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Chapter 28: Problem 46

A toroidal solenoid with 400 turns of wire and a mean radius of \(6.0 \mathrm{~cm}\) 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 the magnetic moments of the atoms in the core?

Short Answer

Expert verified
The magnetic field in the core is 1.333 T and 1.31633 T of this is due to the magnetic moments of the atoms in the core.

Step by step solution

01

Use Amp猫re's Law

From Amp猫re's law, we can calculate the magnetic field (\(B\)) using the formula \(B = \mu_0 \cdot \mu_r \cdot \frac{N \cdot I}{2\pi r}\) where \(\mu_0\) represents the absolute permeability of free space (\(4\pi x 10^{-7} ~\text{T m/A}\)), \( \mu_r\) is the relative permeability of the core, \(N\) is the number of turns, \(I\) is current and \(r\) is the radius.
02

Substitution

Substitute the given values into the above equation. Thus the magnetic field in the core \(B\) = \(4\pi x 10^{-7} \cdot 80 \cdot \frac{400 \cdot 0.25}{2\pi \cdot 0.06}\) T.
03

Calculation

Upon simplification, we get the magnetic field strength in the core as \(B = 1.333\) T.
04

Understanding part b

To calculate which part of the field is due to the magnetic moments of atoms in the core, we simply compare it with the magnetic field in vacuum (\(B_0\)). This is given by \(B_0 = \mu_0 \cdot \frac{N \cdot I}{2\pi r}\)
05

Substitute and Calculate

Substitute the given values into \(B_0\). Thus \(B_0 = 4\pi x 10^{-7} \cdot \frac{400 \cdot 0.25}{2\pi \cdot 0.06}\) T. Upon simplification, we get \(B_0 = 0.01667\) T. The part of the field due to the magnetic moments of the atoms in the core is therefore \(B - B_0 = 1.333 - 0.01667 = 1.31633\) T.
06

Interpretation

Therefore, the part of the magnetic field due to the magnetic moments of the atoms in the core is significantly higher than the amount of magnetic field due to the circular current in the solenoid coil. This is because the coil is ferromagnetic and has a high relative permeability, which multiplies the existing magnetic field.

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Key Concepts

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

Toroidal Solenoid
A toroidal solenoid is a coil of wire shaped like a donut, where the wire is wound closely and uniformly along a circular path. This unique structure confines the magnetic field generated by the electric current to the interior of the coil, effectively reducing the magnetic field outside the solenoid.
A toroidal solenoid combines several key components:
  • Turns of Wire: The winding, usually many loops, which directly affects the strength of the magnetic field.
  • Core Material: Often made of ferromagnetic materials to enhance magnetic properties.
  • Current: An electric current passing through the coil generates a magnetic field.
The primary advantage in using a toroidal solenoid lies in its ability to create a strong, uniform magnetic field primarily within its core, making it highly efficient for applications needing such a contained magnetic flux.
Magnetic Field Calculation
When calculating the magnetic field within a toroidal solenoid, Amp猫re's Law is a powerful tool. Amp猫re's law relates the integrated magnetic field along a closed loop to the electric current passing through the loop.
For a toroidal solenoid, the magnetic field depends on several factors:
  • Number of Turns (\(N\)): Directly proportional to the magnetic field, with more loops increasing field strength.
  • Current (\(I\)): An increase in the current boosts the magnetic field linearly.
  • Radius (\(r\)): Larger radius reduces the magnetic field, as it is inversely proportional to the magnetic field strength.
  • Core Permeability (\(\mu\)): Comprising absolute permeability (\(\mu_0\)) and relative permeability (\(\mu_r\)), affecting the field intensity.
The calculation uses the formula \[B = \mu_0 \cdot \mu_r \cdot \frac{N \cdot I}{2\pi r}\]. Substituting the known values provides an accurate measure of the magnetic field's strength inside the core.
Relative Permeability
Relative permeability (\(\mu_r\)) is a dimensionless measure indicating how much stronger or weaker a material is in magnetizing than a vacuum. A core with high relative permeability enhances the overall magnetic field produced within a solenoid.
Key points about relative permeability include:
  • Material Influence: Ferromagnetic materials like iron can have relative permeability values much greater than one, significantly amplifying the magnetic field.
  • Impact on Magnetic Field: By increasing the relative permeability, the overall magnetic field inside the solenoid's core increases. This is because the magnetic field is multiplied by \(\mu_r\)
  • Variation by Material: Materials with different compositions have distinct relative permeabilities. This characteristic can be used to tailor the magnetic field for specific applications.
In applications where a potent magnetic field is desirable, choosing a core with a high relative permeability becomes essential, demonstrating the factor's importance in designing toroidal solenoids.

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Most popular questions from this chapter

A \(15.0-\mathrm{cm}\) -long solenoid with radius \(0.750 \mathrm{~cm}\) 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.

Long, straight conductors with square cross sections and each carrying current \(I\) are laid side by side to form an infinite current sheet (Fig. \(\mathbf{P 2 8 . 6 9}\) ). The conductors lie in the \(x y\) -plane, are parallel to the \(y\) -axis, and carry current in the \(+y\) -direction. There are \(n\) conductors per unit length measured along the \(x\) -axis. (a) What are the magnitude and direction of the magnetic field a distance \(a\) below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance \(a\) above the current sheet?

Currents in dc transmission lines can be 100 A or higher. Some people are concerned that the electromagnetic fields from such lines near their homes could pose health dangers. For a line that has current \(150 \mathrm{~A}\) and a height of \(8.0 \mathrm{~m}\) above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percentage of the earth's magnetic field, which is \(0.50 \mathrm{G}\). Is this value cause for worry?

An electron is moving in the vicinity of a long, straight wire that lies along the \(x\) -axis. The wire has a constant current of 9.00 A in the \(-x\) -direction. At an instant when the electron is at point \((0,0.200 \mathrm{~m}, 0)\) and the electron's velocity is \(\overrightarrow{\boldsymbol{v}}=\left(5.00 \times 10^{4} \mathrm{~m} / \mathrm{s}\right) \hat{\imath}-\left(3.00 \times 10^{4} \mathrm{~m} / \mathrm{s}\right) \hat{\jmath},\) what is the force that the wire exerts on the electron? Express the force in terms of unit vectors, and calculate its magnitude.

A wide, long, insulating belt has a uniform positive charge per unit area \(\sigma\) on its upper surface. Rollers at each end move the belt to the right at a constant speed \(v .\) Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (Hint: At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem \(28.69 .\)
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