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29. Sources of Magnetic Field

Magnetic Field Produced by Loops andSolenoids

Physics

29. Sources of Magnetic Field

Magnetic Field Produced by Loops andSolenoids

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5:45 minutes

Problem 47b

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?

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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Related Practice

Multiple Choice

The two tightly wound solenoids below both have length 40 cm and current 5 A in the directions shown. The left solenoid has radius 20 cm and 50 m of total wire. The right solenoid has radius 0.5 m and 314 m of total wire. The thinner solenoid is placed entirely inside the wider one so their central axes perfectly overlap. Assume wires don't touch. What is the magnitude and direction of the magnetic field that is produced by a combination of the two solenoids at their central axis?

(Note:your worksheet may have a typo and say "0.5 cm"for the right solenoid's radius; it should be 0.5 m.)

A certain solenoid is

30 cm

long and has 1500 turns of wire. If the magnetic field inside the solenoid is

30 mT

, what is the current in the wire?

599

views

Multiple Choice

A loop of wire has a magnetic dipole moment of

0.80 A m^{2}

_. What is the on-axis magnetic field strength

7.0 cm

A square loop of wire with

12 cm

on a side carries a

10 A

current. What is the size of its magnetic dipole moment?

611

views

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?

935

views

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.

1238

views

Textbook Question

FIGURE CP29.79 is an edge view of a 2.0 kg square loop, 2.5 m on each side, with its lower edge resting on a frictionless, horizontal surface. A 25 A current is flowing around the loop in the direction shown. What is the strength of a uniform, horizontal magnetic field for which the loop is in static equilibrium at the angle shown?

views

Textbook Question

Determine the field strength at the center of a current-carrying square loop having sides of length 2R.

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