Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
7510: Homework # 8 Problem 13. Consider a cylindrical superconductor of radius R (and infinite length) placed in the external magnetic field H which is directed perpendicular to the cylinder axis. The critical magnetic field is Hc . Find the range of magnetic fields H for which intermediate state will occur. [5 points] Solution. Outside of the superconductor magnetic field satisfies ∇ · H=0, ∇ × H = 0. The second condition is taken care of by H = ∇ψ, while the first one gives Laplace equation ∇2 ψ = 0. From the separation of variables it is found that any function rl cos (lθ) with arbitrary l satisfies Laplace equation. The solution should be looked in the form, ψ = Hr cos θ + C cos θ , r where the first term is the applied field, and the second term is the field created by the cylinder (since it vanishes at r → ∞ only l < 0 are allowed, in addition any |l| 6= 1 will not satisfy the boundary condition). The constant C is found from vanishing of the normal component of H at the surface: ∂ψ |r=R = 0, ∂r which gives C = HR2 . Tangential component of magnetic field at the surface of a cylinder, Hθ (R) = 1 ∂ψ |r=R = −2H sin θ. r ∂θ The maximal value of |H(R)| is reached at θ = π/2 and is 2H. When this value exceeds Hc the superconductor goes into intermediate state. Thus, for Hc /2 < H < Hc intermediate state is realized. Problem 14. Find surface density of the induced supercurrents as a function of the position on a cylinder (consider a simple limit of small penetration depth λ/R → 0. Find the total magnetic moment (per unit of length of the cylinder) created by these supercurrents. [5 points] Solution. The (volume) density of electric currents is found from the equation ∇ × H = 4πj/c. Writing z-component of this equation, we get, 1 ∂(rHθ ) 1 ∂Hr 4π − = jz r ∂r r ∂θ c The second term on the left-hand side vanishes at the surface of the superconductor, while the first term has a singularity. Integrating it from r = R − 0 to r = R + 0, we find the density of surface currents R+0 Z jz (r)dr = R−0 Hc c [Hθ (R + 0) − Hθ (R − 0)] = − sin θ. 4π 2π