(I) A 32 -cm-long solenoid, 1.8 cm in diameter, is to...

(I) A 32 -cm-long solenoid, $1.8 \mathrm{~cm}$ in diameter, is to produce a 0.30-T magnetic field at its center. If the maximum current is $4.5 \mathrm{~A}$, how many turns must the solenoid have?

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Key Concepts

Permeability of Free Space (??)

This constant represents the permeability of free space and is essential in electromagnetic calculations, including those for solenoids. It provides the proportionality factor in the relationship between the current and the magnetic field it generates in a vacuum, which is crucial for designing and analyzing electromagnetic devices.

Magnetic Field in a Solenoid

Inside a long solenoid, the magnetic field is uniform and can be calculated using 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. This principle is fundamental in understanding how electric current produces a magnetic field in coiled conductors.

Turn Density and Total Number of Turns

Turn density, defined as the number of turns per unit length (n = N/L), is critical when designing solenoids to produce a desired magnetic field. By knowing the required field strength and maximum current, one can rearrange the field formula to calculate the total number of turns, ensuring that the solenoid meets performance specifications.

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