·A solenoid is designed to produce a 0.0279 T magnetic field...

$\cdot$ A solenoid is designed to produce a 0.0279 T magnetic field near its center. It has a radius of 1.40 $\mathrm{cm}$ and a length of $40.0 \mathrm{cm},$ and the wire carries a current of 12.0 A. (a) How many turns must the solenoid have? (b) What total length of wire is required to make this solenoid?

Key Concepts

Magnetic Field in a Solenoid

This concept describes how the magnetic field inside a long, tightly wound solenoid is uniform and is determined by the current flowing through the wire and the number of turns per unit length. The relation B = ??(n)I is central, where ?? is the permeability of free space, n is the turn density, and I is the current, illustrating the direct link between the solenoid’s geometry and the magnetic field produced.

Ampere's Law

Ampere's Law relates the magnetic field in space to the current that produces it, encapsulated in the integral form ?B·dl = ??I_enc. This law provides the theoretical foundation for deriving the expression of the magnetic field inside a solenoid, thus bridging the geometry and current distribution with magnetic effects.

Turn Density and Number of Turns

Turn density is the number of loops or turns per unit length in a solenoid, and it directly influences the strength of the magnetic field inside. Understanding and calculating the total number of turns required for a given solenoidal length ensures that the designed solenoid produces the requisite magnetic field.

Calculation of Wire Length

When designing a solenoid, it is often necessary to calculate the total length of wire needed. This involves determining the circumference of one loop and multiplying it by the total number of turns, which is critical for both material estimation and understanding the physical layout of the solenoid.

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