   Chapter 19, Problem 61P

Chapter
Section
Textbook Problem

It is desired to construct a solenoid that will have a resistance of 5.00 Ω (at 20°C) and produce a magnetic field of 4.00 × 10−2 T at its center when it carries a current of 4.00 A. The solenoid is to be constructed from copper wire having a diameter of 0.500 mm. If the radius of the solenoid is to be 1.00 cm, determine (a) the number of turns of wire needed and (b) the length the solenoid should have.

a)

To determine
The number of turns in the solenoid.

Explanation

Given info: The required resistance of the solenoid is 5.00Ω . The required magnetic field at its center is 4.00×102T . The current in the coil is 4.00A . The diameter of the copper wire to be used is 0.500mm . The radius of the solenoid is 1.00cm .

Explanation:

The resistance of an wire is given by,

R=ρLA

• ρ is the resistivity of the wire material
• L is the length of the conductor
• A is area of cross-section of the wire

The length required to acquire the required resistance is,

L=RAρ       (1)

The number of turns required to get the necessary length is,

N=L2πr       (2)

• r is the radius of the solenoid

Combining (1) and (2),

N=12πr(RAρ)

The area of cross-section of the copper wire is,

A=πd24

• d is the diameter of the wire

Hence the number of turns is,

N=R2πrρ(πd24)

Substitute 5.00Ω for R , 0

b)

To determine
The required length of the solenoid.

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