In Figure P 31.62, a uniform magnetic field decreases at a...

In Figure $\mathrm{P} 31.62$, a uniform magnetic field decreases at a constant rate $d B / d t=-K$, where $K$ is a positive constant. A circular loop of wire of radius $a$ containing a resistance $R$ and a capacitance $C$ is placed with its plane normal to the field. (a) Find the charge $Q$ on the capacitor when it is fully charged. (b) Which plate is at the higher potential? (c) Discuss the force that causes the separation of charges.

Key Concepts

Electromagnetic Forces and Charge Separation

The separation of charges in a circuit subject to a changing magnetic field is driven by the induced electric field, which acts on the free charges in the conductor. This induced field exerts forces, often described through the Lorentz force in dynamic situations, causing electrons to move and accumulate in a manner that establishes a potential difference across circuit elements like capacitor plates. The resulting spatial distribution of charge is a direct consequence of the interplay between the induced electric forces and the circuit’s geometry.

Capacitor Charging in an Induced Circuit

When a capacitor is placed in a circuit with an induced emf, it charges until the voltage across its plates equals the magnitude of the induced potential difference. The charging process in such circuits follows the basic capacitor relation Q = CV, where Q is the charge, C is the capacitance, and V is the voltage. In the steady state, the capacitor stores energy in its electric field with the charge reflecting the effect of the induced emf.

Faraday's Law of Electromagnetic Induction

This law states that a changing magnetic flux through a closed loop induces an electromotive force (emf) in that loop. It is expressed mathematically as the emf being equal to the negative rate of change of the magnetic flux. This principle underlies how a time-varying magnetic field can produce an electric potential in a circuit, which is central to problems involving induced currents and voltages.

Lenz's Law

Lenz's Law provides the direction of the induced electromotive force and the resulting current. It asserts that the induced emf always acts in such a way as to oppose the change in magnetic flux that produced it. This opposition is evident in the way induced currents generate magnetic fields that counteract the original variation in the external magnetic field.

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