A circular region in an x y plane is penetrated by a uniform...

A circular region in an $x y$ plane is penetrated by a uniform magnetic field in the positive direction of the $z$ axis. The field's magnitude $B$ (in teslas) increases with time $t$ (in seconds) according to $B=$ at. where $a$ is a constant. The magnitude $E$ of the electric field set up by that increase in the magnetic field is given by Fig. $30-57$ versus radial distance $r$; the vertical axis scale is set by $E_{s}=300 \mu \mathrm{N} / \mathrm{C}$, and the horizontal axis scale is set by $r_{s}=4.00 \mathrm{~cm}$. Find $a$.

Key Concepts

Symmetry in Electromagnetic Induction

Due to the symmetry of the problem (a uniform magnetic field perpendicular to a circular region), the induced electric fields are circular (concentric) about the axis of symmetry. This symmetry simplifies the application of Faraday's law by allowing the use of a circular integration path to relate the electric field with the changing magnetic flux.

Induced Electric Field

The induced electric field is nonconservative and arises from the time variation of the magnetic field, as described by Faraday's law. Unlike the electric fields produced by static charges, these electric fields form closed loops, which is a characteristic outcome of a changing magnetic environment.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a given surface area. In problems involving an increasing magnetic field, the rate of change of the magnetic flux (which depends on both the field strength and the area it penetrates) directly influences the magnitude of the induced electric field.

Faraday's Law of Induction

This law states that a changing magnetic field induces an electromotive force (emf) in a closed circuit. It is central to the understanding of how time-varying magnetic fields create electric fields. In a broader sense, it captures the relationship between time-dependent magnetic flux and the resulting induced electric field, a fundamental concept in electromagnetism.

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