor at the

instant. Thus, the active voltage VA = Vs –Vc.

When charging is complete, Vc is equal to Vs and

hence the active voltage is zero. Thus no current

passes through the circuit after the charging process

is over. The circuit is then equivalent to two

batteries of the same output connected opposite to

each other. The behavior of Vc and the current I

passing through the circuit as a function of time is

shown in figure 2.

When a charged capacitor is disconnected from the

battery and then connected across a resistor R, it

becomes discharged. The current flows through R

and discharging follows an exponentially decaying

behavior which is characteristic of R and C. In this

experiment we will study the time behavior of the

charging and discharging of a capacitor in an RC

circuit.

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Theory

Discharging

If a capacitor is connected to a resistor R by closing a switch S, as shown in Figure 3, the positive

and negative charges ±Q stored on the plates are no longer prevented from neutralizing each other.

A current will flow in the resistor until the energy stored in the capacitor is dissipated in the

resistor. Let us assume that at time t = 0 we close the switch. The potential difference across the

capacitor is:

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏: 𝑉𝑐 =
𝑄

𝐶

where V and Q are functions of t. We want to know how fast the capacitor will discharge. Let us

consider the situation some time t after the switch has been closed. We denote by Q (t) the charge

left on the capacitor plates and by I(t) the current through the resistor R. If we consider just the

region enclosed by a surface S as shown in Figure 4, the law of conservation of charge requires

that

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐: −
𝑑𝑄

𝑑𝑡
= 𝐼(𝑡)

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Since the only current anywhere on the surface S is through the resistor, then Ohm’s Law tells us

the potential difference

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑: 𝐼(𝑡)𝑅 = 𝑉𝑐(𝑡)

Substituting the left-hand side of Equation 2 and the right-hand side of Equation 1 into this yields

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒: −
𝑑𝑄

𝑑𝑡
𝑅 =

𝑄(𝑡)

𝐶

Dividing both sides gives the differential equation

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟓: −
𝑑𝑄

𝑑𝑡
=

𝑄(𝑡)

𝑅𝐶

The solution to which is

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟔: 𝑄(𝑡) = 𝑄0𝑒

−𝑡
𝑅𝐶

where Q0 is the charge at t = 0.

This can also be written as

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟕: 𝐼(𝑡) = 𝐼0𝑒
−𝑡
𝑅𝐶

where I0 is the current at t = 0.

Or

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟖: (𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔):

𝑉𝑐(𝑡) = 𝑉0𝑒
−𝑡

𝑅𝐶

where V0 is the voltage across the capacitor at t = 0.

It can be seen from this equation that after time  = R × C the voltage V will drop by a factor of

1/e. This amount of time,  = RC, is called the time constant of the circuit.

Charging

For charging, the time behavior of the voltage Vc across the capacitor is governed by the

following equation:

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟗: (𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝑉𝑐(𝑡) = 𝑉𝑆(1 − 𝑒
−𝑡
𝑅𝐶)

where VS is the source voltage used to charge the capacitor. It is also the limiting value of Vc as

time increases. Equations 8 and 9 for the charging and discharging processes give most of the

information needed to understand the process. One should study them carefully.

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟎: (𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝐼(𝑡) =
𝑉𝑆

𝑅
𝑒

−𝑡
𝑅𝐶

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Pre- Lab

1. A 6 µF capacitor is initially charged to 100 V and then discharged across a 600 ohm resistor.

a. What is the initial charge on the capacitor?

b. What is the time constant of the circuit?

c. How much charge is on the capacitor after 6 ms?

2. A 10 mega ohm resistor is connected in series with a 5 µF capacitor and 12 V battery. The

capacitor is initially uncharged. After a time equal to one time constant find:

a. The charge on the capacitor

b. The current

3. Graph V of a capacitor when it is charging and discharging as a function of time for a RC circuit.

What should this graph look like? No numbers just a qualitative sketch.

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Apparatus

• Regulated DC power supply

• Two switches

• Digital multimeter with 10 MΩ internal resistance

• Electric wires

• Decade Capacitor Box

• Stopwatch- use phones

20 V

20 V

1 µF

1 µF

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Procedure

Part I: charging time constant

1. We will use the circuit in Figure 5 for studying the charging process of a 1 µF capacitor. The

internal resistance of the digital multimeter is 10 mega ohms. This the time constant () of the

circuit is τ = R × C = 10 MΩ × 1µF = 10 s.:

Hence, we want to observe the voltage V across the capacitor at the end of one time constant

(10 seconds), two time constants (20 s), etc. Connect the circuits as shown in Figure 5.

2. Close switch S2. Then close switch S1. Increase the voltage Vs of the power supply until the

multimeter reads 20 V. Now you have set the voltage at 20 V exactly.

3. Open the switch S2 and at the same instant, start your digital clock. When the clock reads

10

seconds, read the voltage from the multimeter. Record the voltage every 10 seconds for

60

seconds. Remember that in this circuit the multimeter reads the active voltage VA.

