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Chapter 20: Problem 80

The alternator in an automobile generates an emf of amplitude $12.6 \mathrm{V}\( when the engine idles at \)1200 \mathrm{rpm}$. What is the amplitude of the emf when the car is being driven on the highway with the engine at 2800 rpm?

Short Answer

Expert verified
Answer: The amplitude of the emf generated at 2800 rpm is approximately 29.4 V.

Step by step solution

01

Understand the information given

We are given the initial emf amplitude \(E_{1} = 12.6 V\), the initial engine speed \(n_{1} = 1200 rpm\), and the final engine speed \(n_{2} = 2800 rpm\). We need to find the final emf amplitude \(E_{2}\).
02

Convert engine speeds to angular velocities

Angular velocity (ω) can be found by multiplying the engine speed (n) by the conversion factor (2π/60), where 2π represents a full circle, and 60 converts minutes to seconds: \(\omega = \frac{2\pi}{60}n\) Calculate the angular velocities ω₁ and ω₂: \(\omega_{1} = \frac{2\pi}{60}n_{1}\) \(\omega_{2} = \frac{2\pi}{60}n_{2}\)
03

Use the proportionality relationship to solve for the final emf amplitude

Since the emf generated by the alternator is directly proportional to its angular velocity, we can write: \(\frac{E_{1}}{\omega_{1}} = \frac{E_{2}}{\omega_{2}}\) Now, solve for \(E_{2}\): \(E_{2} = E_{1} \cdot \frac{\omega_{2}}{\omega_{1}}\)
04

Calculate the final emf amplitude

Now, plug in the given values and the calculated angular velocities to find the final emf amplitude: \(E_{2} = 12.6 V \cdot \frac{\frac{2\pi}{60} \cdot 2800}{\frac{2\pi}{60} \cdot 1200}\) Simplify and solve for \(E_{2}\): \(E_{2} = 12.6 V \cdot \frac{2800}{1200}\) \(E_{2} \approx 29.4 V\) The amplitude of the emf when the car is being driven on the highway with the engine at 2800 rpm is approximately 29.4 V.

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Most popular questions from this chapter

(a) For a particle moving in simple harmonic motion, the position can be written \(x(t)=x_{\mathrm{m}}\) cos \(\omega t .\) What is the velocity \(v_{x}(t)\) as a function of time for this particle? (b) Using the small-angle approximation for the sine function, find the slope of the graph of \(\Phi(t)=\Phi_{0} \sin \omega t\) at \(t=0 .\) Does your result agree with the value of \(\Delta \Phi / \Delta t=\omega \Phi_{0} \cos \omega t\) at \(t=0 ?\)
A transformer with a primary coil of 1000 turns is used to step up the standard \(170-\mathrm{V}\) amplitude line voltage to a \(220-\mathrm{V}\) amplitude. How many turns are required in the secondary coil?
The outside of an ideal solenoid $\left(N_{1} \text { turns, length } L\right.\( radius \)r)\( is wound with a coil of wire with \)N_{2}$ turns. (a) What is the mutual inductance? (b) If the current in the solenoid is changing at a rate \(\Delta I_{1} / \Delta t,\) what is the magnitude of the induced emf in the coil?
The time constant \(\tau\) for an \(L R\) circuit must be some combination of $L, R,\( and \)\mathscr{E} .$ (a) Write the units of each of these three quantities in terms of \(\mathrm{V}, \mathrm{A},\) and \(\mathrm{s}\) (b) Show that the only combination that has units of seconds is \(L / R\)
Suppose you wanted to use Earth's magnetic field to make an ac generator at a location where the magnitude of the field is \(0.50 \mathrm{mT}\). Your coil has 1000.0 turns and a radius of \(5.0 \mathrm{cm} .\) At what angular velocity would you have to rotate it in order to generate an emf of amplitude $1.0 \mathrm{V} ?$
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