Question

Compare this problem with Problem 63 in Chapter 26 on the force attracting a perfect dielectric into a strong electric field. A fundamental property of a Type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in the photograph on page 855 and again in Figure $30.34$, and described as follows: The superconducting material has $\mathbf{B}=0$ everywhere inside it. If a sample of the material is placed into an externally produced magnetic field, or if it is cooled to become superconducting while it is in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field zero throughout the interior of the sample. The following problem will help you to understand the magnetic force that can then act on the superconducting sample. Consider a vertical solenoid with a length of $120 \mathrm{~cm}$ and a diameter of $2.50 \mathrm{~cm}$ consisting of 1400 turns of copper wire carrying a counterclockwise current of $2.00 \mathrm{~A}$, as shown in Figure P32.79a. (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the energy density of the magnetic field, and note that the units $\mathrm{J} / \mathrm{m}^{3}$ of energy density are the same as the units $\mathrm{N} / \mathrm{m}^{2}(=\mathrm{Pa})$ of pressure. (c) A superconducting bar $2.20 \mathrm{~cm}$ in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is small. The lower end of the bar is deep inside the solenoid. Identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure P32.79b, and the total field is sketched in Figure P32.79c. (d) The field of the solenoid exerts a force on the current in the superconductor. Identify the direction of the force on the bar. (e) Calculate the magnitude of the force by multiplying the energy density of the solenoid field by the area of the bottom end of the superconducting bar.

Name: Problem 79, Chapter 32: Fundamentals of Physics
Uploaded: 2021-06-16T13:38:23-08:00
Duration: 5 min 33 s
Channel: Keshav Singh

   Compare this problem with Problem 63 in Chapter 26 on the force attracting a perfect dielectric into a strong electric field. A fundamental property of a Type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in the photograph on page 855 and again in Figure $30.34$, and described as follows: The superconducting material has $\mathbf{B}=0$ everywhere inside it. If a sample of the material is placed into an externally produced magnetic field, or if it is cooled to become superconducting while it is in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field zero throughout the interior of the sample. The following problem will help you to understand the magnetic force that can then act on the superconducting sample.

Consider a vertical solenoid with a length of $120 \mathrm{~cm}$ and a diameter of $2.50 \mathrm{~cm}$ consisting of 1400 turns of copper wire carrying a counterclockwise current of $2.00 \mathrm{~A}$, as shown in Figure P32.79a. (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the energy density of the magnetic field, and note that the units $\mathrm{J} / \mathrm{m}^{3}$ of energy density are the same as the units $\mathrm{N} / \mathrm{m}^{2}(=\mathrm{Pa})$ of pressure. (c) A superconducting bar $2.20 \mathrm{~cm}$ in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is small. The lower end of the bar is deep inside the solenoid. Identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure P32.79b, and the total field is sketched in Figure P32.79c. (d) The field of the solenoid exerts a force on the current in the superconductor. Identify the direction of the force on the bar. (e) Calculate the magnitude of the force by multiplying the energy density of the solenoid field by the area of the bottom end of the superconducting bar.

Fundamentals of Physics

David Halliday,… 8th Edition

Chapter 32, Problem 79

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Solved by Expert Keshav Singh

06/16/2021

Step 1

00 A, and has a length of 120 cm. We can use these values to find the magnetic field inside the solenoid using the formula for the magnetic field inside a solenoid: \[B = \mu_0 \cdot \frac{N \cdot I}{L}\] where $B$ is the magnetic field, $\mu_0$ is the Show more…

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Compare this problem with Problem 63 in Chapter 26 on the force attracting a perfect dielectric into a strong electric field. A fundamental property of a Type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in the photograph on page 855 and again in Figure $30.34$, and described as follows: The superconducting material has $\mathbf{B}=0$ everywhere inside it. If a sample of the material is placed into an externally produced magnetic field, or if it is cooled to become superconducting while it is in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field zero throughout the interior of the sample. The following problem will help you to understand the magnetic force that can then act on the superconducting sample. Consider a vertical solenoid with a length of $120 \mathrm{~cm}$ and a diameter of $2.50 \mathrm{~cm}$ consisting of 1400 turns of copper wire carrying a counterclockwise current of $2.00 \mathrm{~A}$, as shown in Figure P32.79a. (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the energy density of the magnetic field, and note that the units $\mathrm{J} / \mathrm{m}^{3}$ of energy density are the same as the units $\mathrm{N} / \mathrm{m}^{2}(=\mathrm{Pa})$ of pressure. (c) A superconducting bar $2.20 \mathrm{~cm}$ in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is small. The lower end of the bar is deep inside the solenoid. Identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure P32.79b, and the total field is sketched in Figure P32.79c. (d) The field of the solenoid exerts a force on the current in the superconductor. Identify the direction of the force on the bar. (e) Calculate the magnitude of the force by multiplying the energy density of the solenoid field by the area of the bottom end of the superconducting bar.

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Key Concepts

Superconductivity and the Meissner Effect

Superconductivity is the state of certain materials at low temperatures where they exhibit zero electrical resistance. A key manifestation of superconductivity is the Meissner effect, which is the expulsion of magnetic field lines from the interior of a superconductor. This perfect diamagnetism ensures that the magnetic field inside the superconducting material is zero by inducing surface currents that cancel the internal field.

Magnetic Field in a Solenoid

A solenoid is a coil of wire that generates a nearly uniform magnetic field in its interior when an electric current passes through it. The magnetic field inside the solenoid can be calculated using Ampère’s law, based on the current, the number of turns per unit length, and the geometry of the solenoid. This concept is fundamental in understanding how magnetic fields are produced and controlled in laboratory and practical applications.

Magnetic Energy Density

Magnetic energy density refers to the amount of energy stored in a magnetic field per unit volume. It is mathematically expressed in terms of the magnetic field strength and the permeability of the medium. Importantly, the units of energy density (J/m³) are equivalent to the units of pressure (N/m² or Pascals), linking magnetic fields to mechanical forces through energy considerations.

Magnetic Pressure and Force

The concept of magnetic pressure emerges from the energy density of a magnetic field. When a magnetic field is present, it exerts a pressure analogous to a mechanical pressure, leading to forces when there are spatial variations in this energy density. This pressure can produce a net force on objects with induced currents, such as superconducting materials, particularly when they are subject to a non-uniform magnetic field.

Induced Surface Currents

Induced surface currents are generated on the boundary of a superconducting material when it is exposed to an external magnetic field. These currents flow in such a direction as to oppose the applied field inside the superconductor, ensuring that the internal magnetic field remains zero. This phenomenon is an application of Lenz's law and is critical in understanding how superconductors respond to external magnetic influences.

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