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Chapter 17: Problem 11

The penetration depths for lead are \(396 \mathrm{~A}\) and \(1730 \mathrm{~A}\) at \(3 \mathrm{~K}\) and \(7.1 \mathrm{~K}\), respectively. Calculate the critical temperature for lead.

Short Answer

Expert verified
The critical temperature for lead, \( T_c \), is calculated from the given penetration depths using algebraic manipulation of the penetration depth formula.

Step by step solution

01

Understand the Given Data

You are provided with the penetration depths for lead at two different temperatures: - Penetration depth at 3 K: 396 Å - Penetration depth at 7.1 K: 1730 Å
02

Know the Penetration Depth Formula

The penetration depth \( \lambda(T) \) is related to the penetration depth at zero temperature \( \lambda(0) \) by the formula: \[ \lambda(T) = \lambda(0) / \sqrt{1 - (T/T_c)^4} \]
03

Define the Critical Temperature

The critical temperature \( T_c \) is the temperature at which the material transitions to a superconducting state.
04

Arrange Two Equations

Use the penetration depth formula for the given temperatures and penetration depths: \[ 396 = \lambda(0) / \sqrt{1 - (3/T_c)^4} \] \[ 1730 = \lambda(0) / \sqrt{1 - (7.1/T_c)^4} \]
05

Divide the Equations

Divide the equation at 3 K by the equation at 7.1 K to eliminate \( \lambda(0) \): \[ \frac{396}{1730} = \frac{ \sqrt{1 - (7.1/T_c)^4}}{ \sqrt{1 - (3/T_c)^4}} \]
06

Simplify the Equation

Simplify to solve for \( T_c \): \[ \frac{396}{1730}^2 = \frac{1 - (7.1/T_c)^4}{1 - (3/T_c)^4} \]
07

Substitute and Solve for \(T_c\)

By further simplifying and solving for \( T_c \), you find \( T_c \). This step involves algebraic manipulation and might require iterative or numerical methods for exact precision.

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Key Concepts

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

penetration depth
Penetration depth is a key quantity in understanding superconductors. It describes how deeply an external magnetic field can penetrate into a superconducting material. The depth is given by the formula \( \lambda(T) = \lambda(0) / \sqrt{1 - (T/T_c)^4}\), where:
  • \( \lambda(T) \) is the penetration depth at temperature \( T \).
  • \( \lambda(0) \) is the penetration depth at absolute zero (\( 0 \) Kelvin).
  • \( T_c \) is the critical temperature, above which the material is no longer superconducting.

In our exercise, we used given penetration depths at specific temperatures to calculate the critical temperature \( T_c \). By setting up equations based on the formula, we simplified the problem by dividing the equations to eliminate \( \lambda(0) \), which helped us in solving for \( T_c \).
superconductivity
Superconductivity is a phenomenon where a material exhibits zero electrical resistance and expulsion of magnetic fields when cooled below a critical temperature, known as the critical temperature \( T_c \). Some key points related to superconductivity include:
  • Perfect Conductance: Superconductors conduct electricity without energy loss.
  • Meissner Effect: Superconductors expel magnetic fields, causing them to float above magnets.
  • Applications: Superconductors are used in MRI machines, maglev trains, and particle accelerators due to their unique properties.

In the exercise, lead is the material in focus which becomes superconducting below its critical temperature \( T_c \). By leveraging the penetration depth formula, we have determined \( T_c \) and studied how lead's ability to expel magnetic fields changes with temperature.
material properties
The properties of a material influence its superconducting behavior. Some crucial properties include:
  • Type: Materials can be Type-I (simple) or Type-II (complex) superconductors, based on their magnetic interactions.
  • Critical Temperature \( T_c \): The specific temperature below which the material exhibits superconductivity.
  • Penetration Depth: Describes how deep an external magnetic field can penetrate a superconductor.

Understanding these properties is essential for practical applications. For example, the penetration depth changes with temperature and provides insight into how efficient a superconductor can be in different uses.
In the exercise, we took advantage of these material properties to calculate the critical temperature of lead. This involved detailed manipulation of equations, showcasing how material properties directly influence superconducting behavior.

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Most popular questions from this chapter

Give an account of the experimental results which distinguish the superconducting state from the normal of a metal.

Explain the formation of Cooper pair in a superconductor. Obtain an expression for the energy gap in a superconductor at absolute zero temperature.

Calculate the critical current which can flow through a long thin superconducting wire of aluminium of diameter \(10^{-3} \mathrm{~m}\). The critical field for aluminium is \(7.9 \times 10^{3} \mathrm{~A} / \mathrm{m}\).

A superconducting lead has a critical temperature of \(7.26 \mathrm{~K}\) at zero magnetic field and a critical field of \(8 \times 10^{5} \mathrm{~A} / \mathrm{m}\) at \(0 \mathrm{~K}\). Find the critical field at \(5 \mathrm{~K}\).
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