--- title: "Meissner effect types and critical current" subject: "Electrical Engineering Material" chapter: 5 key: "semester_3/electrical_engineering_material:meissner-critical-current" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/meissner-critical-current site: BCT Engineering --- # Meissner effect types and critical current Electrical Engineering Material · Chapter 5: Superconductivity - Asked in: 2082 Kartik (new course), 6 marks - Video: [Meissner Effect (Properties of Superconductor) Impurity Effect & Pressure Effect (Btech 1st year)](https://www.youtube.com/watch?v=vePYlqZTvuk) — Gautam Varde - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/meissner-critical-current - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-5 ## Question How does Meissner effect help to differentiate superconductor as type-I and type-II? Explain in brief. Explain critical current in superconductor with necessary mathematical expression and diagram. ## Short answer Type‑I exhibits complete flux expulsion up to Hc; type‑II is complete only below Hc1 and partial (vortex state) up to Hc2. Critical current is the maximum supercurrent given by Ic = 2πr Hc, where Hc = H0[1 – (T/Tc)²]; with applied field Ha, Ic = 2πr(Hc – Ha). ## Answer #### Meissner effect: type-I vs type-II The Meissner effect is the expulsion of flux ($B = 0$, $M = -H$) below $T_c$. The way it ends as $H$ increases separates the two types: - **Type-I:** complete diamagnetism up to $H_c$, then an abrupt drop to $M = 0$ (normal). Single, low $H_c$. Examples: Pb, Hg. - **Type-II:** complete only up to $H_{c1}$; between $H_{c1}$ and $H_{c2}$ flux partly enters as vortices (mixed state), $-M$ falls gradually; normal above $H_{c2}$. Examples: NbTi, Nb$_3$Sn. ```text -M -M | /| | /\ | / | | / \ | / | | / \_ | / | |/ \__ |/ |____ H +----|------|--- H 0 Hc 0 Hc1 Hc2 Type-I Type-II ``` #### Critical current The **critical current** $I_c$ is the maximum current a superconductor can carry without becoming normal. The self-field of the current must remain below $H_c$ (Silsbee's rule). For a wire of radius $r$, by Ampere's law the surface field is $H = I/(2\pi r)$. Setting $H = H_c$: $$ \begin{aligned} I_c &= 2\pi r H_c = 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \\ J_c &= \frac{I_c}{\pi r^2} = \frac{2H_c}{r} \end{aligned} $$ ```text H = I/(2πr) .-~~~~~-. / .-----. \ | ( I ⊙ ) | wire radius r \ '-----' / '-~~~~~-' ``` With an applied field $H_a$, $I_c = 2\pi r (H_c - H_a)$, so $I_c$ falls with field and with temperature. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the Meissner effect do in type-I as H rises? A. Complete diamagnetism up to Hc then an abrupt drop to M = 0 B. Gradual fall starting at Hc1 C. Remains complete for all fields D. No expulsion at any field Answer: A. Single low Hc with sharp transition. (Key point: Type-I behaviour) 2. What happens in type-II? A. Complete up to Hc2 then normal B. Complete up to Hc1, vortices between Hc1 and Hc2, normal above Hc2 C. Always incomplete D. Normal below Hc1 Answer: B. Mixed state between the two critical fields. (Key point: Type-II behaviour) 3. For a wire of radius r, what is the critical current by Silsbee's rule? A. Ic = pi r^2 Hc B. Ic = Hc / (2 pi r) C. Ic = 2 pi r Hc D. Ic = 2 pi r^2 Hc Answer: C. Ampere's law gives surface field H = I/(2 pi r); setting H = Hc gives Ic = 2 pi r Hc. (Key point: Critical current of a wire) 4. What causes the critical current limit in a superconducting wire? A. Joule heating from I^2R losses in the wire B. The Cooper pairs run out of electrons C. The wire's length exceeds a critical value D. The magnetic field produced by the current itself reaches Hc at the surface Answer: D. Silsbee's rule: superconductivity is lost when the current's own surface field equals Hc. 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