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Chapter 20: Problem 8

The armature of an ac generator is a rectangular coil \(2.0 \mathrm{cm}\) by \(6.0 \mathrm{cm}\) with 80 turns. It is immersed in a uniform magnetic field of magnitude 0.45 T. If the amplitude of the emf in the coil is $17.0 \mathrm{V},$ at what angular speed is the armature rotating?

Short Answer

Expert verified
Answer: The angular speed of the rectangular coil is approximately \(393.52\ \mathrm{rad/s}\).

Step by step solution

01

Write down the formula

We use the formula for the amplitude of the emf in a rotating coil: \(emf = NBA\omega\sin{\omega t}\). Since we are given the amplitude of the emf, we can set \(emf = NBA\omega\).
02

Substitute the given values

We are given the following values: \(emf = 17.0\ \mathrm{V}\) \(N = 80\) turns \(B = 0.45\ \mathrm{T}\) \(A = 2.0\,\mathrm{cm} \times 6.0\,\mathrm{cm} = 12.0\,\mathrm{cm}^2 = 0.0012\,\mathrm{m}^2\) Now we can plug these values into the equation: \(17.0\ \mathrm{V} = (80)(0.45 \mathrm{T})(0.0012\ \mathrm{m}^2)\omega\)
03

Solve for the angular speed \(\omega\)

Now we solve for \(\omega\): $$ \omega = \frac{17.0\ \mathrm{V}}{(80)(0.45 \mathrm{T})(0.0012\ \mathrm{m}^2)} = \frac{17.0}{0.0432} \mathrm{rad/s} $$ $$ \omega \approx 393.52\ \mathrm{rad/s} $$
04

Write the final answer

The angular speed at which the armature rotating is approximately \(393.52\ \mathrm{rad/s}\).

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Most popular questions from this chapter

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 $1.0 \mathrm{V} ?$
An ideal solenoid ( \(N_{1}\) turns, length \(L_{1},\) radius \(r_{1}\) ) is placed inside another ideal solenoid ( \(N_{2}\) turns, length \(L_{2}>L_{1}\), radius \(r_{2}>r_{1}\) ) such that the axes of the two coincide. (a) What is the mutual inductance? (b) If the current in the outer solenoid is changing at a rate \(\Delta I_{2} / \Delta t,\) what is the magnitude of the induced emf in the inner solenoid?
When the emf for the primary of a transformer is of amplitude $5.00 \mathrm{V},\( the secondary emf is \)10.0 \mathrm{V}$ in amplitude. What is the transformer turns ratio \(\left(N_{2} / N_{1}\right) ?\)
Calculate the equivalent inductance \(L_{\mathrm{eq}}\) of two ideal inductors, \(L_{1}\) and \(L_{2},\) connected in parallel in a circuit. Assume that their mutual inductance is negligible. [Hint: Imagine replacing the two inductors with a single equivalent inductor \(L_{\mathrm{eq}} .\) How is the emf in the parallel equivalent related to the emfs in the two inductors? What about the currents? \(]\)
The component of the external magnetic field along the central axis of a 50 -turn coil of radius \(5.0 \mathrm{cm}\) increases from 0 to 1.8 T in 3.6 s. (a) If the resistance of the coil is \(2.8 \Omega,\) what is the magnitude of the induced current in the coil? (b) What is the direction of the current if the axial component of the field points away from the viewer?
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