---
title: "Superconductor, critical field and critical current density"
subject: "Electrical Engineering Material"
chapter: 5
key: "semester_3/electrical_engineering_material:define-critical-field-current"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/define-critical-field-current
site: BCT Engineering
---

# Superconductor, critical field and critical current density

Electrical Engineering Material · Chapter 5: Superconductivity

- Asked in: 2075 Asoj, 4 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/define-critical-field-current
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-5

## Question

Define superconductor, critical magnetic field, and critical current density.

## Short answer

A superconductor is a material with exactly zero resistance and perfect diamagnetism (Meissner effect) below its critical temperature T_c. Critical magnetic field H_c is the minimum field that destroys superconductivity, given by H_c(T)=H_0[1−(T/T_c)^2]. Critical current density J_c is the maximum current per area; for a wire of radius r, J_c=2H_c/r.

## Answer

#### Superconductor
A **superconductor** is a material whose resistance becomes exactly zero below a **critical temperature** $T_c$ and which expels magnetic flux from its interior (Meissner effect, perfect diamagnetism). Examples: Hg ($T_c$ = 4.2 K), Pb (7.2 K), Nb$_3$Sn (18 K), YBCO (92 K).

#### Critical magnetic field
The **critical magnetic field** $H_c$ is the minimum magnetic field that destroys superconductivity at a given temperature. It is maximum ($H_0$) at 0 K and zero at $T_c$:

$$
H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]
$$

```text
  Hc
  H0|*
    |    *    normal
    | super  *
    |         *
    +----------*--- T
               Tc
```

#### Critical current density
The **critical current density** $J_c$ is the largest current density a superconductor can carry without becoming normal. The current's own magnetic field at the surface must stay below $H_c$. For a wire of radius $r$ (Silsbee's rule):

$$
I_c = 2\pi r H_c, \qquad J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}
$$

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement defines a superconductor?
   A. Resistance becomes exactly zero below Tc and it expels flux
   B. It has very small but nonzero resistance at all temperatures
   C. Resistance is zero only in a strong field
   D. It expels flux only above Tc
   Answer: A. Zero resistance plus the Meissner effect. (Key point: Superconductor defn)

2. What are the values of Hc at 0 K and at Tc?
   A. Zero at 0 K and H0 at Tc
   B. H0 (maximum) at 0 K and zero at Tc
   C. H0 at both
   D. Zero at both
   Answer: B. Hc(T) = H0[1 - (T/Tc)^2]. (Key point: Hc at 0 K and Tc)

3. 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 magnetic field produced by the current itself reaches Hc at the surface
   D. The wire's length exceeds a critical value
   Answer: C. Silsbee's rule: superconductivity is lost when the current's own surface field equals Hc. (Key point: Silsbee's rule)

4. What is Jc in terms of Hc and radius r?
   A. Jc = Hc / (2 r)
   B. Jc = 2 pi r Hc
   C. Jc = Hc r / 2
   D. Jc = 2Hc / r
   Answer: D. Jc = Ic/(pi r^2) = 2 pi r Hc/(pi r^2) = 2Hc/r. (Key point: Critical current density)

## Related in Electrical Engineering Material

- [Strong magnetic field and critical current](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acritical-current-strong-field) — Electrical Engineering Material, chapter 5, asked 5×
- [Critical current density, critical field, soft vs hard](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acritical-density-soft-hard) — Electrical Engineering Material, chapter 5
- [Critical magnetic field and critical current](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acritical-field-current-relation) — Electrical Engineering Material, chapter 5
- [Superconductor expels magnetic lines of force](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Asuperconductor-expels-field) — Electrical Engineering Material, chapter 5, asked 2×
- [Critical current decreases linearly with field](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Acritical-current-linear-decrease) — Electrical Engineering Material, chapter 5
- [Meissner effect types and critical current](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ameissner-critical-current) — Electrical Engineering Material, chapter 5

## Related in other subjects

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