A circular coil is rotating in a magnetic field of magnitude 0.25T with angular speed 6 rpm about its diameter. At \( t = 0 \), the coil’s configuration is as shown. If the induced emf after the coil is rotated by an angle of 30° is 1.6 mV, find the radius of the coil (in cm). (\( \pi^2 = 10 \))
Show Hint
Use the formula for induced emf and remember to convert angular speed to radians per second. The induced emf depends on the area of the coil, magnetic field strength, and the rate of rotation.
Step 1: Formula for induced emf.
The induced emf \( \epsilon \) in a coil rotating in a magnetic field is given by:
\[
\epsilon = N A B \omega \sin \theta,
\]
where:
- \( N \) is the number of turns (assumed to be 1 for this case),
- \( A \) is the area of the coil (\( A = \pi R^2 \)),
- \( B \) is the magnetic field strength,
- \( \omega \) is the angular speed (in radians per second),
- \( \theta \) is the angle of rotation.
Step 2: Convert angular speed to radians per second.
Given that the angular speed is 6 rpm, we convert it to radians per second:
\[
\omega = \frac{6 \times 2 \pi}{60} = \frac{\pi}{5} \, \text{rad/s}.
\]
Step 3: Calculate the radius.
Substitute the known values into the induced emf formula:
\[
1.6 \times 10^{-3} = \pi R^2 \times 0.25 \times \frac{\pi}{5} \times \sin 30^\circ.
\]
Since \( \sin 30^\circ = \frac{1}{2} \), we solve for \( R \) and find \( R = 8 \, \text{cm} \).
Final Answer:
\[
\boxed{8 \, \text{cm}}.
\]
