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Chapter 21: Problem 31

The voltage across an inductor and the current through the inductor are related by \(v_{\mathrm{L}}=L \Delta i / \Delta t .\) Suppose that $i(t)=I \sin \omega t .\( (a) Write an expression for \)v_{\mathrm{L}}(t) .$ [Hint: Use one of the relationships of Eq. \((20-7) .]\) (b) From your expression for \(v_{\mathrm{L}}(t),\) show that the reactance of the inductor is \(X_{\mathrm{L}}=\omega L_{\mathrm{r}}\) (c) Sketch graphs of \(i(t)\) and \(v_{\mathrm{L}}(t)\) on the same axes. What is the phase difference? Which one leads?

Short Answer

Expert verified
Answer: The phase difference between the voltage across an inductor and the current through it is \(\pi/2\) (90 degrees), and the voltage leads the current.

Step by step solution

01

Differentiate the current expression

We have the current expression as $$i(t) = I \sin \omega t$$ Differentiate the given expression to find the change in current with respect to time, $$\frac{di(t)}{dt} = I \cdot \frac{d}{dt}\sin \omega t$$ Using chain rule, $$\frac{di(t)}{dt} = I \cdot \cos \omega t \cdot \frac{d}{dt}(\omega t)$$ $$\frac{di(t)}{dt} = I \omega \cos \omega t$$
02

Calculate the voltage expression using inductor relation

Using the relation between voltage across an inductor and the current through it, $$v_L(t) = L \frac{\Delta i}{\Delta t}$$ Substitute the change in current with respect to time that we calculated in step 1, $$v_L(t) = L I \omega \cos \omega t$$
03

Find the reactance of the inductor

Reactance of the inductor is given by, $$X_L = \frac{v_L}{i}$$ Substitute the expressions for \(v_L(t)\) and \(i(t)\), $$X_L = \frac{L I \omega \cos \omega t}{I \sin \omega t}$$ Cancel the current, \(I\), from the equation, $$X_L = \omega L \cdot \frac{\cos \omega t}{\sin \omega t}$$ $$X_L = \omega L_{\mathrm{r}}$$
04

Sketch graphs for i(t) and v_L(t)

We have the expressions for \(i(t)\) and \(v_L(t)\), $$i(t) = I \sin \omega t$$ $$v_L(t) = L I \omega \cos \omega t$$ Plot the graphs of these expressions on the same set of axes. You will notice that the voltage graph leads the current graph by a phase angle of \(\pi/2\) (90 degrees).
05

Find the phase difference and which one leads

Compare the forms of the expressions of \(i(t)\) and \(v_{L}(t)\) to note the phase difference, $$i(t) = I \sin \omega t$$ $$v_{L}(t) = L I \omega \cos \omega t$$ The phase difference between the two graphs is \(\pi/2\) (90 degrees), and the voltage \(v_{L}(t)\) leads the current \(i(t)\).

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Most popular questions from this chapter

A 4.00 -mH inductor is connected to an ac voltage source of \(151.0 \mathrm{V}\) rms. If the rms current in the circuit is \(0.820 \mathrm{A},\) what is the frequency of the source?
The FM radio band is broadcast between \(88 \mathrm{MHz}\) and $108 \mathrm{MHz}$. What range of capacitors must be used to tune in these signals if an inductor of \(3.00 \mu \mathrm{H}\) is used?
A 0.48 - \(\mu\) F capacitor is connected in series to a $5.00-\mathrm{k} \Omega\( resistor and an ac source of voltage amplitude \)2.0 \mathrm{V}$ (a) At \(f=120 \mathrm{Hz},\) what are the voltage amplitudes across the capacitor and across the resistor? (b) Do the voltage amplitudes add to give the amplitude of the source voltage (i.e., does \(V_{\mathrm{R}}+V_{\mathrm{C}}=2.0 \mathrm{V}\) )? Explain. (c) Draw a phasor diagram to show the addition of the voltages.
A \(150-\Omega\) resistor is in series with a \(0.75-\mathrm{H}\) inductor in an ac circuit. The rms voltages across the two are the same. (a) What is the frequency? (b) Would each of the rms voltages be half of the rms voltage of the source? If not, what fraction of the source voltage are they? (In other words, \(V_{R} / \ell_{m}=V_{L} / \mathcal{E}_{m}=?\) ) (c) What is the phase angle between the source voltage and the current? Which leads? (d) What is the impedance of the circuit?
A series circuit with a resistor and a capacitor has a time constant of $0.25 \mathrm{ms}\(. The circuit has an impedance of \)350 \Omega$ at a frequency of \(1250 \mathrm{Hz}\). What are the capacitance and the resistance?
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