Ch 28: Sources of Magnetic Field

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 28: Sources of Magnetic Field

Problem 47b

Chapter 28, Problem 47b

A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 cm and length 40.0 cm, and the wire can carry a maximum current of 12.0 A. What total length of wire is required?

Verified step by step guidance

First, understand that the magnetic field inside a solenoid is given by the formula:

B = μ

₀nI, where

B

is the magnetic field,

μ

₀ is the permeability of free space,

n

is the number of turns per unit length, and

I

is the current.

Rearrange the formula to solve for

n

n = \frac{B}{μ}

₀I. Substitute the given values:

B = 0.0270

I = 12.0

A, and

μ

₀ = 4π × 10^-7 T·m/A.

Calculate the number of turns per unit length

n

using the rearranged formula.

Once

n

is found, calculate the total number of turns

N

by multiplying

n

by the length of the solenoid:

N = n × 0.40

Finally, calculate the total length of wire required by multiplying the number of turns

N

by the circumference of the solenoid:

L = N × 2π × 0.014

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Field of a Solenoid

The magnetic field inside a solenoid is uniform and parallel to its axis, given by B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. Understanding this formula is crucial for calculating the required parameters of the solenoid.

Number of Turns in a Solenoid

The number of turns per unit length, n, is a key factor in determining the magnetic field strength of a solenoid. It is calculated as n = N/L, where N is the total number of turns and L is the length of the solenoid. This concept helps in understanding how the solenoid's design affects its magnetic properties.

Length of Wire in a Solenoid

The total length of wire required for a solenoid is determined by the number of turns and the circumference of the solenoid. It is calculated using the formula: total wire length = N × 2πr, where N is the number of turns and r is the radius of the solenoid. This concept is essential for solving the problem of wire length needed to achieve the desired magnetic field.

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Designing a Solenoid (Total Length of Wire)

Related Practice

Textbook Question

A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c (Fig. E28.43). The central conductor and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field at points outside the tube (r > c).

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Textbook Question

A solenoid is designed to produce a magnetic field of 0.0270 T at its center. It has radius 1.40 cm and length 40.0 cm, and the wire can carry a maximum current of 12.0 A. What minimum number of turns per unit length must the solenoid have?

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Textbook Question

A 15.0 cm long solenoid with radius 0.750 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.

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