/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 47 The compound \(\mathrm{SiV}_{3}\... [FREE SOLUTION] | 91影视

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Chapter 29: Problem 47

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{\mathrm{cl}}=55.0 \mathrm{mT}\) and \(B_{\mathrm{c} 2}=15.0 \mathrm{T}\) . The normal phase of \(\mathrm{Si} \mathrm{V}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an external magnetic field \(\vec{\boldsymbol{B}}_{0}\) in the \(+x\) -direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the \(x\) -axis. At a temperature near absolute zero, the external magnetic field is slowly increased from zero. What are the resultant magnetic field \(\vec{\boldsymbol{B}}\) and the magnetization \(\vec{M}\) inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal?

Short Answer

Expert verified
(a) \( \vec{B} = 0 \), \( \vec{M} = -\frac{B_{cl}}{\mu_0} \). (b) \( \vec{B} = B_0 \), \( \vec{M} \approx 0 \).

Step by step solution

01

Understanding Type-II Superconductor

A type-II superconductor allows partial penetration of magnetic fields between its two critical fields, \( B_{cl} \) and \( B_{c2} \). Below \( B_{cl} \), it expels all magnetic fields (Meissner state), and above \( B_{c2} \), it becomes normal and allows all magnetic fields to penetrate.
02

Analyzing the Initial State

Initially, when the external magnetic field \( \vec{B}_0 \) is below \( B_{cl} = 55.0 \) mT, the magnetic field inside the superconductor is completely expelled, meaning \( \vec{B} = 0 \) inside.
03

Calculating Resultant Magnetic Field at \( B_{cl} \)

Just before the flux begins to penetrate, when \( B_0 = B_{cl} = 55.0 \) mT, the external magnetic field is exactly countered by the superconductor, so \( \vec{B} = 0 \) inside.
04

Determining Magnetization at \( B_{cl} \)

The magnetization \( \vec{M} \) is defined such that \( \vec{B} = \mu_0(\vec{H} + \vec{M}) \), where \( \vec{H} \) is the applied magnetic field. Since \( \vec{B} = 0 \) and \( \vec{H} = \frac{\vec{B}_0}{\mu_0} \), the magnetization \( \vec{M} = -\vec{H} = -\frac{B_{cl}}{\mu_0} \).
05

Analyzing the Final State

Once \( B_0 \) exceeds \( B_{c2} = 15.0 \) T, the superconductor becomes normal. The magnetic field inside equals the external field, so \( \vec{B} = B_0 \).
06

Determining Magnetization at \( B_{c2} \)

In the normal state, the magnetic susceptibility is nearly zero, so magnetization \( \vec{M} \approx 0 \). Thus, \( \vec{B} = \mu_0 \vec{H} = \vec{B}_0 \).

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Key Concepts

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

Type-II superconductors
Type-II superconductors are fascinating materials that behave uniquely when exposed to magnetic fields. Most superconductors are categorized into two types, I and II. Type-I superconductors completely expel all magnetic fields when they are below their critical temperature. This expulsion is known as the Meissner effect. However, Type-II superconductors behave a bit differently.
  • They have two critical magnetic fields, denoted as \(B_{cl}\) and \(B_{c2}\).
  • In the region between \(B_{cl}\) and \(B_{c2}\), called the "mixed state" or "vortex state," magnetic flux penetrates the superconductor in quantized units known as vortices.
  • Below \(B_{cl}\), the material is in the Meissner state, expelling all magnetic fields.
  • Above \(B_{c2}\), the material reverts to a normal state, where magnetic fields penetrate freely.
Understanding these phases is crucial when working with type-II superconductors, as it impacts their applications in magnetic field environments.
Meissner effect
The Meissner effect is a fundamental property of superconductors, which allows them to expel magnetic fields completely when cooled below a certain critical temperature. This effect provides the superconductor with one of its essential characteristics, the ability to maintain zero resistance.
  • In Type-II superconductors, the Meissner effect is observed below the first critical field \(B_{cl}\).
  • This state results in the magnetic field inside the superconductor being zero, ensuring effective expulsion of magnetic flux.
  • Because of the Meissner effect, superconductors can "levitate" above a magnet, a phenomenon visible in several real-world applications.
In Type-II superconductors, the Meissner effect is succeeded by a mixed state as the field strengths reach \(B_{cl}\) and then penetrates through vortices, demonstrating a unique relationship with external magnetic fields.
Magnetic susceptibility
Magnetic susceptibility is an important concept when dealing with superconductors. It is a measure of how a material reacts to an external magnetic field, indicating how much a material will become magnetized in response.
  • In the Meissner state of a superconductor, the magnetic susceptibility is highly negative, indicating complete expulsion of the magnetic field.
  • For the normal phase of some superconductors, like \(\text{SiV}_3\), the susceptibility is close to zero. This means they do not significantly attract or repel the magnetic field.
  • During the transitions between the fully superconducting states and the normal state, the susceptibility changes, reflecting the material's internal adjustments to external fields.
Understanding magnetic susceptibility in superconductors is essential for controlling their magnetic environments and optimizing their performance in applications.
Critical magnetic fields
Critical magnetic fields are thresholds that define the distinct magnetic phases within superconductors, especially for Type-II superconductors. These are crucial for understanding how a superconductor behaves under varying magnetic field intensities.
  • The first critical field, \(B_{cl}\), represents the maximum external magnetic field strength that a superconductor can completely expel.
  • Beyond \(B_{cl}\) but before reaching \(B_{c2}\), the superconductor allows magnetic flux to enter in discrete vortices, maintaining partial superconductivity.
  • The second critical field, \(B_{c2}\), marks the transition to a normal state where superconductivity is lost entirely, and the material can no longer repel magnetic fields.
Knowing these values is pivotal for using superconductors in devices that require their unique properties, such as magnetic levitation and advanced MRI machines.