4. Open S1 and close S2. You have just discharged the capacitor.

5. Close S2 and then S1. Make sure the meter reads 20 V.

6. Repeat steps 2 through 5 for a total of 3 trials. Calculate the average.

t

(s)

Active voltage 𝑉𝐴(V) Experimental

Ave, Vc

(V)

Computed Vc

(V)

Percent error

Trial 1 Trial 2 Trial 3 Avg 20 V – 𝑉𝑎𝑣
𝑉0(1 − 𝑒

−𝑡
𝑅𝐶 )

10

20

30

40

50

60

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Part II: Discharging Time constant

1. Connect the circuit as shown in Figure 6. The theoretical  again is 10 s. You want to first

charge the capacitor and then discharge it through the multimeter (R = 10 mega ohms)

2. Close S2 and then S1. Set Vs to 20 V and wait until the multimeter reads 20 V.

3. Open S1 and at the same instant start the digital clock. The capacitor has started discharging

through the 10 mega ohm resistor. Take readings every 10 s.

4. Repeats steps 2 and 3 for a total of 3 trials. Remember that this time the meter directly reads

the voltage Vc across the capacitor.

t

(s)

Vc (V) Computed Vc Percent error

Trial 1 Trial 2 Trial 3 Avg Vc 𝑉𝑐(𝑡) = 𝑉0𝑒
−𝑡
𝑅𝐶

10

20

30

40

50

60

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Questions

1. A way of discharging a fully charged capacitor is to short the two plates with an electrical

wire. Explain why this is useful way.

2. To graphically represent the value of , based on your data, consider equation 8:

𝑉𝑐(𝑡) = 𝑉0𝑒
−𝑡

𝑅𝐶

Because RC = , this is equivalent to

𝑉𝑐(𝑡) = 𝑉0𝑒
−𝑡

𝜏 .

Taking the natural logarithm of both sides yields

ln(𝑉𝑐) = (−
1

𝜏
) 𝑡 + ln (𝑉0)

(Recall that ln(𝑒𝑥) = 𝑥 and ln(𝑎𝑏) = ln(𝑎) + ln (𝑏)).

Observe that this has the form of a linear equation 𝑦 = 𝑚𝑥 + 𝑏 for which y = ln(Vc), x = t,

the slope m = (−
1

𝜏
), and the intercept b = ln(V0).

Graph the natural log of your measured values Vc from part II versus t and draw a line of best fit

for the plotted points. Determine the slope m of the line you drew and use it to make an estimate

of the time constant by calculating τest = -1/m. Compare this estimated value to the theoretical

value of τ = 10 s.

1

Laboratory Report on RC Time Constants

Student’s Name

Professor’s Name

Course Name

Date

Abstract:

A resistor-capacitor (RC) circuit or RC line or RC organization is an electrical circuit consisting of a resistor and a capacitor. These can be driven very well by voltage or current sources, and they will produce different responses. This experiment seeks to examine how RC circuits can be used to transmit signals by jamming at certain frequencies and across other frequencies. RC circuits can be used to transmit signals by jamming at certain frequencies and across other frequencies. The two most normal RC channels are high-frequency and low-frequency channels; Band channels and band termination channels usually require RLC channels, but unrefined channels can be created with RC channels.

Objectives:

The experiment aimed to focus on the RC circuit and measure the time constant for the main RC circuit.

Background Theory:

When showing the possible electrical contrast V between the two terminals of a capacitor of capacitance C, the two terminals accumulate charges of opposite sign but equal magnitude, q = V/C. In a romantic situation for a fully isolated capacitor, this changes q as soon as V changes. One can charge or discharge the capacitor immediately! Practically speaking, you can never test on a single capacitor. Also, isolated capacitors are not very useful.

Now consider a large RC circuit (Figure 1) consisting of a capacitor with capacitance C and a resistor with resistance R. Due to resistance, the current (I) in the circuit cannot be infinitely large. Thus, the capacitor discharge/charge is not instantaneous (Wilson & Hernández-Hall, 2014). At such a voltage, a larger R results in a simpler I and a longer discharge/charge time in that direction. To change the voltage across the capacitor by the same degree, a larger C causes a larger difference in q and therefore a longer discharge/charge time. Therefore, a larger R or C requires more series time for the RC chain. Such a connetion makes stable weather a controlled and useful frontier. I tried to shorten too long.

Some of the useful equation for RC time constants are:

𝑉𝑐 =𝑄/C

𝐼(𝑡)𝑅 = 𝑉𝑐(𝑡)

𝑄(𝑡)= 𝑄0𝑒^−𝑡/RC

𝐼(𝑡)= 𝐼0𝑒^−t/ RC

(𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝑉𝑐(𝑡)= 𝑉o𝑒^−𝑡/RC

(𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝑉𝑐(𝑡)= 𝑉𝑆(1−𝑒^−𝑡/𝑅𝐶)

(𝑐ℎ𝑎𝑟𝑔𝑖𝑛𝑔): 𝐼(𝑡)= 𝑉𝑆/R e^-t/RC

Experimental Procedure:

1. Set up the RC circuit with R=100 Ω
(Figure 2(a))

Connect the shift lead S1 (charge) S1 to the positive battery lead (approximately 3.0V) charging as DC power supply, and connect the switching lead S2 (discharge) to the battery ground wire. Warning: above 5.00 V, the capacitor dissolves.

