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Chapter 12: Problem 12

In the August 1992 space shuttle flight, only 250 m of the conducting tether considered in Example could be let out. A 40.0 V motional emf was generated in the Earth鈥檚 5.00脳10?5T field, while moving at 7.80脳103m/s. What was the angle between the shuttle鈥檚 velocity and the Earth鈥檚 field, assuming the conductor was perpendicular to the field?

Short Answer

Expert verified
The angle between the shuttle's velocity and the Earth's magnetic field, assuming the conductor is perpendicular to the field, is approximately 24.1 degrees.

Step by step solution

01

Understanding the given information

We're given a motional emf (electromotive force) of 40.0 V generated by a conducting tether let out from a space shuttle moving at a velocity of 7.80x10^3 m/s in the Earth鈥檚 magnetic field of 5.00x10^-5 T. The length of the conducting tether is 250 m. We are to assume that the conducting tether is perpendicular to the magnetic field. The task is to find the angle between the shuttle's velocity and the Earth's magnetic field.
02

Use the formula for motional emf

The formula for the motional emf (\(\varepsilon\)) induced in the conductor moving in a magnetic field is \(\varepsilon = Bvl \sin(\theta)\), where \(B\) is the magnetic field strength, \(v\) is the velocity, \(l\) is the length of the conductor, and \(\theta\) is the angle between the velocity and the magnetic field. Here, the emf is given as 40.0 V, \(B = 5.00 \times 10^{-5} T\), \(v = 7.80 \times 10^{3} m/s\), and \(l = 250 m\).
03

Rearrange the formula to solve for the angle

Rearrange the formula for motional emf to solve for \(\theta\) by isolating \(\sin(\theta)\) on one side: \(\sin(\theta) = \frac{\varepsilon}{Bvl}\). Substitute the given values into the rearranged equation.
04

Substitute the given values and solve for 胃

Substitute the given values into the formula to find \(\sin(\theta)\): \(\sin(\theta) = \frac{40.0 V}{(5.00 \times 10^{-5} T) (7.80 \times 10^{3} m/s) (250 m)}\). Calculate the value of \(\sin(\theta)\) and then use the inverse sine function (also known as arcsin) to determine the angle \(\theta\).
05

Calculate the value of 胃

Perform the calculation: \(\sin(\theta) = \frac{40.0}{(5.00 \times 10^{-5}) (7.80 \times 10^{3}) (250)}\). Then, \(\sin(\theta) = \frac{40}{0.0975}\), which gives \(\sin(\theta) = 410.256\). Because the sine of an angle cannot be greater than 1, we've likely made a calculation error. Let us re-evaluate our calculation.
06

Correcting the error and recalculating

After re-evaluating the calculations, we find \(\sin(\theta) = \frac{40.0}{(5.00 \times 10^{-5}) (7.80 \times 10^{3}) (250)} = 0.410256\). Now we can calculate the angle using the inverse sine function: \(\theta = \arcsin(0.410256)\).
07

Determine the final angle

Using a calculator, calculate \(\theta = \arcsin(0.410256)\), which results in an angle of approximately 24.1 degrees (ensure the calculator is set to the correct mode, either degrees or radians as required).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Induction
Electromagnetic induction is the process by which a conductor moving through a magnetic field develops an electromotive force (emf). This phenomenon is central to the generation of electricity and is described by Faraday's Law of Induction. It's effectively what was occurring on the space shuttle with the conducting tether. When the tether cuts across the Earth's magnetic field lines at a particular angle, a voltage, which we refer to as motional emf, is produced across the ends of the tether.

The amount of emf generated depends on several factors, including the speed at which the conductor moves, the strength of the magnetic field, the length of the conductor, and importantly, the angle between the conductor's motion and the magnetic field. In the exercise, by understanding these principles, we could infer that the 40.0 V emf resulted from the specific arrangement and motion of the conducting tether within the Earth's magnetic field.
Magnetic Field Strength
Magnetic field strength, denoted by the symbol B, is a measure of the magnitude of the magnetic field at a given location and is a critical factor in determining the motional emf. The unit of magnetic field strength is the Tesla (T). In our exercise, the Earth's magnetic field strength was given as 5.00 脳 10^-5 T, a fairly typical value for the Earth's surface.

The stronger the magnetic field, the higher the emf induced in a given conductor. In practical applications, for example in electrical generators, powerful magnets are used to create strong magnetic fields to increase the efficiency of electromagnetic induction. Knowing how to calculate the emf using the magnetic field strength and other given variables allows one to engineer systems that maximize the output voltage for a given set of circumstances.
Angle Calculation
Angle calculation plays a pivotal role in understanding the orientation of a conductor with regard to the magnetic field for determining the induced emf. In the context of our exercise, the angle calculation involved solving for \( \theta \) when given the emf, magnetic field strength, velocity, and length of the conductor.

To find this angle using the relationship \( \varepsilon = Bvl \sin(\theta) \), we isolate \( \sin(\theta) \) to calculate its value using given emf, field strength, velocity, and length. Afterward, applying the inverse sine function, or arcsin, reveals the measure of the angle in question. Calculating angles is vitally important in designing systems involving electromagnetic induction, as the emf can be maximized by optimizing the angle at which conductors move through magnetic fields.

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Most popular questions from this chapter

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