Problem 52 - University Physics with Modern Physics

Question

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could uperate using a generalor but you have nu maguels. The eardis 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 mumber 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 this device is feasible? Explain.

Step-by-Step Solution

Answer

The coil must have an area of 17.9 m². Max translational speed: 7.51 m/s. It's not feasible due to size and effort.

Step 1: Understand the Required Equation

To find the area, we need to use Faraday's Law of Electromagnetic Induction. The expression for the peak emf induced in a rotating coil is given by:\[ \text{EMF} = NBA\omega \cos(\theta) \] where N is the number of turns, B is the magnetic field, A is the area of the coil, and \(\omega\) is the angular velocity of the coil. \(\theta\) is the angle between the plane of the coil and the magnetic field, which is 90 degrees for maximum EMF, making \(\cos(\theta) = 1\).

Step 2: Calculate Angular Velocity

The angular velocity (\(\omega\)) can be calculated using the relation:\[ \omega = \frac{2\pi \times \text{revolutions per minute}}{60} \] Given that the coil rotates at 30 rpm, \(\omega = \frac{2\pi \times 30}{60} = \pi \) radians per second.

Step 3: Rearrange the Equation to Solve for Area

Rearrange the expression for the peak EMF to solve for the area (A):\[ A = \frac{\text{EMF}}{NB\omega} \]Substitute the known values: EMF = 9.0 V, N = 2000, B = 8.0 \times 10^{-5} T, \(\omega = \pi \) rad/s.

Step 4: Substitute the Values and Calculate Area

Substitute the given values into the equation to calculate the area:\[ A = \frac{9.0}{2000 \times 8.0 \times 10^{-5} \times \pi} \approx 17.9 \text{ m}^2 \].

Step 5: Determine Maximum Translational Speed

The maximum translational speed (v) of a point on the coil is given by \( v = r \cdot \omega \), where r is the radius of the coil. If the coil is circular, the radius r is derived from the area using \( \pi r^2 = 17.9 \). Hence,\[ r = \sqrt{\frac{17.9}{\pi}} \approx 2.39 \text{ m}\].Then, the maximum translational speed is:\[ v = 2.39 \cdot \pi \approx 7.51 \text{ m/s} \].

Step 6: Evaluate Feasibility

Given the large required area and the resulting maximum speed, constructing such a coil by spinning around a crank seems impractical due to its bulk and the physical effort required to maintain rotation. Thus, the idea, while theoretically possible, might not be feasible in a practical scenario.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental concept in physics that describes how a changing magnetic field induces an electromotive force (emf) in a conductor. The principle is best captured by Faraday's Law, which states that the induced emf is directly proportional to the rate of change of the magnetic flux linked with the circuit. In the context of the generator problem from the textbook exercise, this involves rotating a coil within a magnetic field, which is achieved by the Earth's horizontal magnetic field on the island.

• The key factor here is the change in the magnetic field experienced by the coil as it rotates.
• Without a magnet, utilizing Earth's magnetic field for generation becomes a unique challenge.

Understanding this concept helps explain why different coil dynamics (like speed and number of turns) affect the overall emf produced.

Angular Velocity

Angular velocity is an essential aspect to understand how fast an object rotates around an axis. It is particularly vital when dealing with rotating systems like generators. In this exercise, the generator's efficiency relies significantly on how fast the coil is rotated.
To find the angular velocity (\( \omega \)), the formula \[ \omega = \frac{2\pi \times \text{revolutions per minute}}{60} \]is used. Here, it translates the rotation speed from revolutions per minute (rpm) to radians per second, which is a standard measurement in physics.

• For the coil rotating at 30 rpm, the angular velocity is calculated as \( \pi \) radians per second.
• This measure allows us to further calculate the generated emf and the translational speed of the coil.

EMF Calculation

Calculating the electromagnetic force (emf) is the core challenge in this exercise, as it requires balancing multiple factors to reach the desired peak emf of 9.0V. The formula from Faraday's Law \(\text{EMF} = NBA\omega \cos(\theta)\)helps calculate this. Here,\( N \) is the number of turns, \( B \) is the magnetic field strength, \(\omega\) is the angular velocity, and \(\cos(\theta)\) accounts for the orientation of the coil relative to the magnetic field, which is 1 for the maximum since \(\theta = 90^\circ\).
To find the area (\( A \)), the equation is rearranged: \[ A = \frac{\text{EMF}}{NB\omega} \].By substituting the known values and solving, we determine the coil must have an area of approximately 17.9 \( m^2 \) to achieve the desired emf.

Circular Coil Design

The design of the coil greatly affects the functionality of the generator. A circular coil is practical but requires careful consideration of size. From the calculated area of 17.9 \( m^2 \), the radius, \( r \), is determined using the area formula for a circle, \(\pi r^2\). Solving for \( r \) gives approximately 2.39 meters.
This large radius impacts the generator's operation:

• The maximum translational speed of a point on the rotating coil is calculated as \( v = r \cdot \omega \), approximately 7.51 \( m/s \), highlighting potential feasibility issues.
• Such a large, fast-spinning coil may not be easily managed manually on a deserted island, rendering the setup impractical even if theoretically sound.