--- title: "Critical magnetic field and critical current" subject: "Electrical Engineering Material" chapter: 5 key: "semester_3/electrical_engineering_material:critical-field-current-relation" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/critical-field-current-relation site: BCT Engineering --- # Critical magnetic field and critical current Electrical Engineering Material · Chapter 5: Superconductivity - Asked in: 2070 Chaitra, 8 marks - Video: [Properties of Superconductors Part 2 | Critical Magnetic Field Hc and Critical Current Density Jc](https://www.youtube.com/watch?v=QSDfViUaoLs) — Dr. Vishnu Awasthi - 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-field-current-relation - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-5 ## Question Define Critical magnetic field and Critical current in a super-conductor with mathematical relation involved. ## Short answer Critical magnetic field H_c is the minimum field that destroys superconductivity, varying as H_c(T)=H_0[1−(T/T_c)^2]. Critical current I_c is the maximum current a superconductor can carry; from Ampere's law the self-field at the surface must not exceed H_c, giving I_c=2πr H_c and critical current density J_c=I_c/(πr^2)=2H_c/r. ## Answer #### Critical magnetic field The **critical magnetic field** $H_c$ is the minimum magnetic field which, applied to a superconductor at a temperature below $T_c$, destroys its superconductivity and returns it to the normal state. - It is largest ($H_0$) at 0 K and falls to zero at $T = T_c$. - The temperature variation is nearly parabolic: $$ H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] $$ ```text Hc H0|* | * | supercon- * normal | ducting * | * +----------------*--- T 0 Tc ``` - Points under the curve are superconducting; points above it are normal. - Type-I materials have a single $H_c$ (about 0.01 to 0.2 T). Type-II have $H_{c1}$ (flux starts entering) and $H_{c2}$ (fully normal), with $H_{c2}$ as high as tens of tesla. - Example: lead has $H_0 \approx 0.08$ T and $T_c = 7.2$ K. #### Critical current The **critical current** $I_c$ is the maximum current a superconductor can carry without losing superconductivity. A current produces its own magnetic field; when this field at the surface reaches $H_c$, the material turns normal (**Silsbee's rule**). For a long straight wire of radius $r$ carrying current $I$, Ampere's law gives the surface field: $$ H \cdot 2\pi r = I \;\Rightarrow\; H = \frac{I}{2\pi r} $$ Setting $H = H_c$: $$ \begin{aligned} I_c &= 2\pi r H_c \\ &= 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \end{aligned} $$ The critical current density is $$ J_c = \frac{I_c}{\pi r^2} = \frac{2 H_c}{r} $$ With an external field $H_a$ adding to the self-field, $I_c = 2\pi r (H_c - H_a)$. So the critical current depends on wire size, temperature and applied field. It falls as $T$ rises towards $T_c$ and becomes zero at $T_c$. A thicker wire carries a larger $I_c$ (though a smaller $J_c$). Practical superconducting cables use many fine filaments in a copper matrix, mainly for stability against flux jumps and quenching. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which expression gives the critical field at temperature T? A. Hc(T) = H0[1 - (T/Tc)^2] B. Hc(T) = H0[1 - (T/Tc)] C. Hc(T) = H0[(T/Tc)^2 - 1] D. Hc(T) = H0 (T/Tc)^2 Answer: A. Hc is maximum (H0) at 0 K and falls parabolically to zero at Tc. (Key point: Temperature dependence of Hc) 2. On an Hc versus T plot, what do points under the curve represent? A. Normal state B. Superconducting state C. A mixed state of two metals D. Forbidden values Answer: B. Points below the Hc(T) curve are superconducting; above it are normal. (Key point: Meaning of the curve) 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. How does Ic change with temperature and wire size? A. Rises with T and falls with radius B. Independent of T and r C. Rises with T and with radius D. Falls to zero as T approaches Tc, and larger for thicker wires Answer: D. Ic = 2 pi r Hc(T), proportional to r and falling with T. 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