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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 \(0.400-\mu \mathrm{F}\) capacitor is connected across the terminals of a variable frequency oscillator. (a) What is the frequency when the reactance is \(6.63 \mathrm{k} \Omega ?\) (b) Find the reactance for half of that same frequency.
Finola has a circuit with a \(4.00-\mathrm{k} \Omega\) resistor, a \(0.750-\mathrm{H}\) inductor, and a capacitor of unknown value connected in series to a \(440.0-\mathrm{Hz}\) ac source. With an oscilloscope, she measures the phase angle to be \(25.0^{\circ} .\) (a) What is the value of the unknown capacitor? (b) Finola has several capacitors on hand and would like to use one to tune the circuit to maximum power. Should she connect a second capacitor in parallel across the first capacitor or in series in the circuit? Explain. (c) What value capacitor does she need for maximum power?
Transformers are often rated in terms of kilovolt-amps. A pole on a residential street has a transformer rated at $35 \mathrm{kV} \cdot \mathrm{A}$ to serve four homes on the strect. (a) If each home has a fuse that limits the incoming current to \(60 \mathrm{A}\) rms at \(220 \mathrm{V}\) rms, find the maximum load in \(\mathrm{kV} \cdot \mathrm{A}\) on the transformer. (b) Is the rating of the transformer adequate? (c) Explain why the transformer rating is given in kV.A rather than in kW.
A series \(R L C\) circuit has a \(0.20-\mathrm{mF}\) capacitor, a \(13-\mathrm{mH}\) inductor, and a \(10.0-\Omega\) resistor, and is connected to an ac source with amplitude \(9.0 \mathrm{V}\) and frequency \(60 \mathrm{Hz}\) (a) Calculate the voltage amplitudes \(V_{L}, V_{C}, V_{R},\) and the phase angle. (b) Draw the phasor diagram for the voltages of this circuit.
A capacitor (capacitance \(=C\) ) is connected to an ac power supply with peak voltage \(V\) and angular frequency \(\alpha\) (a) During a quarter cycle when the capacitor goes from being uncharged to fully charged, what is the average current (in terms of \(C . V\), and \(\omega\) )? [Hint: \(\left.i_{\mathrm{av}}=\Delta Q / \Delta t\right]\) (b) What is the rms current? (c) Explain why the average and rms currents are not the same.
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