A long solenoid, with its axis along the x axis, consists of...

A long solenoid, with its axis along the $x$ axis, consists of 200 turns per meter of wire that carries a steady current of $15.0 \mathrm{~A}$. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of $8.00 \mathrm{~cm} .$ The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and coincides with the $y$ axis. The coil is then rotated with an angular speed of $4.00 \pi \mathrm{rad} / \mathrm{s}$. (The plane of the coil is in the $y z$ plane at $t=0 .$ ) Determine the emf developed in the coil as a function of time.

Key Concepts

Rotational Dynamics of a Coil

When the coil rotates at a constant angular speed about an axis, the angle between the coil's normal and the magnetic field changes with time. If the coil rotates about the y-axis and its normal rotates in the x-z plane, the cosine term in the flux expression becomes a cosine function of time. This relationship is key to expressing the induced emf as a sinusoidal function of time.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic flux through a circuit induces an electromotive force (emf) in the circuit. Mathematically, emf = -N (d?/dt), where N is the number of turns. The negative sign represents Lenz's Law, indicating that the induced emf opposes the change in flux. This concept explains how the rotation of the coil in the solenoid’s magnetic field results in a time-varying emf.

Coil Geometry and Area Calculation

The geometry of the coil defines the area through which the magnetic field permeates. For a circular coil, the area A is calculated as A = ?r², where r is the radius of the coil. Knowing the area is essential for determining the magnetic flux through the coil.

Magnetic Field of a Solenoid

A solenoid with a steady current produces a nearly uniform magnetic field within its interior. The magnetic field inside a long solenoid is given by B = ?? n I, where ?? is the permeability of free space, n is the number of turns per unit length, and I is the current. This concept is crucial because it allows us to compute the uniform field that interacts with the rotating coil.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given area and is defined as ? = B · A · cos?, where B is the magnetic field strength, A is the area, and ? is the angle between the magnetic field and the normal to the area. This concept underpins the calculation of the induced emf when the coil rotates.

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