At what temperature, in terms of the critical temperature 𝑇 sub 𝑐, is the critical field of a superconductor one-half its value at 𝑇 equals zero K?
In this exercise, we want to solve for a temperature we’ll call capital 𝑇 sub one-half. To begin our solution, let’s recall the relationship for the critical field of a superconductor. 𝐻 sub 𝑐, the critical magnetic field of a superconductor, is equal to 𝐻 sub zero, the critical field when the temperature is equal to zero Kelvin, times one minus the quantity 𝑇 over 𝑇 sub 𝑐.
𝑇 sub 𝑐, the critical temperature, is the temperature at which a material effectively loses its electrical resistance, that is, the temperature at or below which the material becomes a superconductor.
Applying this relationship to our scenario, we’re told that the critical field at our temperature, 𝑇 sub one-half, is equal to half the critical field at a temperature of zero Kelvin. That means that 𝐻 sub zero divided by two equals 𝐻 sub zero times one minus the quantity 𝑇 sub one-half over 𝑇 sub 𝑐 squared.
We see we now have a relationship where 𝐻 sub zero can cancel out, since it appears on both sides, leaving us with the expression one-half equals one minus the quantity 𝑇 sub one-half over 𝑇 sub 𝑐 squared.
We want to rearrange to solve for 𝑇 sub one-half. When we do, we find that 𝑇 sub one-half squared equals 𝑇 sub 𝑐 squared over two. Taking the square root of both sides, 𝑇 sub one-half equals the square root of 𝑇 sub c squared over two, equivalently one over the square root of two times 𝑇 sub 𝑐.
We want our answer in terms of 𝑇 sub 𝑐, so we only need to evaluate one over the square root of two. To three significant figures, it equals 0.707. This is the temperature at which the critical field of the semiconductor is one-half the value of the field when the temperature is at zero Kelvin.