Problem 43 - University Physics with Modern Physics

Question

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{c 1}=\) 55.0 \(\mathrm{mT}\) and \(B_{c 2}=15.0 \mathrm{T}\) . The normal phase of \(\mathrm{SiV}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an extermal magnetic field \(\vec{B}_{0}\) 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 \(\overrightarrow{\boldsymbol{B}}\) 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?

Step-by-Step Solution

Answer

Before flux penetration: \\overrightarrow{\boldsymbol{B}} = \vec{0}, \ \vec{M} = -\vec{B}_{0}. After material becomes normal: \\overrightarrow{\boldsymbol{B}} = \vec{B}_{0}, \ \vec{M} = \vec{0}.

Step 1: Understanding Type-II Superconductors

Type-II superconductors, like \mathrm{SiV}_{3}, have two critical magnetic fields: \(B_{c1}\) and \(B_{c2}\). Below \(B_{c1}\), the material is in the Meissner state (completely superconducting with no magnetic flux inside). Between \(B_{c1}\) and \(B_{c2}\), the material is in the mixed state, where magnetic flux penetrates the material in the form of vortices. Above \(B_{c2}\), the material becomes normal.

Step 2: Analyzing Condition Just Before Flux Penetration (a)

Just before the magnetic flux begins to penetrate the \mathrm{SiV}_{3} cylinder, the external magnetic field \(B_{0}\) is less than \(B_{c1} = 55.0 \, ext{mT}\). The cylinder is in the Meissner state, meaning \overrightarrow{oldsymbol{B}} = \vec{0}\ and \vec{M} = -\vec{B}_{0} \, ext{(complete expulsion of field)}\.

Step 3: Analyzing Condition Just After Material Becomes Normal (b)

When the external magnetic field exceeds \(B_{c2} = 15.0 \, T\), the \mathrm{SiV}_{3} becomes completely normal. Near absolute zero, the normal phase of \mathrm{SiV}_{3} has a magnetic susceptibility close to zero, meaning \overrightarrow{oldsymbol{B}} = \vec{B}_{0}\, and \vec{M} = \vec{0} \, ext{(no magnetization)}\.

Key Concepts

Magnetic Fields

Magnetic fields are crucial in understanding the behavior of superconductors. They are represented by the vector \( \vec{B} \), which indicates the direction and strength of the field. In superconductors, the response to magnetic fields can vary dramatically depending on the material's phase and the external conditions.

- **Type-II Superconductors**: These have two critical fields. At low magnetic fields (below \( B_{c1} \)), they expel all magnetic lines, behaving as perfect diamagnets. This is known as the Meissner effect. As the field increases beyond \( B_{c1} \) but remains below \( B_{c2} \), magnetic fields begin to penetrate the superconductor in quantized vortices.

Understanding how magnetic fields interact with superconductors is key to utilizing them in applications like MRI machines and maglev trains, which rely on high magnetic flux capacities.

Meissner State

The Meissner state is a defining feature of superconductors when they are below their first critical magnetic field \( B_{c1} \). In this state, superconductors expel all external magnetic fields, exhibiting an expulsion known as the "Meissner effect."

- **Characteristics of Meissner State**:
- **Zero Magnetic Field Inside**: The internal magnetic field \( \vec{B} \) is zero. This is due to the complete expulsion of magnetic flux lines.
- **Surface Currents**: These currents flow on the surface of the superconductor and generate a field that exactly cancels the external field.

This creates a situation where the superconductor appears to "levitate" a magnet placed above it, providing a visible demonstration of quantum mechanical principles at play.

Critical Magnetic Fields

Critical magnetic fields in type-II superconductors provide information on when and how superconductivity is maintained or lost.

- **First Critical Field (\( B_{c1} \))**:
- **Value**: For \( \text{SiV}_3 \), \( B_{c1} = 55.0 \, \text{mT} \).
- **Significance**: Below this value, the superconductor is in the Meissner state with no magnetic flux inside.

- **Second Critical Field (\( B_{c2} \))**:
- **Value**: For \( \text{SiV}_3 \), \( B_{c2} = 15.0 \, \text{T} \).
- **Significance**: Beyond this threshold, the superconductor becomes normal and loses all superconducting properties, allowing the magnetic field to fully permeate.

These thresholds help determine the operational limits of materials in magnetic environments, crucial for engineering superconducting technologies.

Magnetization

Magnetization \( \vec{M} \) measures how much a material becomes magnetized in an external magnetic field. It gives insight into the internal magnetic responses of superconductors.

- **In the Meissner State**:
- The magnetization is equal and opposite to the external field, \( \vec{M} = -\vec{B}_0 \). This results from the expulsion of the field so that no internal field remains.

- **When Becomes Normal (Above \( B_{c2} \))**:
- The superconducting properties vanish, and the material's magnetization disappears, \( \vec{M} = 0 \). The magnetic field internal to the material is equal to the applied external field, \( \vec{B} = \vec{B}_0 \).

Understanding magnetization is vital to creating applications where controlling and manipulating magnetic fields is necessary, especially in advanced materials and devices.