To solve this problem, we need to apply Faraday's law of electromagnetic induction, which states that the electromotive force (emf) induced in a coil is proportional to the rate of change of magnetic flux through the coil. Let's break down the components of the problem step by step.
Understanding the Components
We have the following parameters:
- Number of turns in the coil (N): 100 turns
- Area of the coil (A): 0.1 m²
- Magnetic field strength (B): 0.01 T (tesla)
- Rotation rate: 50 revolutions per minute (rpm)
Calculating Maximum EMF
The maximum emf (E_max) can be calculated using the formula:
E_max = N * B * A * ω
Where:
- ω (angular velocity): This is in radians per second. To convert from rpm to radians per second, we use the conversion factor:
- ω = (50 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds) = (50 * 2π / 60) rad/s
Calculating ω:
ω = (50 * 2π / 60) ≈ 5.24 rad/s
Now, substituting the values into the maximum emf formula:
E_max = 100 turns * 0.01 T * 0.1 m² * 5.24 rad/s
E_max = 100 * 0.01 * 0.1 * 5.24 = 0.0524 V
Finding Average EMF
The average emf (E_avg) over one complete cycle of rotation can be calculated as:
E_avg = E_max / 2
This is because the emf varies sinusoidally as the coil rotates, reaching its maximum value and then returning to zero.
So, substituting the maximum emf we calculated:
E_avg = 0.0524 V / 2 = 0.0262 V
Summary of Results
To summarize, we have:
- Maximum EMF: 0.0524 V
- Average EMF: 0.0262 V
This example illustrates how the rotation of a coil in a magnetic field induces an electromotive force, which is fundamental to many applications in electrical engineering and physics. Understanding these principles can help in various fields, including renewable energy and electrical machinery design.
