A coil of 15 turns and radius 10.0 cm surrounds a long...

A coil of 15 turns and radius $10.0 \mathrm{~cm}$ surrounds a long solenoid of radius $2.00 \mathrm{~cm}$ and $1.00 \times 10^{3}$ turns per meter (Fig. $\mathrm{P} 31.10$ ). If the current in the solenoid changes as $I=(5.00 \mathrm{~A}) \sin (120 t)$, find the induced emf in the 15 -turn coil as a function of time.

Key Concepts

Multi-Turn Coils

In coils with multiple turns, the net magnetic flux is the sum of the flux through each turn. Consequently, when applying Faraday's Law, the total induced emf is multiplied by the number of turns in the coil. This amplifies the effect of the changing magnetic flux and is crucial in many electromagnetic induction applications.

Time-Varying Currents and Their Effects

When the current in a circuit changes with time, as described by a sinusoidal function, the resulting magnetic field also varies with time. This time-varying magnetic field leads to a changing magnetic flux through nearby circuits, thereby inducing an emf. Calculating the derivative of the time-dependent current function is essential for finding the rate of change of the magnetic flux.

Magnetic Field in a Solenoid

A long solenoid produces a nearly uniform magnetic field inside its winding, which is proportional to both the current and the density of turns per unit length. This concept is critical in determining the magnitude of the magnetic field that links the surrounding coil, as the field inside the solenoid serves as the source of the changing flux.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given area. It is calculated as the integral of the magnetic field over that area. For uniform fields and simple geometries, it simplifies to the product of the magnetic field strength and the area perpendicular to the field, a key step when applying Faraday's Law to determine induced emf.

Faraday's Law of Induction

This principle states that an electromotive force (emf) is induced in a circuit when the magnetic flux through the circuit changes in time. The induced emf is equal to the negative rate of change of the magnetic flux multiplied by the number of turns in the coil, encapsulating both the magnitude and direction (via Lenz's Law) of the induced emf.

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