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B4 Lab Report

I.

Data A. Table 1: Measured Across

1.

Resistor (𝝮) Predicted 𝜏c (s) Measured 𝜏cc (s)

𝛥V (V)

T 1/2 (s)

Resistor

15,000

0.0034

0.0038 +/- 2.9E-4

7.2

0.0026

Capacitor (0.22 µF)

15,000

0.0034

0.0035 +/- 2.9E-4

7.38

0.0024

Capacitor (0.22 µF)

1500

0.00046

0.00049 +/- 5.4E-5

7.46

3.37E-04

Error:

(5% Error)

𝛿𝜏c = 10%

See above

𝛿V= +/- 156 mV

𝛿T=+/- 200 µs

Table 1 Legend: Summary of the results of part I excluding derived quantities, where the voltage was monitored across either the capacitor or resistor in series to determine the capacitive time constant. All of the predicted and measured time constants agree within error. See sketches of oscilloscopes in the in-lab photos.

B. Table 2: Measured Across Resistor (𝝮)

1.

II.

Predicted 𝜏L (s)

Measured 𝜏L (s)

Delta V (V)

T 1/2 (s)

Resistor

1,500

4.37E-05 +/- 4E-6

2.89E-05 +/- 2E-6

7.52

2E-5s

Inductor

1,500

4.37E-05

NA

NA

NA

Table 2 Legend: Summary of the results of part II, excluding derived quantities, where the voltage was monitored across the resistor in series with an inductor to calculate the inductive time constant. The plot of voltage across the inductor was not used as it is not an accurate depiction of the change in voltage vs time as the inductor is resistant to changes in current, and voltage The predicted and measured inductive time constants do not agree within experimental error. See sketches of oscilloscopes in the in-lab photos.

Figures A. Figure 1:

1. 2.

Figure 1 Legend: Image of the Oscilloscope From Part I- RC Circuits, Step 2 where the voltage was measured as a function of time across the larger of the two resistors (15,000 𝝮), the plot is essentially a visualization of the RC circuit discharging equation, i(t) = -𝜖/R x e-t/Tc. From this plot the t(½) and later the capacitive time constant was determined.

B. Figure 2:

1. 2.

Figure 2 Legend: Image of the Oscilloscope From Part I- RC Circuits, Step 3 where the voltage was measured as a function of time across the larger of the capacitor (0.22 µF), the plot is essentially a visualization of the RC circuit charging/discharging equation, q(t) = q0 x e-t/Tc. From this plot the t(½) and later the capacitive time constant was determined. C. Figure 3:

1. 2.

Figure 3 Legend: Image of the Oscilloscope From Part I- RC Circuits, Step 3 where the voltage was measured as a function of time across the smaller of the two resistors (1500 𝝮), the plot is essentially a visualization of the RC circuit charing/discharging equation, i(t) = -𝜖/R x e-t/Tc. From this plot the t(½) and later the capacitive time constant was determined. D. Figure 4:

1. 2.

III.

Figure 4 Legend: Image of the Oscilloscope From Part II- RL Circuits, Step 5 where the voltage was measured as a function of time across the larger of the two resistors (15,000 𝝮), the plot is essentially a visualization of the RL circuit charging/discharging equation, i(t) = -𝜖/R x (1- e-t/TL). From this plot the t(½) and later the capacitive time constant was determined.