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Most popular questions from this chapter

CALC A dielectric of permittivity \(3.5 \times 10^{-11} \mathrm{F} / \mathrm{m}\) completely fills the volume between two capacitor plates. For \(t>0\) the electric flux through the dielectric is \(\left(8.0 \times 10^{3} \mathrm{V} \cdot \mathrm{s} / \mathrm{s}^{3}\right) t^{3}\) . The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21\(\mu \mathrm{A} ?\)

CP CALC A square, conducting, wire loop of side \(L\) , total mass \(m,\) and total resistance \(R\) initially lies in the horizontal \(x y\) -plane, with corners at \((x, y, z)=(0,0,0),(0, L, 0),(L, 0,0)\) and \((L, L, 0) .\) There is a uniform, upward magnetic field \(\vec{\boldsymbol{B}}=\hat{B hat{\boldsymbol{k}}}\) in the space within and around the loop. The side of the loop that extends from \((0,0,0)\) to \((L, 0,0)\) is held in place on the \(x\) -axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the net torque (magnitude and direction) that acts on the loop when it has rotated through an angle \(\phi\) from its original orientation and is rotating downward at an angular speed \(\omega\) . (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through \(90^{\circ} ?\) Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.

A Magnetic Exercise Machine. You have designed a new type of exercise machine with an extremely simple mechanism (Fig. E29.28). A vertical bar of silver (chosen for its low resistivity and because it makes the machine look cool) with length \(L=3.0 \mathrm{m}\) is free to move left or right without friction on silver \(L=3.0 \mathrm{m}\) is free to move left or right without friction on silver rails. The entire apparatus is placed in a horizontal, uniform magnetic field of strength 0.25 \(\mathrm{T}\) . When you push the bar to the left or right, the bar's motion sets up a current in the circuit that includes the bar. The resistance of the bar and the rails can be neglected. The magnetic field exerts a force on the current-carrying bar, and this force opposes the bar's motion. The health benefit is from the exercise that you do in working against this force, (a) Your design goal is that the person doing the exercise is to do work at the rate of 25 watts when moving the bar at a steady 2.0 \(\mathrm{m} / \mathrm{s}\) . What should be the resistance \(R ?\) (b) You decide you want to be able to vary the power required from the person, to adapt the machine to the person's strength and fitness. If the power is to be increased to 50 \(\mathrm{W}\) by altering \(R\) while leaving the other design parameters constant, should \(R\) be increased or decreased? Calculate the value of \(R\) for 50 \(\mathrm{W} .\) (c) When you start to construct a prototype machine, you find it is difficult to produce a \(0.25-\mathrm{T}\) magnetic field over such a large area. If you decrease the length of the bar to 0.20 \(\mathrm{m}\) while leaving \(B, v,\) and \(R\) the same as in part (a), what will be the power required of the person?

CALC In Fig. 29.22 the capacitor plates have area 5.00 \(\mathrm{cm}^{2}\) and separation 2.00 \(\mathrm{mm}\) . The plates are in vacuum. The charging current \(i_{\mathrm{C}}\) has a constant value of 1.80 \(\mathrm{mA} .\) At \(t=0\) the charge on the plates is zero. (a) Calculate the charge on the plates, the electric field between the plates, and the potential difference between the plates when \(t=0.500 \mu \mathrm{s}\) (b) Calculate \(d E / d t\) , the time rate of change of the electric field between the plates. Does \(d E / d t\) vary in time? (c) Calculate the displacement current density \(j_{\mathrm{D}}\) between the plates, and from this the total displacement current \(i_{\mathrm{D}} .\) How do \(i_{\mathrm{C}}\) and \(i_{\mathrm{D}}\) compare?

A long, thin solenoid has 900 turns per meter and radius 2.50 \(\mathrm{cm} .\) The current in the solenoid is increasing at a uniform rate of 60.0 \(\mathrm{A} / \mathrm{s}\) . What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 \(\mathrm{cm}\) from the axis of the solenoid; (b) 1.00 \(\mathrm{cm}\) from the axis of the solenoid?
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