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Johns Hopkins University

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Problem 36

Generator for space station Astronauts on a space station decide to use Earth's magnetic field to generate electric current. Earth's $\vec{B}$ field in this region has the magnitude of $3.0 \times 10^{-7} \mathrm{T}$ . They have a coil that rotates $90^{\circ}$ in 1.2 $\mathrm{s}$ . The area inside the coil measures 5000 $\mathrm{m}^{2}$ . Estimate the number of loops needed in the coil so that during that $90^{\circ}$ turn it produces an average induced emf of about 120 V. Indicate any assumptions you

made. Is this a feasible way to produce electric energy?

Answer

96000

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## Discussion

## Video Transcript

in this question, we have been given a very interesting scenario where astronauts and space are trying to use induction to produce electricity. So in this case we are given the Earth's magnetic field in that region to be three times 10 to the negative seventh test. Sula and the astronauts are equipped with a coil that rotates 90 degrees in 1.2 seconds. So that tells me the change in angle is 90 degrees overall. And how much time did that take That took 1.2 seconds. They also tell us the area inside the coil measures 5000 meters squared, so area is 5000 meters squared. Now, that is a pretty large oil. Um, and here we're going to estimate the number of loops needed in the coils so that during the 90 degree turn, it produces an average induced E m f about 120 volts. So they do tell us about did you tell us the target? Yemma And they are asking for us for the number of turns. Um, so this we are here. We are going to use Faraday's law of induction again, and that is equal to end times to change in magnetic flux over time. So in this case, we know that, um, Ennis, what we're looking for and we do have be we do have a what is changing is the angle. So remember, if I is equal to be a co sign of the angle So for the purpose of simplicity, I am just going to, um, say that the final angle of zero and the initial angle is 90 degrees. Either way, the changes 90 degrees, and that's what they're asking for right all over the changing time. And to simplify this further. I know co sign of zero is one, and the coastline of 90 is zero, So that will make our calculation much simpler. We have The MF is equal to n B A divided by Delta t So now we just have to solve for n and so and we'll equal to a e m f times change in time over B times A. So let's go ahead and plug in everything. We have 1 20 bolts times the changing time, which is 1.8 seconds divided by B, which is three times 10 to the negative. Seventh, uh, times a, which is five 1000 and plugging all of that into the calculator. We have how many turns 96,000 turns, and that's how you saw for the unknown in this question.

## Recommended Questions

As a new electrical engineer for the local power company, you are assigned the project of designing a generator of sinusoidal ac voltage with a maximum voltage of 120 $\mathrm{V}$ . Besides plenty of wire, you have two strong magnets that can produce a constant uniform magnetic field of 1.5 T over a square area of 10.0 $\mathrm{cm}$ on a side when they are 12.0 $\mathrm{cm}$ apart. The basic design should consist of a square coil turning in the uniform magnetic field. To have an acceptable coil resistance, the coil can have at most 400 loops. What is the minimum rotation rate (in $\mathrm{rpm} )$ of the coil so it will produce the required voltage?

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The magnetic field $B$ at the center of a circular coil of wire

carrying a current $I$ (as in Fig. 20-9) is

$$B = $\frac{\mu_0 NI}{2r},$$

where $N$ is the number of loops in the coil and $r$ is its radius. Imagine a simple model in which the Earth's magnetic field of about 1 G $(= 1 \times 10^{-4}$ T$)$ near the poles is produced by a single current loop around the equator. Roughly estimate the current this loop would carry.

The strength of Earth's magnetic field, as measured on the surface, is approximately $6.0 \times 10^{-5} \mathrm{T}$ at the poles and $3.0 \times 10^{-5} \mathrm{T}$ at the equator. Suppose an alien from outer space were at the North Pole with a single loop of wire of the same circumference as his space helmet. The diameter of his helmet is $20.0 \mathrm{cm} .$ The space invader wishes to cancel Earth's magnetic field at his location. (a) What is the current required to produce a magnetic ficld (due to the current alone) at the center of his loop of the same size as that of Earth's field at the North Pole? (b) In what direction does the current circulate in the loop, CW or CCW, as viewed from above, if it is to cancel Earth's field?

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and has magnitude $8.0 \times 10^{-5} \mathrm{T},$ and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 $\mathrm{V}$ and estimate that you can rotate the coil at 30 $\mathrm{rpm}$ by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of

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A 100-turn square wire coil of area 0.040 $\mathrm{m}^{2}$ rotates about a vertical axis at 1500 $\mathrm{rev} / \mathrm{min}$ , as indicated in Figure $\mathrm{P} 20.32 .$ The horizontal component of Earth's magnetic field at the location of the loop is $2.0 \times 10^{-5} \mathrm{T}$ . Calculate the maximum emf induced in the coil by Earth's field.

Calculate the induced electric field in a 50-turn coil with a diameter of $15 \mathrm{cm}$ that is placed in a spatially uniform magnetic field of magnitude $0.50 \mathrm{T}$ so that the face of the coil and the magnetic field are perpendicular. This magnetic field is reduced to zero in 0.10 seconds. Assume that the magnetic field is cylindrically symmetric with respect to the central axis of the coil.

An astronaut is connected to her spacecraft by a 25.0-m-long tether cord as she and the spacecraft orbit the Earth in a circular path at a speed of $7.80 \times 10^{3} \mathrm{m} / \mathrm{s}$ . At one instant, the emf between the ends of a wire embedded in the cord is measured to be 1.17 $\mathrm{V}$ . Assume the long dimension of the cord is perpendicular to the Earth's magnetic field at that instant. Assume also the tether's center of mass moves with a velocity perpendicular to the Earth's magnetic field. (a) What is the magnitude of the Earth's field at this location? (b) Does the emf change as the system moves from one location to another? Explain. (c) Provide two conditions under which the emf would be zero even though the magnetic field is not zero.