--- title: "J = σE, resistivity and superconductors" subject: "Engineering Physics" chapter: 8 key: "semester_1/engineering_physics:resistivity-j-sigma-e-superconductor" source: https://www.bctengineer.com/q/semester_1/engineering_physics/resistivity-j-sigma-e-superconductor site: BCT Engineering --- # J = σE, resistivity and superconductors Engineering Physics · Chapter 8: Electromagnetism - Asked in: 2069 Chaitra, 5 marks - Video: [Relation between Current density, Conductivity and electric field | Superconductivity | 12th #cbse](https://www.youtube.com/watch?v=5p4b9VFQ5XI) — PHYSICS with Umesh Rajoria - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/resistivity-j-sigma-e-superconductor - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8 ## Question Derive an equation J = σE. Explain why resistivity of a conductor increases with increasing temperature; plot a graph between Rθ (resistance at any temperature θ) and temperature. Based on the graph, explain what are superconductors? How do they differ from perfect conductor? Describe the characteristics of superconductors. ## Short answer From drift velocity v_d=eτE/m, current density J=ne v_d gives J=(ne²τ/m)E=σE. Resistivity ρ=1/σ rises with temperature because stronger lattice vibrations reduce relaxation time τ. Superconductors have zero resistance below T_c and exhibit the Meissner effect (perfect diamagnetism), unlike a perfect conductor that retains flux. Key characteristics: zero resistance, Meissner effect, critical field and current. ## Answer #### Derivation of J = σE Let n be the free-electron density and τ the average time between collisions. 1. Acceleration: a = eE/m. 2. Drift velocity: v_d = aτ = eτE/m. 3. J = n e v_d = (ne²τ/m) E. 4. Writing σ = ne²τ/m (conductivity), **J = σE**. The resistivity is ρ = 1/σ = m/(ne²τ). #### Why resistivity rises with temperature When temperature increases, the lattice ions vibrate more strongly. Electrons collide more often, so τ decreases. Since ρ = m/(ne²τ), ρ increases. For a metal, n stays almost constant. Near room temperature: ρ = ρ₀[1 + α(θ − θ₀)], and R_θ = R₀(1 + αθ). #### Graph of R_θ against temperature ```text R | / normal metal | / | / | _ / | ____________ -- <- residual resistance at 0 K | | / superconductor | / | |------/ | | +------------+------------------> T 0 Tc ``` For a normal metal, R falls steadily as T is lowered and levels off at a small residual value at 0 K. For a superconductor, R drops abruptly to **zero** at the critical temperature T_c and stays zero below it. #### Superconductors A superconductor is a material whose resistance falls to exactly zero below T_c (e.g. Hg 4.2 K, Nb 9.3 K, YBCO ~92 K). **Difference from a perfect conductor:** a perfect conductor (ρ = 0) only keeps the magnetic flux it already has, so its state depends on history. A superconductor also shows the **Meissner effect**: it expels magnetic flux (B = 0 inside) whatever the order of cooling and field application. It is a perfect diamagnet (χ = −1). #### Characteristics 1. Zero electrical resistance below T_c, so a persistent current flows without loss. 2. Meissner effect: perfect diamagnetism. 3. Critical field H_c(T) = H₀[1 − (T/T_c)²]; above it superconductivity is destroyed. 4. Critical current density: a large enough current also destroys it. 5. Type I (sharp transition) and Type II (mixed state between H_c1 and H_c2). 6. Uses: MRI magnets, maglev, SQUIDs, power cables. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The conductivity in J = σE is: A. σ = m/(ne²τ) B. σ = ne²τ/m C. σ = neτ/m² D. σ = ne²m/τ Answer: B. J = n e v_d with v_d = eτE/m. (Key point: J = σE) 2. Why does the resistivity of a metal increase with temperature? A. Lattice vibrations grow, collisions are more frequent and τ decreases B. The electron density n falls sharply C. τ increases D. The electrons become heavier Answer: A. ρ = m/(ne²τ), so smaller τ means larger ρ. (Key point: Temperature) 3. In the R vs T graph, a superconductor shows: A. resistance falling slowly to a small residual value B. resistance rising at T_c C. resistance dropping abruptly to zero at the critical temperature T_c D. resistance constant at all temperatures Answer: C. A normal metal drops to a residual value; a superconductor goes to exactly zero. (Key point: Superconductor graph) 4. What distinguishes a superconductor from a perfect conductor? A. A superconductor has non-zero resistance B. A perfect conductor expels flux; a superconductor does not C. There is no difference D. The Meissner effect: it expels magnetic flux regardless of the order of cooling and field Answer: D. A perfect conductor only keeps existing flux, so it is history-dependent; a superconductor is a perfect diamagnet. 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