Ch 28: Sources of Magnetic Field

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 28: Sources of Magnetic Field

Problem 43a

Chapter 28, Problem 43a

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?

Verified step by step guidance

Start by recalling the formula for the magnetic field inside a solenoid:

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.

Identify the given values:

B = 0.0270

I = 12.0

A, and

μ

₀ = 4π × 10^-7 T·m/A.

Rearrange the formula to solve for

n

n = B / (μ

₀I).

Substitute the known values into the rearranged formula:

n = 0.0270 / (4π × 10

^-7 × 12.0).

Calculate the expression to find the minimum number of turns per unit length required for the solenoid to produce the desired magnetic field.

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Key Concepts

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

Magnetic Field in a Solenoid

The magnetic field inside a solenoid is uniform and parallel to the axis of the solenoid. It is given by the formula 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 relationship is crucial for calculating the required number of turns per unit length to achieve a specific magnetic field.

Permeability of Free Space

Permeability of free space, denoted as μ₀, is a fundamental physical constant that describes the ability of a vacuum to support a magnetic field. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is essential in calculating the magnetic field produced by a solenoid, as it directly influences the strength of the field for a given current and number of turns.

Turns per Unit Length

Turns per unit length, represented by n, is a measure of how many loops of wire are present per unit length of the solenoid. It is a critical factor in determining the magnetic field strength inside the solenoid. To find the minimum number of turns per unit length required to produce a specific magnetic field, one must rearrange the solenoid magnetic field formula to solve for n, considering the given current and desired field strength.

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Related Practice

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 total length of wire is required?

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Textbook Question

As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near the center of the solenoid. You have enough wire for 4000 circular turns. This solenoid must be 55.0 cm long and 2.80 cm in diameter. What current will you need to produce the necessary field?

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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 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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