Suppose the loop in Fig. P29.50 is (a) rotated about the y-axis; (b) rotated about the x-axis; (

step 1 For Part A, we're asked to find the induced EMF maximum rotated about the Y axis. And then for X

Suppose the loop in Fig. P29.50 is (a) rotated about the...

Suppose the loop in $\textbf{Fig. P29.50}$ is (a) rotated about the $y$-axis; (b) rotated about the x-axis; (c) rotated about an edge parallel to the $z$-axis. What is the maximum induced emf in each case if $A =$ 600 cm$^2$, $\omega$ = 35.0 rad/s, and $B =$ 0.320 T?

Key Concepts

Rotational Motion and Axis Orientation

When a loop rotates in a magnetic field, the orientation of its area vector with respect to the field changes, altering the magnetic flux through the loop. The axis about which the loop rotates determines the pattern and amplitude of this change. Different rotation axes yield different functional dependencies of flux on time, leading to variations in the maximum induced emf.

Faraday's Law of Electromagnetic Induction

This law states that a changing magnetic flux through a loop induces an electromotive force (emf) in the loop. The magnitude of the induced emf is proportional to the rate of change of the magnetic flux, with the sign determined by Lenz's law. Understanding this principle is essential when analyzing how the loop's changing orientation in a magnetic field generates a voltage.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing perpendicular to an area. It is calculated as the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the magnetic field and the normal to the loop. This concept is crucial for determining how the flux changes as the loop rotates, thereby affecting the induced emf.

Maximum Induced EMF Calculation

The maximum induced emf occurs when the rate of change of the magnetic flux is at its greatest. For a uniformly rotating loop, this is closely related to the product of the angular frequency, the magnetic field strength, and the effective area component that contributes to the flux variation. This concept ties together the effects of rotational dynamics, the geometry of the loop, and Faraday's law.

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