A solenoid used to produce magnetic fields for research...

A solenoid used to produce magnetic fields for research purposes is $2.0 \mathrm{m}$ long, with an inner radius of $30 \mathrm{cm}$ and 1000 turns of wire. When running, the solenoid produces a field of $1.0 \mathrm{T}$ in the center. Given this, how large a current does it carry?

Key Concepts

Magnetic Field of a Solenoid

This concept deals with the relationship between the magnetic field inside a solenoid and its physical characteristics. The field inside an ideal solenoid, which is assumed to be uniform, is given by the formula B = ?? n I, 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 flowing through the coil. This relationship is central to understanding how solenoids generate magnetic fields and is derived from Ampère’s law.

Ampère’s Law

Ampère’s law relates the magnetic field in space to the electric current that produces it. In the context of a solenoid, it provides the theoretical foundation for the formula B = ?? n I. Ampère's law simplifies the analysis by considering a path that follows the symmetry of the solenoid, allowing for the derivation of the uniform field inside and the negligible field outside the solenoid.

Turn Density

Turn density is the measure of how many turns of the wire are contained in a given unit length of the solenoid. It is calculated by dividing the total number of turns by the length of the solenoid. This parameter is crucial because it directly affects the magnetic field produced in the solenoid, as seen in the formula B = ?? n I.

Vacuum Permeability (??)

Vacuum permeability, denoted as ??, is a fundamental physical constant that quantifies the ability of a vacuum to support the formation of a magnetic field. Its value, approximately 4? × 10?? T·m/A, is used in electromagnetic formulas including the magnetic field calculation for a solenoid. Understanding this constant is important for calculating the magnetic response of free space when currents generate fields.

Transcript

Please give Ace some feedback