Suppose the loop in Fig. 29.45 is (a) rotated about the y -...

Suppose the loop in Fig. 29.45 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 cach case if $A=600 \mathrm{cm}^{2}, \omega=35.0 \mathrm{rad} / \mathrm{s}$ , and $B=0.450 \mathrm{T} ?$

Suppose the loop in Fig. 29.45 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 cach case if
$A=600 \mathrm{cm}^{2}, \omega=35.0 \mathrm{rad} / \mathrm{s}$ , and $B=0.450 \mathrm{T} ?$

Key Concepts

Axis of Rotation and Geometric Considerations

The axis about which the loop rotates affects how the magnetic flux changes over time. Different axes, such as rotations about the y-axis, x-axis, or an edge parallel to the z-axis, alter the relationship between the loop’s normal vector and the magnetic field. This geometric dependence is essential for calculating the maximum induced emf, since the effective modulation of the flux will vary with the rotation axis.

Rotation of a Loop in a Magnetic Field

When a conducting loop rotates in a constant magnetic field, the orientation of the area vector (normal to the loop) with respect to the magnetic field changes. This change alters the magnetic flux through the loop over time, creating a time-dependent emf that typically varies sinusoidally. The maximum emf occurs when the rate of change of the flux is greatest.

Faraday's Law of Electromagnetic Induction

This law states that a change in magnetic flux through a loop induces an electromotive force (emf) in the loop. The induced emf is proportional to the rate of change of the magnetic flux, providing the foundational principle for understanding how electric generators and many other electromagnetic devices work.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a surface 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 field and the normal to the surface. Its variation with time, especially in response to rotations or movements, is the key factor in inducing an emf.

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