Calculations A. Part I: RC Circuits 1. Calculation of the Predicted Capacitor Time Constant a) For Circuit Containing 15000 Ω Resistor (1) 𝜏C = RC = (15000 Ω + 600 Ω) (0.22µF) = 0.0034 s b) For Circuit Containing 1500 Ω Resistor (1) 𝜏C = RC = (1500 Ω + 600 Ω) (0.22µF) = 0.00046 s 2. Calculation of the Measured Capacitor Time Constant a) For Measurement Across 15000 Ω Resistor (1) 𝜏C = t1/2 / ln(2) = 0.0026 / ln(2) = 0.0038 s b) For Measurement Across Capacitor with 15000 Ω Resistor (1) 𝜏C = t1/2 / ln(2) = 0.0024 / ln(2) = 0.00346 s c) For Measurement Across Capacitor with 1500 Ω Resistor (1) 𝜏C = t1/2 / ln(2) = 336.75E-6 / ln(2) = 0.00049 s 3. Measured Maximum and Minimum Values of Current a) Maximum: (1) As imax = 𝜀 / R before exponentially decreasing to zero in an RC circuit the maximum current flowing through the circuit can be determined via the equivalent resistance of the circuit and the measured change in voltage during discharging. (2) As the potential difference was measured to be 7.2 V, the maximum current imax is: (3) imax = 7.2 V / (15,600 Ω) = 4.62 x 10-4 amperes b) Minimum: (1) As the current exponentially decreases to zero in an RC circuit, the minimum current is 0 Amperes (which occurs after the capacitor becomes fully charged or in the case of discharging after the capacitor has become completely discharged.

4.

Calculation of Maximum Charge on The Capacitor a) Through the relation of capacitance with charge and potential, the maximum charge on the capacitor can be calculated b) Q = CV where C is the capacitance and V is the maximum potential difference (1) Q = 0.22µF x 7.46 V = 1.64 x 10-6 C B. Part II: RL Circuits 1. Calculation of Predicted Inductive Time Constant a) 𝜏L = L/R = 100 mH / (1500 Ω + 190 Ω + 600 Ω)= 4.37 x10-5 s 2. Calculation of Measured Inductive Time Constant a) As T(½) was measured to be 2x10-4 seconds and 𝜏L = t(½) / ln(2) b) 𝜏L = (2x10-5s ) / ln(2) = 2.89E-5 s 3. Minimum and Maximum Currents in Circuit:

a) Maximum Current: (1) imax = 𝜀/R → imax = 7.52 / 1690 Ohms = 0.0044 Amperes b) Minimum Current: (1) imin = 0 A (a) The current is zero at the start of the rise phase (b) The current decreases to zero during the decay phase IV.

Error Analysis A. Part I: RC Circuits 1. Error Associated With Calculation of the Predicted Capacitor Time Constant a) As the procedure stated to assume that the error associated with capacitance and resistance were 5% the error associated with the predicted capacitor time constant can be determined via the product rule, where 𝛿𝜏C = 𝜏C (𝛿R/R + 𝛿C/C) b) For Circuit Containing 15000 Ω Resistor (1) 𝛿𝜏C = 0.0034 s * [(780/15600 Ω) + (0.011/ 0.22µF)] = (a) 𝛿𝜏C = +/- 0.00034 or 10% c) For Circuit Containing 1500 Ω Resistor (1) 𝛿𝜏C= 0.00046 +/- 0.000046 or 10% 2. Error Associated With Calculation of the Measured Capacitor Time Constant a) This was calculated for each scenario through the error associated with the t(½) measurements. As the time constant is calculated using the measured value the error associated with it can be measured via: 𝛿𝜏C = 𝜏C (𝛿t/ t) b) For Measurement Across 15000 Ω Resistor (1) 𝛿𝜏C = 0.0038 x (200E-6 s / 0.0026 s) = +/- 2.9E-4 s c) For Measurement Across Capacitor with 15000 Ω Resistor (1) 𝛿𝜏C = 0.0035 x (200E-6 s / 0.0024 s) = +/- 2.9E-4 s d) For Measurement Across Capacitor with 1500 Ω Resistor (1) 𝛿𝜏C = 0.00049 x (3.74E-5/ 336.75E-6) = +/- 5.4E-5 s 3. Error Associated With Maximum Value of Current a) As the maximum current was determined via the equation: imax = 𝜀 / R and the error associated with the potential difference and resistance are known the error associated with the maximum current can be determined via the product rule. (1) 𝛿imax = imax (𝛿V/V + 𝛿R/R) (2) 𝛿imax = 4.62E-4 (156E-3/7.2 + 780/15600) = +/- 3.3 x 10-5 amperes 4. Error Associated With Calculation of Maximum Charge on The Capacitor a) As the procedure stated to assume that the error associated with capacitance was 5%, and the error associated with the measurement of voltage was experimentally determined, the error associated with the maximum charge on

