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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 (蠅) can be found by multiplying the engine speed (n) by the conversion factor (2蟺/60), where 2蟺 represents a full circle, and 60 converts minutes to seconds: \(\omega = \frac{2\pi}{60}n\) Calculate the angular velocities 蠅鈧� and 蠅鈧�: \(\omega_{1} = \frac{2\pi}{60}n_{1}\) \(\omega_{2} = \frac{2\pi}{60}n_{2}\)
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 coil has an inductance of \(0.15 \mathrm{H}\) and a resistance of 33 \(\Omega\). The coil is connected to a 6.0 - \(V\) battery. After a long time elapses, the current in the coil is no longer changing. (a) What is the current in the coil? (b) What is the energy stored in the coil? (c) What is the rate of energy dissipation in the coil? (d) What is the induced emf in the coil?
An ideal inductor of inductance \(L\) is connected to an ac power supply, which provides an emf \(\mathscr{E}(t)=\mathscr{E}_{\mathrm{m}}\) sin \(\omega t\) (a) Write an expression for the current in the inductor as a function of time. [Hint: See Eq. \((20-7) .]\) (b) What is the ratio of the maximum emf to the maximum current? This ratio is called the reactance. (c) Do the maximum emf and maximum current occur at the same time? If not, how much time separates them?
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The primary coil of a transformer has 250 turns; the secondary coil has 1000 turns. An alternating current is sent through the primary coil. The emf in the primary is of amplitude \(16 \mathrm{V}\). What is the emf amplitude in the secondary? (tutorial: transformer)
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