Connect the center wire SO of the double throw change to the positive pole of the ammeter, connect the unfavorable pole of the ammeter to the positive pole of the 1.0 F capacitor and connect the negative pole of the capacitor to the open wire of one of the two 100ω resistors (Wilson & Hernández-Hall, 2014). Another resistor wire is connected to S2 and through this wire goes to battery ground.

Finally, connect the voltmeter and capacitor according to the correct polarity. Ask your vet to thoroughly check the circuit.

Charge the capacitor connected to the 100Ω resistor

Set correct ammeter and voltmeter awareness.

One backup looks at the clock and the other at the ammeter and voltmeter and records the readings in Table below. When finished, start the clock at t=0 and reverse the double change from S0 to S1. This starts the loading system.

TABLE 1: Charging capacitor connected to R = 100ω

t (s
)

I (A
)

V

t (s
)

I (A
)

V

(V
)

(V
)

5

25.2

0.29

210

3.0

2.70

30

19.6

0.98

250

2.3

2.78

60

14

1.55

300

1.4

2.87

90

10

1.96

350

0.9

2.92

120

7.2

2.25

400

0.6

2.95

150

5.3

2.46

500

0.3

2.98

180

4.0

2.59

600

.2

3

3. Disconnect the capacitor connected to the 100Ω resistor

Hold for another 2 minutes. Again, one backup will check the clock and the other the voltmeter and record the readings in Table 2. Now that you are ready, start the clock at t=0 and simultaneously reverse the doubled change in S2. This will start the release system.

TABLE 2 Discharge of the capacitor connected to R = 100ω

t (s
)

V

t (s
)

V

(V
)

(V
)

5

210

0.29

30

250

60

300

90

350

120

400

150

500

180

600

2.7

2.06

0.19

1.48

0.12

1.03

0.08

0.74

0.05

0.54

0.02

0.39

0.01

4. Set the RC circuit with R=50 (Figure 2(b))

Change step 1 unless you replace R=100Ω with R=50Ω (by connecting two 100Ω resistors evenly) and remove the ammeter from the circuit. Ask your vet to take a close look at the circuit.

5. Charge the capacitor connected to the 50Ω resistor

One backup will check the clock and the other the voltmeter and record the readings in Table 3. Now that you are ready, start the clock at t=0 and reverse the change in double step S1 all the time. This starts the loading system.

TABLE 3 Charging the capacitor connected to R = 50

t (s
)

V

t (s
)

V

(V
)

(V
)

5

0.98

120

30

2.87

150

60

180

2.95

210

2.98

90

250

3.0

 0.42 105 2.75 15 2.82 1.6 135 45 2.02 2.91 2.33 75 2.52 2.64

Data and Graphs

1. Charging the RC circuit to R = 100

Using Table 1, draw VC-versus-t and install the bend according to equation (5) VC(t) A (1 e Bt) with assembly limits A V0 and B 1 / RC.

Using Table 1, plot I-versus-t and consider bending according to Equation (3′) I(t)

Ae with adjacent boundaries A V0/R and B 1/RC.

Notice assembled V0/R = 26.6/100 ohms = 0.266 01 = RC = 94.46

2. Discharging the RC element with R = 100

Using Table 2, draw a graph of VC – versus t and adjust the deflection according to Equatio; VC(t) Ae Bt with installation limitations A V0 and B 1 / RC.

Note assembly 01 = RC = 90.3

3. Loading RC circuit for R = 50

Using Table 3, draw VC-versus-t and assemble the bend according to Equation (5).

VC(t) A(1 e Bt) with adjacent boundaries A V0 and B 1/RC.

Note assembly 02 = RC = 39.2

Results conclusion

The information obtained is very broad because it is still within normal limits. When the capacitor is charged, the voltage increases dramatically in 600 seconds and rises to the ends. On release the opposite occurs, there is a dramatic reduction from the start and smoothing towards the end. This level indicates that the battery is fully charged or empty. At the point where the resistance is lower, the battery can be charged faster as shown in Table 3. This shows that if we assume that there is a higher R or C, the steady time of the RC circuit will also be greater. Most likely the error in this lab may be due to human error. Result and conclusion is separate

Reference

Wilson, J. D., & Hernández-Hall, C. A. (2014).
Physics laboratory experiments. Cengage Learning.

The graph I got for discharging. I do not understand how you got 4 graphs there should be 2 graph one for charging and one for discharging according to my table

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