PHYS121A_ Lab 218

Physics Laboratory Report Lab Number and Title : Lab 223: Faraday’s Law Name : Katie Nguyen Group ID : 4 Date of Experiment : 11/30/2023 Date of Report Submission : 12/6/2023 Course and Section Number : PHYS121A - 011 Instructor’s Name : Keitarou Matsumoto Partner’s Names : Xin Jin, Caden Hendrickson, Woojeong Yoo 1. Introduction 1.1. Objectives - To understand how electromotive force (EMF) works in an inductor and look into what happens with the current in an RL circuit when it starts and stops. 1.2. Theoretical Background The equation describing the self-induced electromotive force (ε) across an inductor is ε L = -L(di/dt), where L represents the inductance of the coil, and di/dt is the rate of change of current. If the switch remains in any position long enough, establishing a constant current, and there is no change in current (di/dt = 0), the induced electromotive force across the inductor is zero. However, when there is a change in current (di/dt ≠ 0), an electromotive force (emf) exists. Kirchhoff's Voltage Law is applicable when the switch is shifted to position A, expressed as V0 - iR - L(di/dt) = 0. In this equation, V0 signifies the battery voltage, R is the loop resistance, and L denotes the coil inductance. The equation can be reformulated as i(t) = (V0/R)(1 - e^(-t/τ)), where τ is the circuit time constant (in seconds), given by L/R. Since εL = -L(di/dt), the equation VL(t) = V0e^(-t(R/L)) can be derived. Taking the natural logarithm of both sides results in ln[1 - (i(t)/i0)] = -(R/L)t, yielding a straight line. Current initiates when the switch moves from B to A, but the inductor's generated electromotive force prevents the current from reaching its maximum instantaneously. The rate of current increase is contingent on the circuit's resistance and inductance, gradually reaching its peak over time from 0. The given equation holds true when t < 0 in position A, depicting a constant current (i = V0/R): εL = -L(di/dt) = 0. At t = 0, with the switch shifted to position B, Kirchhoff’s Law of Voltages is applicable: iR + L(di/dt) = 0. This can be reexpressed as: i(t) = (V0/R)(e^(-t/τ)). Since εL = -L(di/dt) is valid, the ensuing equation can be formulated: VL(t) = V0e^(-t(R/L)). Taking the natural logarithm of both sides results in ln[i(t)/i0] = -(R/L)t, forming a linear relationship. The current ceases its flow when the switch is moved from A to B, but the inductor's generated electromotive force ensures a gradual current flow over time. The reduction in circuit current is contingent on the circuit’s resistance and inductance. 2. Experimental Procedure The instructions in the lab manual were followed with no manipulations to the experimental procedure 3. Results Data Table 1: Inductor Inductance (mH) Resistance (Ω) Current Loop 20.7 6.3 Solenoid 7.2 1.7

Solenoid: Figure showing the max voltage Figure showing the slopes of the graphs which are used to find the time constant

4. Calculation Solenoid: - R total = 10 + 1.7 = 11.7 - τ = L/R total = (7.2 x 10^-3)/11.7 = 0.000615 - Maximum voltage = 5.2589 - (-1.7515) = 7.0104 - τ [ln(i(t)/i 0 ) vs. time] = 1/4340 = 0.000230 - τ [ln(1- i(t)/i 0 ) vs. time] = 1/1300 = 0.000769 - Percent error [ln(i(t)/i 0 ) vs. time] = (|0.000230 - 0.000615|/0.000615) x 100 = 62.6% - Percent error [ln(1- i(t)/i 0 ) vs. time] = (|0.000769 - 0.000615|/0.000615) x 100 = 25.0% Current Loop: - R total = 10 + 6.3 = 17.3 - τ = L/R total = (20.7 x 10^-3)/17.3 = 0.00120 - Maximum voltage = 4.0386 - (-1.3339) = 5.3725 - τ [ln(i(t)/i 0 ) vs. time] = 1/2160 = 0.000463 - τ [ln(1- i(t)/i 0 ) vs. time] = 1/608 = 0.00164 - Percent error [ln(i(t)/i 0 ) vs. time] = (|0.000463 - 0.00120|/0.00120) x 100 = 61.4% - Percent error [ln(1- i(t)/i 0 ) vs. time] = (|0.00164 - 0.00120|/0.00120) x 100 = 36.7% 5. Analysis and Discussion When comparing the time constants in both ln(i(t)/i0) vs. time and ln(1 - i(t)/i0) vs. time graphs, the percentage errors ranged from moderately to high, with the lowest error at 25.0% and the highest at 62.6%. Notably, a higher percentage error was observed in the ln(i(t)/i0) vs. time graphs for both the solenoid and the current loop. This may be due to the slopes for the two graphs varying greatly. Several factors may contribute to the higher-than-desired percentage errors. During the experiment, an anomaly occurred where the voltage vs. time graph displayed values below zero, necessitating the subtraction of the upper and lower curves to determine the maximum voltage. No matter how many times we tried to redo the reading, the result of the graph remained the same and could not be fixed. Any miscalculation in this process becomes a potential source of error for both ln(i(t)/i0) vs. time and ln(1 - i(t)/i0) vs. time graphs, as this value was utilized in generating the graphs. Additionally, potential errors in the computer system and the equipment used, such as the LCR meter and digital multimeter, may have impacted accuracy. Fluctuating numbers during readings required averaging or consideration of the most frequently settled value. Human error, such as improper circuit assembly or loose connections, could also contribute to inaccuracies. 6. Conclusion Throughout this laboratory experiment, we graphically depicted the relationship between ln[1 - (i(t)/i0)] and ln[i(t)/i0] over time. Analyzing the graph allowed us to determine the slope, revealing the negative correlation between R/L. Understanding this correlation enabled us to establish the inverse relationship, leading to the identification of the time constant represented by τ, also known as L/R. Kirchhoff's Law of Voltages played a crucial role in this experiment, as we derived the equation on the vertical axis of the graph from the original equation. Additionally, we observed different equations employed to assess the time constant during current flow and when the current ceases. In essence, this laboratory experience provided insights into constructing and analyzing current in an R. Enhancements to the experiment could have been achieved through conducting multiple trials or employing improved equipment. Additionally, more thorough troubleshooting of the computer system could have contributed to better results. This experiment raised inquiries into the impact of other inductors, beyond the current loop and solenoid, on the behavior of an RL circuit.

7. Raw Data