Question

The level of liquid helium (temperature ≈ 4K) in its storage tank can be monitored using a vertically aligned niobium–titanium (NbTi) wire, whose length ℓ spans the height of the tank. In this level-sensing setup, an electronic circuit maintains a constant electrical current I at all times in the NbTi wire and a voltmeter monitors the voltage V across this wire. The NbTi wire is superconducting ( R = 0) if below its transition temperature of 10 K, so the portion of the wire immersed in the liquid helium is in the superconducting state, while the portion above the liquid (in helium vapor with temperature above 10 K) is in the normal state. Define ƒ = x/ℓ to be the fraction of the tank filled with liquid helium (Fig. 25–40) and V₀ to be the value of the voltage V when the tank is empty (ƒ = 0) . Determine the relation between f and V (in terms of V₀).

Accepted Answer

Step 1: Understand the physical setup. The NbTi wire is superconducting (R = 0) below its transition temperature of 10 K, meaning no voltage drop occurs in the portion of the wire immersed in liquid helium (temperature ≈ 4 K). The portion of the wire above the liquid helium, in helium vapor (temperature \u003e 10 K), is in the normal state and contributes to the voltage drop. Step 2: Define the fraction of the tank filled with liquid helium, ƒ = x/ℓ, where x is the length of the wire immersed in liquid helium and ℓ is the total length of the wire spanning the tank. When the tank is empty (ƒ = 0), the entire wire is in the normal state, and the voltage across the wire is V₀. Step 3: Recognize that the voltage V across the wire is proportional to the resistance of the portion of the wire in the normal state. Since the resistance is proportional to the length of the wire in the normal state, the voltage V will decrease as more of the wire becomes superconducting (immersed in liquid helium). Step 4: Express the length of the wire in the normal state as ℓ(1 - ƒ), where (1 - ƒ) represents the fraction of the wire above the liquid helium. The voltage V is proportional to this length, so V = V₀(1 - ƒ). Step 5: Conclude the relationship between ƒ and V. The voltage V across the wire is given by V = V₀(1 - ƒ), where V₀ is the voltage when the tank is empty (ƒ = 0). This equation shows how the voltage decreases as the fraction of the tank filled with liquid helium increases.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Superconductivity

Voltage and Resistance Relationship

Fraction of Liquid Helium (ƒ)