Answer :
To find the maximum electromotive force (emf) induced in a coil due to its rotation in a magnetic field, we can use Faraday's Law of Electromagnetic Induction, which states that the emf induced is equal to the rate of change of the magnetic flux through the coil.
Here's a step-by-step breakdown of the calculation:
Determine the Number of Turns (N):
The wire is wound into a single coil, and the length of the wire limits the number of turns possible.
The circumference of one turn is given by the formula:[tex]C = 2\pi r[/tex]
where [tex]r = 3.30 \ cm = 0.0330 \ m[/tex]. Therefore, the circumference of one coil is:
[tex]C = 2 \times \pi \times 0.0330 \approx 0.207 \ m[/tex]
The number of turns [tex]N[/tex] is then the total length of the wire divided by the circumference of one loop:
[tex]N = \frac{1.40}{0.207} \approx 6.76[/tex]
Since the number of turns must be an integer, approximately 6 full turns can be made.
Calculate the Area (A) of One Turn:
The area of one turn of the coil is given by the formula for the area of a circle:[tex]A = \pi r^2[/tex]
Substituting the given radius:
[tex]A = \pi (0.0330)^2 \approx 0.00342 \ m^2[/tex]
Determine the Angular Velocity ([tex]\omega[/tex]):
The coil rotates at [tex]97.4 \ rpm[/tex]. We need to convert this to radians per second since that's the SI unit for angular velocity:[tex]\omega = 97.4 \times \frac{2\pi}{60} \approx 10.20 \ rad/s[/tex]
Compute the Maximum Emf ([tex]\text{emf}_{max}[/tex]):
The maximum emf can be found using the formula:[tex]\text{emf}_{max} = N \cdot B \cdot A \cdot \omega[/tex]
where [tex]B = 0.0683 \ T[/tex] (the magnetic field strength). Substituting the known values:
[tex]\text{emf}_{max} = 6 \times 0.0683 \times 0.00342 \times 10.20 \approx 0.0142 \ V[/tex]
Therefore, the maximum emf induced in the coil is approximately [tex]0.0142 \ V[/tex].