the capacitor can be determined via the product rule, where 𝛿Q = Q (𝛿V/V + 𝛿C/C) b) 𝛿Q = 1.64E-6 (156E-3/7.2 + 0.011/0.22) = +/- 1.2 x 10-7 C B. Part II: RL Circuits 1. Error Associated with Calculation of Predicted Inductive Time Constant a) The error associated with 𝜏L can be determined via the product rule. b) 𝜏L = L/R → 𝛿𝜏L = 𝜏L (𝛿L/L + 𝛿R/R) c) 𝛿𝜏L = 4.37 x10-5 (0.005/0.1 + 114.5/2290)= +/- 4.37 x 10-6 s 2. Error Associated with Calculation of Measured Inductive Time Constant a) 𝛿𝜏L = 𝜏L (𝛿t/ t) → 𝛿𝜏L = (2.89x10-5s ) x (1.538E-6 / 2E-5) = +/- 2.22E-6 s

3. Error Associated Maximum Current a) 𝛿imax = i *[(𝛿𝜀/𝜀) + (𝛿R/R)] b) 𝛿imax = 0.0044 A x (0.156/7.52V + 0.05) = +/- 3.11E-4 A V. 1.

2.

3.

4.

Questions Did your measured values of time constants experimental error with the predictions? What are the possible sources of systematic errors in this experiment? a. For part I RC circuits, the measured and predicted experimental time constants agreed with one another for all cases, however, for part II, RL circuits, the measured and predicted time constants did not agree with one another. The error is likely due to the steepness of the graph on the oscilloscope where it was extremely difficult to determine an accurate value of t(½). In your first RC circuit, what was the maximum charge on the capacitor? How do the maximum voltage across the resistor and, separately, across the capacitor, compare with the output of the function generator? Explain a. For the RC circuit with a resistor of 15600 ohms and a capacitor of 0.22 microF the maximum charge on the capacitor was 1.64 x 10-6 C via the equation Qmax = VC. The maximum voltage across the resistor is given by the equation V= imaxR and for the capacitor is V = Qmax/R. i. Resistor: V= 4.62 x 10-4 * 15,000 = 6.93 V ii. Capacitor: V = 1.64 x 10-6 C / 0.22E-6 F = 7.45 V In the RL circuit, what was the maximum rate at which the current changed? Did the voltage drop through the resistor plus the voltage drop through the inductor always equal the voltage applied by the function generator? a. The inductor equation → V= L x di/dt where the potential difference is 7.52 V and the Inductance is L = 0.1 H, therefore the maximum rate at which current changes is equal to V / L b. di/dt = V / L = 7.52 V / 0.1 H = 75.2 Amperes per second c. As Kirchoff’s Loop Rule supplies the equation Vt - (VR+VL) = 0 the voltage drop across the resistor and the inductor should be equal the the total voltage supplied by the function generator. Additionally, the voltage drop across the resistor is equal to iR whereas the voltage drop across the inductor is L*di/dt, therefore KLR can be re-written as Vt = iR + Ldi/dt. Comment on energy in one of your RC and the RL circuits. What was the maximum power dissipated by the resistor? When the resistor was not dissipating energy at the maximum rate, where was the rest of the energy going? a. For the RC circuit, the maximum power dissipated by the resistor is given by the equation P = i2R and imax = 𝜀 / R → for the measurement across the 15,000 ohm resistor. b. P = (𝜀 / R)2x R → P = (7.2 / 15000)2 x 15000 = 0.00346 Watts c. When the resistor was not dissipating energy at the maximum rate, the remaining energy was being delivered to the capacitor resulting in the buildup of charge on the capacitor.

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