Physics Past Paper PDF - Dipôle RL

## Exercice n°1: - The electrical circuit in figure 1 consists of a coil with inductance L = 0.6H and resistance r, a resistor with resistance R, and a DC voltage generator with electromotive force E. - The origin of time is taken at the instant when the switch K is closed. - Using a memory oscillo...

## Exercice n°1: - The electrical circuit in figure 1 consists of a coil with inductance L = 0.6H and resistance r, a resistor with resistance R, and a DC voltage generator with electromotive force E. - The origin of time is taken at the instant when the switch K is closed. - Using a memory oscilloscope, the voltage across the generator and u_R(t) across the resistor are visualized. - The curves 1 & 2 of figure 2 are obtained. ### Figure 1: - A schematic of the circuit with a switch, a resistor, a coil, and a DC voltage generator. ### Figure 2: - A graph of voltage (V) against time (ms) with two curves. - One curve labelled ζ1 and the other labelled ζ2. ### Questions: 1. a- Indicate on figure 1 the connections required to use the oscilloscope to visualize the voltage u_R(t) on channel 1 and the generator voltage on channel 2. b- Assign the corresponding curve to the voltage u_R(t) across the resistor. Justify your answer. c- Explain the delay in the establishment of the current in the coil and name the physical phenomenon involved. 2. Determine from the curves: a- the electromotive force (EMF) E of the generator. b- the time constant τ of the electrical circuit. c- the values of the voltages u_R across the resistor and u_L across the coil in steady state. 3. In steady state: a- Show that U_R= R*E*T / (r+R). b- Deduce the values of R and r. 4. In transient mode: a- Show that the differential equation governing the variation of the current i(t) in the circuit is written as: di/dt + i(t)/τ = E/L. b- Verify that i(t)=I_0(1-e^(-t/τ)) is a solution to this differential equation. c- Specify the physical meaning of I_0 and calculate its value. ## Exercice n°2: - It is proposed to determine the exact nature of an electrical dipole D which can be either a coil with inductance L and resistance r, or a capacitor with capacitance C. - The circuit in figure 1 is used for this purpose. - The circuit has a voltage generator providing a voltage E=6V across its terminals, a resistor with resistance R_0=100Ω, the dipole D and a switch, all connected in series. ### Figure 1: - A schematic of the circuit with a voltage generator, a resistor, a switch, and a dipole. ### Figure 2: - A graph of voltage (V) against time (ms). ### Questions: 1. When the switch K is closed, the voltage U_BA across the resistor is visualized on the oscilloscope. The resulting chronogram is represented in figure 2. a. Reproduce the schematic of figure 1 and represent the connections needed to use an oscilloscope. b. Show that the dipole D is a coil and explain the delay in reaching steady state in the circuit. 2. a. Applying Kirchhoff's voltage law on the circuit, show that the voltage U_BA across the resistor satisfies the following differential equation: dU_BA/dt + U_BA/τ = E/(R_0+r) with τ = L/(R_0+r). b. Knowing that U_BA(t) = E*R_0/(R_0+r)*(1 - e^(-t/τ )), determine graphically the value of τ. c. Determine the values of r and L. ## Exercice n°3: - A coil with inductance L and internal resistance r=10Ω, a voltage generator with EMF E and a resistor with resistance R_0 are connected in series, along with a switch K (figure 1). - In order to record the voltages U_AM(t) and U_MB(t), the Y1 and Y2 inputs of an oscilloscope are connected to points A and B of the circuit, respectively, with the Y2 input in inverted mode. - At time t=0, switch K is closed. - The oscilloscope simultaneously records the curves (C1) and (C2) shown in figure 2. ### Figure 1: - A schematic of the circuit with a voltage generator, a resistor, a switch, and a coil. ### Figure 2: - A graph of voltage (V) against time (ms), with two curves labelled (C1) and (C2). ### Questions: 1. Justify why the Y2 input of the oscilloscope is inverted. 