A metallic ring of radius R moves in a vertical plane in the...

A metallic ring of radius $R$ moves in a vertical plane in the presence of a uniform magnetic field $B$ perpendicular to the plane of the ring. At any given instant of time, its centre of mass moves with a velocity $v$ while ring rotates in its COM frame with angular velocity $\omega$ as shown in Fig. $16.50 .$ The magnitude of induced EMF between points $O$ and $P$ is (A) Zero (B) $v B R \sqrt{2}$ (C) $v B R$ (D) $2 v B R$

A metallic ring of radius $R$ moves in a vertical plane in the presence of a uniform magnetic field $B$ perpendicular to the plane of the ring. At any given instant of time, its centre of mass moves with a velocity $v$ while ring rotates in its COM frame with angular velocity $\omega$ as shown in Fig. $16.50 .$ The magnitude of induced EMF between points $O$ and $P$ is
(A) Zero
(B) $v B R \sqrt{2}$
(C) $v B R$
(D) $2 v B R$

Key Concepts

Motional Electromotive Force (EMF)

This concept involves the generation of an emf when a conductor moves through a magnetic field. The charge carriers within the conductor experience a force due to their motion through the magnetic field (as given by the Lorentz force), which causes a separation of charges, and consequently, a potential difference is established along the conductor.

Lorentz Force

The Lorentz force is fundamental in understanding how moving charges behave in electromagnetic fields. It states that a charge moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field. This force is responsible for pushing charges in different directions in a moving or rotating conductor, leading to an induced emf.

Reference Frame Analysis

When a system involves both translational and rotational motion, analyzing the problem from an appropriate reference frame (such as the centre-of-mass frame) is crucial. This analysis helps in determining the net velocity experienced by different parts of the conductor, thereby clarifying how charge separation and the corresponding emf are generated due to the combined motions.

Uniform Magnetic Field

A uniform magnetic field is one that has constant magnitude and direction over the region of interest. In problems involving induction, this uniformity simplifies the analysis since the induced emf is determined solely by the motion of the conductor relative to the magnetic field, rather than by spatial variations in the field itself.

Faraday's Law of Induction

Although the situation might be approached via the concept of motional emf and the Lorentz force, Faraday’s law of electromagnetic induction underlies the overall principle that a changing magnetic flux through a conductor induces an emf. In cases where the system’s motion results in a time-varying configuration with respect to the magnetic field, Faraday’s law provides a comprehensive framework for understanding the resulting emf.

Transcript

Please give Ace some feedback