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Magazine Chapter 9: Problem 80
What is the critical magnetic field for lead at \(T=2.8 \mathrm{K} ?\)
Short Answer
Expert verified
The critical magnetic field for lead at \(T = 2.8 \mathrm{K}\) is approximately \(5.51 \times 10^4 \mathrm{A/m}\).
Step by step solution
01
Set up the formula
Write out the formula for the critical magnetic field in terms of temperature:
\[B_c(T) = B_{c0}\Biggl[1-\Biggl(\frac{T}{T_c}\Biggr)^2\Biggr]\]
02
Plug in the values
Plug the given values for \(B_{c0}\), \(T_c\), and \(T\) into the formula:
\[B_c(2.8) = 6.5 \times 10^4 \mathrm{A/m} \Biggl[1-\Biggl(\frac{2.8}{7.2}\Biggr)^2\Biggr]\]
03
Compute the fraction
Compute the fraction inside the brackets:
\[\frac{2.8}{7.2} \approx 0.3889\]
04
Square the fraction
Square the calculated fraction:
\[(0.3889)^2 \approx 0.1513\]
05
Calculate the expression inside the brackets
Subtract the squared fraction from 1:
\[1 - 0.1513 \approx 0.8487\]
06
Multiply the result by \(B_{c0}\)
Multiply the calculated expression by the value of \(B_{c0}\):
\[B_c(2.8) = 6.5 \times 10^4 \mathrm{A/m} \times 0.8487 \approx 5.51 \times 10^4 \mathrm{A/m}\]
The critical magnetic field for lead at \(T = 2.8 \mathrm{K}\) is approximately \(5.51 \times 10^4 \mathrm{A/m}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Temperature Dependence in Superconductors
Temperature plays a crucial role in superconductors, affecting how they behave. Superconductors are special because they can carry electricity without any resistance. This amazing property only happens below a certain temperature, called the "critical temperature."
But what if the temperature changes? Let's explore:
But what if the temperature changes? Let's explore:
- Above the critical temperature, superconductors lose their magic and behave like normal conductors with resistance.
- As the temperature gets closer to this critical point, the properties of superconductors, like their ability to exclude magnetic fields, also change.
- The relationship between the critical magnetic field and temperature is given by a specific formula. This helps to understand exactly how temperature affects superconductivity.
Understanding Superconductivity
Superconductivity is quite a fascinating phenomenon, discovered over a century ago. It refers to the state of a material where it can conduct electricity without any resistance.
Superconductors also have a neat property called the Meissner effect, which means they can repel magnetic fields entirely. This is why they are useful in creating powerful magnets and other technologies.
Superconductors also have a neat property called the Meissner effect, which means they can repel magnetic fields entirely. This is why they are useful in creating powerful magnets and other technologies.
- Not all materials are superconductors. Only certain substances exhibit this property under specific conditions.
- Cooling the material below its critical temperature is crucial for it to become a superconductor.
- Superconductivity is greatly influenced by factors like the type of material, impurities, and structural defects.
Exploring Critical Temperature
The critical temperature is a key concept in understanding superconductivity. It's the maximum temperature at which a material remains in the superconducting state. Beyond this temperature, the material goes back to having electrical resistance.
This temperature varies from material to material. For example:
This temperature varies from material to material. For example:
- Lead, a well-known elemental superconductor, has a critical temperature of about 7.2 Kelvin.
- High-temperature superconductors, like some ceramics, have higher critical temperatures, making them useful for more practical applications.
Lead as a Superconducting Material
Lead is one of the classic examples of superconducting materials. When cooled below its critical temperature of 7.2 Kelvin, lead shows superconducting properties.
Here's what makes lead interesting:
Here's what makes lead interesting:
- It is an "elemental" superconductor, meaning it doesn't require a complicated compound structure to exhibit superconductivity.
- Lead can support significant currents without losing its superconducting qualities, making it a useful material for research and some technology applications.
- Its critical magnetic field, which varies with temperature, determines how strong a magnetic field it can exclude completely.
