Problem 52 - University Physics with Modern Physics

Question

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\(^{-5}\) 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 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

Step-by-Step Solution

Answer

The coil area must be approximately 17.9 m², with a translational speed of 7.51 m/s. It is likely impractical due to mechanical constraints.

Step 1: Understand the Problem

You need to produce a peak emf of 9.0 V using the rotation of a coil in the Earth's magnetic field, which has a strength of 8.0 \( \times \) 10\(^{-5}\) T. The frequency of coil rotation is 30 rpm, and you can use up to 2000 turns in the coil to achieve this.

Step 2: Convert Rotational Speed

Convert the angular speed from revolutions per minute (rpm) to radians per second (rad/s) for calculation convenience. Since 1 revolution = \( 2\pi \) radians and there are 60 seconds in a minute, the angular speed \( \omega \) is calculated as follows:\[\omega = 30 \times \left(\frac{2\pi}{60}\right) = \pi\, \text{rad/s}\].

Step 3: Use Faraday's Law for Peak EMF

The magnitude of peak emf, \( \varepsilon \), generated in a coil rotating in a magnetic field is given by Faraday's law of electromagnetic induction:\[\varepsilon = NAB\omega\] where \( N = 2000 \) turns, \( B = 8.0 \times 10^{-5} \) T. Solving for area \( A \), given \( \varepsilon = 9.0 \) V and \( \omega = \pi \) rad/s:\[A = \frac{9.0}{2000 \times 8.0 \times 10^{-5} \times \pi}= \frac{9.0}{0.503}\approx 17.9 \text{ m}^2\].

Step 4: Determine Maximum Translational Speed

For a circular coil of radius \( r \), the area \( A \) is \( \pi r^2 \). Calculate \( r \) from \( A = 17.9 \) m\(^2\):\[r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{17.9}{\pi}} \approx 2.39 \text{ m}\]The maximum translational speed of a point on the coil is the edge speed, \( v = r\omega \):\[v = 2.39 \times \pi \approx 7.51 \text{ m/s}\].

Step 5: Evaluate Feasibility

To evaluate the feasibility, consider mechanical and practical constraints. Although theoretically possible, the large coil area and speed may not be feasible with materials and construction capabilities on a deserted island.

Key Concepts

Electric Generators

Electric generators are fascinating devices that convert mechanical energy into electrical energy. They operate on the principle of electromagnetic induction, which was discovered by Michael Faraday. At the core of a generator's function is a coil of wire. When this coil rotates within a magnetic field, it induces a current. This is how most of our electricity is generated, whether in large power stations or smaller devices.
The key idea here is that motion between a magnetic field and a conductor creates electricity. In the case of a deserted island scenario, the Earth's magnetic field can act as the magnetic source, which is usually provided by strong magnets in a typical generator setup. While building a generator using the Earth's magnetic field is innovative, the strength of this field is usually weaker than artificial magnets, making it a challenging but creative solution.

Earth's Magnetic Field

Earth's magnetic field is a natural protective shield that extends from the planet's core out into space. It not only protects us from harmful solar radiation but can also be harnessed for practical uses such as navigation and electricity generation.
This magnetic field at the surface of the Earth is relatively weak, with a common strength around 8.0 \( \times \) 10^{-5} T (teslas). In this exercise, utilizing Earth's magnetic field for generating electricity highlights its role beyond its usual mechanical benefits. However, keep in mind that due to its weak strength, generating a significant amount of electricity using Earth's magnetic field alone is quite difficult without optimization and precise design in the generator's components.

Electromotive Force (EMF)

Electromotive force, or EMF, is a crucial concept when discussing generators and electromagnetic induction. EMF is essentially the voltage generated by a changing magnetic field around a conductor, such as a coil of wire. According to Faraday's Law of Electromagnetic Induction, the induced EMF in a coil is determined by factors such as the number of turns in the coil, the rate of the coil's rotation (angular velocity), and the strength of the magnetic field.
The formula, \( \varepsilon = NAB\omega \), shows how these factors interact: \( N \) is the number of turns in the coil, \( A \) is the area of the coil, \( B \) is the magnetic field strength, and \( \omega \) is the angular speed of rotation. The goal in this exercise was to produce an EMF of 9 V, which required careful balancing of these variables, especially considering the constraints provided by the materials at hand.

Coil Design Parameters

Designing a coil for optimal performance in an electric generator comes down to several key parameters. First, the number of turns in the coil is important as more turns can increase the EMF, according to Faraday's Law. However, too many turns can increase resistance and practical issues in handling on a deserted island.
The area of the coil is another critical parameter. A larger area can increase the EMF, allowing more of the magnetic field lines to be cut by the rotating coil. In our problem's solution, we calculated the ideal area for the coil to be approximately 17.9 m extsuperscript{2}.
Lastly, the shape of the coil, which is often circular due to ease of rotating, must be optimized given the material constraints. These parameters aid in maximizing efficiency and balancing practical considerations in remote settings.