Question

The ac generator in Fig. $31-39$ supplies 120 $\mathrm{V}$ at 60.0 $\mathrm{Hz}$ . With the switch open as in the diagram, the current leads the generator emf by $20.0^{\circ} .$ With the switch in position $1,$ the current lags the generator emf by $10.0^{\circ} .$ When the switch is in position $2,$ the current amplitude is 2.00 A. What are (a) $R,$ (b) $L,$ and $(\mathrm{c}) C ?$

   The ac generator in Fig. $31-39$ supplies 120 $\mathrm{V}$ at 60.0 $\mathrm{Hz}$ . With the switch open as in the diagram, the current leads the generator emf by $20.0^{\circ} .$ With the switch in position $1,$ the current lags the generator emf by $10.0^{\circ} .$ When the switch is in position $2,$ the current amplitude is 2.00 A. What are (a) $R,$ (b) $L,$ and $(\mathrm{c}) C ?$
 
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Fundamentals of Physics
Fundamentals of Physics
David Halliday,… 10th Edition
Chapter 31, Problem 87 ↓
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The ac generator in Fig. $31-39$ supplies 120 $\mathrm{V}$ at 60.0 $\mathrm{Hz}$ . With the switch open as in the diagram, the current leads the generator emf by $20.0^{\circ} .$ With the switch in position $1,$ the current lags the generator emf by $10.0^{\circ} .$ When the switch is in position $2,$ the current amplitude is 2.00 A. What are (a) $R,$ (b) $L,$ and $(\mathrm{c}) C ?$
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Key Concepts

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RLC Circuit Analysis
RLC circuit analysis involves examining circuits that contain resistors, inductors, and capacitors. The interplay between resistive and reactive components determines the overall impedance and the phase relationship between voltage and current. Analyzing these circuits often requires setting up equations based on the known relationships for resistance, inductive reactance, and capacitive reactance, which is critical in solving for unknown values within the circuit.
Capacitive Reactance
Capacitive reactance is the opposition provided by capacitors to changes in voltage within AC circuits. It is also frequency dependent and is given by XC = 1/(2?fC), with f being the frequency and C the capacitance. Unlike inductive reactance, capacitive reactance causes the current to lead the voltage, a behavior that is essential when working with circuits containing capacitors.
Inductive Reactance
Inductive reactance arises from the presence of inductors in an AC circuit and opposes changes in current. It is frequency dependent and calculated using the formula XL = 2?fL, where f is the frequency and L is the inductance. This reactance causes the current to lag behind the voltage, a principle that is crucial in designing and analyzing circuits with inductors.
Phasor Representation
Phasors are a tool used to represent sinusoidal functions as rotating vectors in the complex plane. They simplify the analysis of AC circuits by converting differential equations into algebraic ones, making it easier to handle phase differences between voltage and current. This representation is fundamental in solving problems that involve phase angles and reactive components.
AC Circuit Analysis
This concept involves understanding circuits that are powered by sinusoidal sources. In these circuits, voltages and currents vary with time, typically described by their amplitude, frequency, and phase. Analyzing AC circuits requires the use of phasor representations, which simplify calculations involving sinusoidally varying quantities by converting them into complex numbers.
Impedance and Phase Angle
Impedance is the total opposition a circuit presents to AC current, combining both resistance (the real part) and reactance (the imaginary part: inductive and capacitive). The phase angle indicates the phase difference between the voltage and current, which is determined by the relative magnitudes of the resistive and reactive components. Understanding this relationship is key to analyzing how current leads or lags the voltage in AC circuits.

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Transcript

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00:01 For this problem on the topic of alternating current, we are shown an rlc circuit in which the ac generator is supplying 120 volts at 60 hz.
00:12 With the switch open, the current leaves the generator emf by 20 degrees, and with the switch in position 1, the current lags the generator emf by 10 degrees.
00:23 When the switch is in position 2, we are told that the current amplitude is 2 ampires and are asked to find the resistance r, the inductors, the inductors.
00:31 L and capacitance c.
00:34 Now when the switch is open, we have a series lrc circuit involving just the one capacitor near the upper right corner.
00:42 And so we have omega d times l minus 1 over omega d times c, all divided by r is equal to the tan of phi n, which is the tan of minus 20 degrees, which is minus 10 20 degrees.
01:12 Now when the switch is in position 1, the equivalent capacitance in the circuit is 2c.
01:16 And so in this case, we have omega -d times l minus 1 over 2 omega -d times c, all divided by r is equal to the tan of 5 -1, which is the 10.
01:40 Of 10 degrees.
01:43 And finally with the switch in position 2 the circuit is simply an lc circuit with the current amplitude i2 equal to the emf epsilon m over the inductance zlc not inductance but impedance at lc and this is equal to epsilon m divided by the square root of omega d times l minus 1 over omega d times c which is that's all squared so this is epsilon m divided by 1 over omega d times c minus omega d times l now we use the fact here that 1 over omega d times c is greater than omega d times l in simplifying the square root...
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