2. Show that the current i flowing in the circuit is governed by the differential equation: di/dt + i/τ = E/R with τ = L/R and R=R_0+r. 3. a. Verify that i(t) = I_p(1-e^(-t/τ)) is a solution to the differential equation, where I_p is a constant that needs to be determined. b. Deduce the expression of each of the voltages U_AM(t) and U_MB(t). c. Identify the curve (C1) or (C2) representing U_MB(t). 4. Using the curves (C1) and (C2) of figure 2, determine the values of: - The EMF E of the generator. - The current I_0 flowing in the circuit in steady state. - The resistance R_0 of the resistor. - The time constant τ and deduce the value of the inductance L. 5. In the previous circuit, one of the parameters L or R_0 is modified. - The recording of the voltage U_MB is represented by the curve (C3). - Identify the quantity whose value has been changed and compare the new value with the original value. ## Exercice n°4: - An electrical circuit AM is built with a resistor of resistance R=50Ω, a coil (B1) with inductance L and negligible resistance, and a switch K (figure 1). - The circuit AM is powered by a voltage generator with EMF E (figure 1 ). - A suitable acquisition system is used to monitor the voltage U_AM(t) and U_DM(t) over time. - The time-varying curves of U_AM(t) and U_DM(t) are shown in figure 2. ### Figure 1: - A schematic of the circuit with a switch, a resistor, and a coil. ### Figure 2: - A graph of voltage (V) against time (ms), with two curves labelled "Courbe 1" and "Courbe 2". ### Questions: 1. a. Show that curve 1 corresponds to U_DM(t). b. Give the value of the EMF E of the generator. 2. a. At time t1=10ms, determine graphically the value of the voltage U_B1 across the coil (B1) and deduce the value of the voltage U_R across the resistor. b. At time t2=90ms, show that the current flowing in the circuit is I_0=0.12A. 3. a. Determine graphically the value of the time constant τ of the RL circuit. b. Knowing that τ = L/R, determine the inductance L of the coil (B1). c. Calculate the energy stored in the coil (B1) in steady state. 4. The coil (B1) is replaced by a coil (B2) with the same inductance L but with non-zero resistance r. - The time-varying curves of U_AM(t) and U_DM(t) correspond to figure 3. ### Figure 3: - A graph of voltage (V) against time (ms). ### Questions: a. Show that in steady state, the voltage across the coil (B2) is: U_B2 = r*E/(r+R). b. Deduce the value of the resistance r of the coil. ## Exercice n°5: - In a laboratory experiment, it is proposed to experimentally determine the values of the resistance r and the inductance L of a coil. - Two experiments are carried out. ### Experiment n°1: Determination of the resistance r of the coil: - The coil is powered by a DC voltage generator. - Two multimeters are used to measure the current flowing through the coil and the voltage across its terminals in steady state. - Measurements give 250 mA and 3.5V. - Give the expression of the instantaneous voltage U_AB(t) across the coil when it carries a current i(t). - What does this expression become when steady state is reached? - Deduce the value of the resistance r of the coil. ### Experiment n°2: Determination of the inductance L of the coil: - The circuit in figure 1 is built. - The circuit consists of a coil (L, r), a resistor with resistance R=262Ω, and a DC voltage generator (figure 1). - An appropriate system is used to record the time variation of the current i(t) flowing through the circuit. - The origin of time is taken at the instant when the switch K is closed. - The recorded curve is shown in figure 2. ### Figure 1: - A schematic of the circuit with a voltage generator, a resistor, a switch, and a coil. ### Figure 2: - A graph of current (A) against time (ms). ### Questions: 1. Establish the differential equation governing the variation of i(t). 2. On the curve of figure 2, indicate the regions corresponding to transient and steady state regimes. 3. a. Determine the value of the time constant of the circuit. b. Verify that L=20mH. 4. Determine the value of E.