The magnetic field perpendicular to the plane of conducting...

The magnetic field perpendicular to the plane of conducting ring of radius $r$ changes at the rate $\frac{d B}{d t}=\alpha$. Then (A) EMF induced in the ring is $\pi r^{2} \alpha$. (B) EMF induced in the ring is $2 \pi r \alpha$. (C) The potential difference between diametrically opposite points on the ring is half of induced EMF. (D) All points on the ring are at same potential.

The magnetic field perpendicular to the plane of conducting ring of radius $r$ changes at the rate $\frac{d B}{d t}=\alpha$. Then
(A) EMF induced in the ring is $\pi r^{2} \alpha$.
(B) EMF induced in the ring is $2 \pi r \alpha$.
(C) The potential difference between diametrically opposite points on the ring is half of induced EMF.
(D) All points on the ring are at same potential.

Key Concepts

Non-conservative Electric Fields

When an EMF is induced by a changing magnetic field, the resulting electric field is non-conservative. This means that the work done by the field around a closed loop is non-zero, and the concept of a unique scalar electric potential cannot be globally defined. As a consequence, potential differences within the loop are path-dependent, complicating straightforward voltage assignments between two points.

Induced EMF

The induced electromotive force (EMF) is the voltage generated in a conductor as a result of a changing magnetic flux. Its magnitude is computed using Faraday's Law and directly depends on how rapidly the magnetic field and consequently the magnetic flux is changing over time.

Faraday's Law

Faraday's Law of electromagnetic induction asserts that a changing magnetic flux through a circuit induces an electromotive force (EMF) in that circuit. The law quantitatively relates the induced EMF to the negative rate of change of the flux, forming the foundational principle for understanding how time-varying magnetic fields generate voltage in conductive loops.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing perpendicularly through a given area. It is calculated as the product of the magnetic field strength and the area it penetrates. Changes in magnetic flux, whether due to variations in the magnetic field strength or in the geometry of the area exposed, play a critical role in inducing an EMF according to Faraday's Law.

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