A steel guitar string vibrates (see Fig. 31.5). The...

A steel guitar string vibrates (see Fig. $31.5)$. The component of the magnetic field perpendicular to the area of a pickup coil nearby is given by $$B=50.0 \mathrm{mT}+(3.20 \mathrm{mT}) \sin (2 \pi 523 t / \mathrm{s}).$$ The circular pickup coil has 30 turns and radius $2.70 \mathrm{~mm}$. Find the emf induced in the coil as a function of time.

Key Concepts

Faraday's Law of Induction

This law states that a time-varying magnetic field through a closed loop induces an electromotive force (emf) in the loop that is proportional to the rate of change of the magnetic flux through the loop. It is mathematically expressed as emf = -N d?/dt, where N is the number of turns in the coil and ? is the magnetic flux.

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given area. When considering a coil in a magnetic field, the flux is calculated by multiplying the component of the magnetic field perpendicular to the coil’s plane by the area of the coil. In the context of induced emf, changes in this flux over time are crucial.

Time-Varying Magnetic Fields

A magnetic field that changes with time, such as a sinusoidally varying field, leads to a change in magnetic flux through a coil. The induced emf in the coil is directly related to the time rate of change of this field, which is inherent in dynamic systems like vibrating objects producing periodic magnetic variations.

Coil Geometry and Multiple Turns

The physical characteristics of the coil, including its area and the number of turns it contains, are essential in determining the magnitude of the induced emf. Each turn of the coil contributes to the total voltage induced, making the total emf proportional to both the per-turn flux change and the total number of turns.

Please give Ace some feedback