Consider the following situation are uniform magnetic field b point t ...
Understanding the Situation
In this scenario, we have a uniform magnetic field \( \mathbf{B} \) directed into the plane of the paper within a rectangular region. A conducting loop is placed inside this region and is rotated counterclockwise with a uniform angular velocity \( \Omega \).
Electromotive Force (EMF) Induction
According to Faraday's Law of Electromagnetic Induction, the EMF (\( \mathcal{E} \)) induced in the loop is given by:
\[
\mathcal{E} = -\frac{d\Phi_B}{dt}
\]
where \( \Phi_B \) is the magnetic flux through the loop.
Magnetic Flux Calculation
- The magnetic flux \( \Phi_B \) through the loop can be expressed as:
\[
\Phi_B = B \cdot A \cdot \cos(\theta)
\]
- Here, \( A \) is the area of the loop, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the magnetic field direction and the normal to the loop's surface.
Effect of Rotation
- As the loop rotates, the angle \( \theta \) changes, leading to a time-dependent magnetic flux:
\[
\theta = \Omega t
\]
- Thus, the magnetic flux becomes:
\[
\Phi_B = B \cdot A \cdot \cos(\Omega t)
\]
Graph of EMF
- The EMF induced will then be:
\[
\mathcal{E} = -\frac{d\Phi_B}{dt} = B \cdot A \cdot \Omega \cdot \sin(\Omega t)
\]
- This indicates that the EMF varies sinusoidally with time, oscillating between \( -B \cdot A \cdot \Omega \) and \( B \cdot A \cdot \Omega \).
Conclusion
- The correct graph for the EMF will be a sine wave starting from zero, oscillating between positive and negative values, reflecting the periodic change in magnetic flux due to the rotating loop.
This behavior perfectly illustrates the principles of electromagnetic induction in a rotating system within a magnetic field.