A 15.0-cm -long solenoid with radius 0.750 cm is closely wound with 600 turns of wire. The curr

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A 15.0-cm -long solenoid with radius 0.750 cm is closely...

A $15.0-\mathrm{cm}$ -long solenoid with radius $0.750 \mathrm{~cm}$ is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

Key Concepts

Solenoid

A solenoid is a coil of wire designed to produce a uniform magnetic field when carrying an electric current. Its structure, typically characterized by a long length relative to its diameter, ensures that the field inside is nearly constant and directed along the axis of the coil. The uniformity of the magnetic field is a key reason solenoids are widely used in electromagnet applications.

Magnetic Field Equation for a Solenoid

The magnetic field inside a long solenoid is determined by the relationship B = ??nI, where B is the magnetic field, ?? is the permeability of free space, n is the number of turns per unit length, and I is the current. This equation succinctly captures how the field strength depends linearly on the current and the density of the coil turns.

Turn Density

Turn density, represented by n in the magnetic field equation for a solenoid, is the number of wire turns per unit length of the solenoid. It is crucial because a higher turn density results in a proportionally stronger magnetic field when current flows through the coil.

Permeability of Free Space

The permeability of free space, denoted by ??, is a fundamental physical constant that quantifies the ability of a vacuum to support a magnetic field. It appears in the magnetic field equation for solenoids and is essential for relating magnetic field strength to the current and turn density in a medium without magnetic material.

Ampere’s Law

Ampere’s Law is a fundamental principle in electromagnetism that relates the magnetic field circulating around a closed loop to the electric current passing through the loop. It provides the theoretical foundation for deriving the expression for the magnetic field inside a solenoid, thereby linking current, geometry, and the generated magnetic field.

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