--- title: "Critical current density, critical field, soft vs hard" subject: "Electrical Engineering Material" chapter: 5 key: "semester_3/electrical_engineering_material:critical-density-soft-hard" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/critical-density-soft-hard site: BCT Engineering --- # Critical current density, critical field, soft vs hard Electrical Engineering Material · Chapter 5: Superconductivity - Asked in: 2081 Bhadra, 2+6 marks - Video: [Critical Current & Critical Current Density (Superconductivity) PHYSICS | BE Btech 1st year](https://www.youtube.com/watch?v=8g8-3JCJLhI) — 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/critical-density-soft-hard - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-5 ## Question What is critical current density and critical magnetic field in superconductors? Differentiate between soft and hard superconductors. ## Short answer Critical current density J_c = 2H_c/r is the maximum current per area a superconductor can carry; critical field H_c varies as H_0[1−(T/T_c)^2]. Soft (type-I) have one low H_c, sharp transition, low current; hard (type-II) have two critical fields, a mixed state with flux pinning, high H_c2 and high J_c. ## Answer #### Critical magnetic field The **critical magnetic field** $H_c$ is the minimum field that destroys superconductivity at a given temperature. It is largest at 0 K and becomes zero at $T_c$: $$ H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] $$ #### Critical current density The **critical current density** $J_c$ is the maximum current density a superconductor can carry without losing superconductivity. The current's own magnetic field must stay below $H_c$. For a wire of radius $r$ (Silsbee's rule): $$ I_c = 2\pi r H_c, \qquad J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r} $$ #### Soft vs hard superconductors Soft superconductors are type-I; hard superconductors are type-II. ```text -M -M | /| | /\ | / | | / \ | / | | / \_ | / | |/ \__ |/ |____ H +----|------|--- H 0 Hc 0 Hc1 Hc2 Type-I Type-II ``` | Point | Soft (Type-I) | Hard (Type-II) | |---|---|---| | Critical fields | One, $H_c$ | Two, $H_{c1}$ and $H_{c2}$ | | Field value | Low (below about 0.2 T) | $H_{c2}$ high (up to tens of T) | | Meissner effect | Complete | Complete only below $H_{c1}$ | | Transition | Sharp | Gradual, through mixed (vortex) state | | Magnetization | Reversible, no hysteresis | Irreversible, large hysteresis (flux pinning) | | Critical current | Small | Large | | Composition | Mostly pure metals | Alloys and compounds | | Examples | Pb, Hg, Sn, Al | NbTi, Nb$_3$Sn, V$_3$Ga, YBCO | | Uses | Few practical | MRI and accelerator magnets, cables | Hard superconductors are "hard" because flux vortices get pinned at defects, so they resist flux movement, show hysteresis like hard magnetic materials, and can carry large currents in strong fields. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the critical magnetic field Hc? A. The minimum field that destroys superconductivity at a given temperature B. The field at which resistance first appears at 0 K only C. The field produced by the maximum current D. The field that creates the Meissner effect Answer: A. Hc is the smallest field that returns the material to normal; it is zero at Tc. (Key point: Critical field definition) 2. What is Jc in terms of Hc and radius r? A. Jc = Hc / (2 r) B. Jc = 2Hc / r C. Jc = 2 pi r Hc D. Jc = Hc r / 2 Answer: B. Jc = Ic/(pi r^2) = 2 pi r Hc/(pi r^2) = 2Hc/r. (Key point: Critical current density) 3. Which is a soft superconductor? A. Type-II such as NbTi with hysteresis B. An alloy with very high Hc2 C. Type-I such as pure Pb or Hg, with low fields and reversible magnetization D. A compound such as YBCO Answer: C. Soft means type-I: single low Hc, reversible, pure metals. (Key point: Soft vs hard) 4. What makes hard superconductors carry large currents in strong fields? A. They have no vortices at all B. They are pure and defect-free C. They have lower Hc2 than soft ones D. Flux vortices pinned at defects resist movement Answer: D. Pinning prevents vortex motion, so Jc is large and magnetization is hysteretic. 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