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Chapter 28: Problem 40

A 15.0 -cm-long solenoid with radius 2.50 \(\mathrm{cm}\) 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.

Short Answer

Expert verified
The magnetic field near the center of the solenoid is approximately 0.0402 T.

Step by step solution

01

Understand the Formula for Magnetic Field in a Solenoid

The magnetic field inside a long solenoid can be calculated using the formula \( B = \mu_0 \cdot n \cdot I \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A} \)), \( n \) is the number of turns per unit length, and \( I \) is the current.
02

Calculate the Number of Turns per Unit Length

Calculate \( n \), the number of turns per unit length, using \( n = \frac{N}{L} \), where \( N = 600 \) is the total number of turns, and \( L = 15.0 \; \text{cm} = 0.150 \; \text{m} \) is the length of the solenoid. Thus, \( n = \frac{600}{0.150} = 4000 \; \text{turns/m} \).
03

Substitute the Values into the Magnetic Field Formula

Substitute \( n = 4000 \; \text{turns/m} \) and \( I = 8.00 \; \text{A} \) into the formula \( B = \mu_0 \cdot n \cdot I \). Calculate \( B = (4\pi \times 10^{-7}) \cdot 4000 \cdot 8.00 \).
04

Calculate the Magnetic Field

Carry out the calculation: \( B = 4\pi \times 10^{-7} \cdot 4000 \cdot 8.00 = 4.02 \times 10^{-2} \; \text{T} \). Thus, the magnetic field at a point near the center of the solenoid is approximately \( 0.0402 \; \text{T} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solenoid
A solenoid is a long coil of wire wound in a helix shape, often used to generate uniform magnetic fields or create electromagnets.
When a current flows through this coil, it generates a magnetic field inside and around the solenoid. This field is typically strongest and most uniform inside the solenoid. The strength of the magnetic field in a solenoid is influenced by several factors:
  • Number of Turns: Increasing the number of turns in the wire coil can enhance the magnetic field strength.
  • Current: A higher current passing through the wire will result in a stronger magnetic field.
  • Length of the Solenoid: The length of the solenoid determines the field's distribution; a longer solenoid will have a more uniform field.
In our exercise, the solenoid has 600 turns and a specific length and radius, all contributing to determining the resulting magnetic field strength when a current of 8.00 A is applied. Understanding the structure and purpose of a solenoid is crucial in calculating its magnetic field.
Electromagnetism
Electromagnetism is the branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. It is a fundamental interaction in nature and is responsible for practically all phenomena encountered in daily life, aside from gravity.
When current flows through a conductor, like the wire in a solenoid, it produces a magnetic field, an essential principle in electromagnetism.
  • Electric Current: The flow of electric charge, generally in the form of electrons flowing through a wire, generating a magnetic field.
  • Magnetic Fields: Created by moving charges (currents), such as those in a solenoid, these fields can exert forces on moving charges and other magnetic fields.
  • Interaction: The interaction between electric currents and magnetic fields is exploited in various technologies, such as electric motors and transformers.
In our case, electromagnetism explains why a magnetic field is induced in the solenoid when an electric current is applied through its windings.
Permeability of Free Space
The permeability of free space, denoted as \( \mu_0 \), is a physical constant essential in electromagnetism. It describes how well a material (or vacuum) supports the formation of a magnetic field. In mathematical terms, it relates the magnetic field strength to the magnetic induction.
The value of the permeability of free space is \( 4\pi \times 10^{-7} \; \text{T}\cdot\text{m/A} \). It appears in various fundamental equations of electromagnetism, such as the formula for calculating the magnetic field inside a solenoid: \( B = \mu_0 \cdot n \cdot I \).
  • Magnetic Field: The magnetic field inside a solenoid is directly proportional to the permeability of free space, the number of turns per unit length, and the current flowing through the solenoid.
  • Vacuum: While \( \mu_0 \) applies to a vacuum, it also serves as a reference point for other materials with different magnetic properties.
Understanding the role of \( \mu_0 \) is vital for predicting and calculating the strength of magnetic fields in a variety of contexts, particularly in our solenoid calculation exercise.

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Most popular questions from this chapter

A closed curve encircles several conductors. The line integral \(\phi \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} .\) (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 cm carries a current of 0.25 A. The relative permeability of the core is \(80 .\) (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to atomic currents?

A closely wound, circular coil with radius 2.40 \(\mathrm{cm}\) has 800 turns. a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 \(\mathrm{T}\) ? b) At what distance \(x\) from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 \(\mathrm{cm}\) . The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 \(\mathrm{T}\) . Calculate (a) the relative permeability and \((b)\) the magnetic susceptibility of the material that fills the toroid.

A long, straight wire with a circular cross section of radius \(R\) carries a current \(L\) . Assume that the current density is not constant across the cross section of the wire, but rather varies as \(J=\alpha r,\) where \(\alpha\) is a constant. (a) By the requirement that \(J\) integrated over the cross section of the wire gives the total current \(I,\) calculate the constant \(\alpha\) in terms of \(I\) and \(R\) (b) Use Ampere's law to calculate the magnetic field \(B(r)\) for (i) \(r \leq R\) and (ii) \(r \geq R\) . Express your answers in terms of \(I\) .
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