CALC A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad / s about an axis th

step 1 In the given problem, length of the rod is 0 .240 meter. It is rotating with an angular velocity

CALC A slender rod, 0.240 m long, rotates with an angular...

CALC A slender rod, 0.240 m long, rotates with an angular speed of 8.80 $\mathrm{rad} / \mathrm{s}$ about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 $\mathrm{T}$ . (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 $\mathrm{rad} / \mathrm{s}$ about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

Key Concepts

Motional EMF

This concept describes the electromotive force induced in a conductor when it moves through a magnetic field. The induced EMF arises from the Lorentz force acting on the free charge carriers within the conductor, causing a separation of charges that results in a potential difference.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is given by the cross-product of the particle's velocity and the magnetic field, and it provides the fundamental mechanism behind the induction of EMF in moving conductors.

Rotational Motion in Magnetic Fields

In systems where a rod or conductor rotates in a magnetic field, each point along the rod has a linear velocity that depends on its distance from the axis of rotation. This variation must be taken into account when determining the induced EMF across the conductor.

Integration in Non-uniform Motion

When a conductor rotates, different segments move at different speeds. Calculating the total induced EMF typically involves integrating the local contributions to the EMF along the length of the conductor, reflecting the non-uniform velocity distribution.

Effect of Axis of Rotation

The position of the axis of rotation, whether at the end of the rod or at its center, changes the velocity distribution along the conductor. This, in turn, affects both the magnitude and the spatial distribution of the induced potential difference, highlighting the importance of geometry in electromagnetic induction problems.

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