A closely wound search coil (see Exercise 29.3) has an area...

A closely wound search coil (see Exercise 29.3) has an area of 3.20 cm$^2$, 120 turns, and a resistance of 60.0 $\Omega$. It is connected to a charge-measuring instrument whose resistance is 45.0 $\Omega$. When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of 3.56 $\times$ 10$^{-5}$ C. What is the magnitude of the field?

Key Concepts

Magnetic Flux

Magnetic flux represents the total magnetic field passing through a given area. It is calculated as the product of the magnetic field strength, the area of the surface, and the cosine of the angle between the field and the normal to that surface. This concept is central in problems of electromagnetic induction, where changes in flux lead to induced currents.

Faraday's Law of Electromagnetic Induction

Faraday's Law explains that a change in magnetic flux through a circuit induces an electromotive force (emf) proportional to the rate of change of the flux and the number of turns in the coil. This principle forms the basis for determining how variations in orientation or field magnitude yield measurable electrical signals.

Impulse in an Inductive Circuit

When a coil is rotated rapidly, the magnetic flux changes almost instantaneously, resulting in a brief pulse of induced emf. Integrating this pulse over time in a resistive circuit yields the total charge transferred. This impulse method connects the transient behavior of induced currents to measurable quantities like charge, providing a way to determine the magnetic field.

Series Resistance in a Circuit

In a series circuit, the total resistance is simply the sum of the individual resistances. This total resistance is crucial in determining the magnitude of the current produced by an induced emf, and thus the total charge transferred. Correctly accounting for series resistance is essential in solving electromagnetic induction problems where circuit response is measured via integrated charge.

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