A circular coil of radius 10 cm and a separate square coil of side 20 cm are both rotated in a magnetic field of 1.5 T.If the circular coil is rotated at a frequency of 60 Hz,then at what frequency must the square coil be rotated in order for both coils to have the same maximum induced voltage?
A)47 Hz
B)60 Hz
C)76 Hz
D)30 Hz
E)19 Hz

Question

Quizplus · Accepted Answer

The maximum induced voltage in a coil rotating in a magnetic field is given by the formula $EMF = NBA\omega$, where $N$ is the number of turns (assumed to be the same for both coils for this comparison), $B$ is the magnetic field strength, $A$ is the area of the coil, and $\omega$ is the angular velocity in radians per second. The frequency $f$ is related to $\omega$ by $\omega = 2\pi f$.For the circular coil, the area $A_{circle} = \pi r^2 = \pi (0.1)^2 = 0.01\pi$ m$^2$.For the square coil, the area $A_{square} = side^2 = (0.2)^2 = 0.04$ m$^2$.To have the same maximum induced voltage, we set the expressions for EMF equal to each other, leading to the relationship between their areas and angular velocities (or frequencies).Given that the circular coil is rotated at 60 Hz, we find the frequency for the square coil by ensuring the product of the area and frequency (which is proportional to angular velocity) is the same for both coils.$\frac{A_{circle}}{A_{square}} = \frac{f_{square}}{f_{circle}}$$\frac{0.01\pi}{0.04} = \frac{f_{square}}{60}$Solving for $f_{square}$, we get:$f_{square} = 60 \times \frac{0.01\pi}{0.04} = 60 \times \frac{\pi}{4} \approx 47$ Hz.Therefore, the square coil must be rotated at approximately 47 Hz to have the same maximum induced voltage as the circular coil rotated at 60 Hz.

A Circular Coil of Radius 10 Cm and a Separate