---
title: "Atomic view of resistivity, ρ = m/ne²τ"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:current-density-mobility-atomic-resistivity"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/current-density-mobility-atomic-resistivity
site: BCT Engineering
---

# Atomic view of resistivity, ρ = m/ne²τ

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2072 Kartik, 5 marks
- Video: [EXPRESSION FOR ELECTRICAL CONDUCTIVITY | NCERT CLASS 12 | CHAPTER 3 | MOST IMPORTANT |](https://www.youtube.com/watch?v=10JcFcQb3Hw) — Shreyas Havaldar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/current-density-mobility-atomic-resistivity
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

What are current density and mobility? Explain the atomic view of the resistivity and show that ρ = m/(ne²τ), where symbols have their usual meanings.

## Short answer

Current density J = I/A = nqv_d (A/m²); mobility μ = v_d/E (m²/V·s). Resistivity arises from electron-lattice collisions; drift velocity v_d = eEτ/m leads to J = (ne²τ/m)E, so ρ = m/(ne²τ).

## Answer

#### Current density
Current density **J** is the current per unit area, taken perpendicular to the flow. J = I/A (unit A/m²). It is a vector along the direction of positive charge flow. For carriers of density n, charge q and drift velocity v_d, **J = nq v_d**.

#### Mobility
Mobility μ is the drift velocity per unit electric field: **μ = v_d/E** (unit m²V⁻¹s⁻¹). It tells how easily carriers move through the material.

#### Atomic view of resistivity
In a metal the free electrons move randomly at high speed (about 10⁶ m/s) and collide with the lattice ions and impurities. With no field the average velocity is zero. A field E gives each electron an acceleration a = eE/m between collisions. After a collision the electron loses its memory of the earlier motion, so the average time between collisions is the **relaxation time τ**.

```text
 field E: ----------------------------->

 electron:  <--.     .--.     .--o (start)
                \   /    \   /
                 . .      . .        . = collision
 net drift: <--------  (opposite to E, as the electron is negative)
```

#### Derivation of ρ = m/(ne²τ)
1. Acceleration of an electron: a = eE/m.
2. Average (drift) velocity gained in time τ: v_d = aτ = eEτ/m.
3. Current density: J = n e v_d = n e (eEτ/m) = (ne²τ/m) E.
4. Ohm's law in vector form is J = σE, so σ = ne²τ/m.
5. Resistivity is the reciprocal: **ρ = 1/σ = m/(ne²τ)**.

Also μ = v_d/E = eτ/m, so σ = neμ. Resistivity rises with temperature because lattice vibrations increase, which reduces τ.

## Check yourself

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

1. For carriers of density n, charge q and drift velocity v_d, the current density is:
   A. J = n q / v_d
   B. J = I A
   C. J = n q v_d
   D. J = q v_d / n
   Answer: C. J is current per unit area, and the current is I = n q v_d A, so J = n q v_d. (Key point: Current density)

2. Mobility of charge carriers is defined as:
   A. drift velocity per unit electric field, v_d/E
   B. electric field per unit drift velocity, E/v_d
   C. random thermal speed per unit field
   D. current per unit area
   Answer: A. mu = v_d/E, measured in m^2 V^-1 s^-1; it shows how easily carriers move. (Key point: Mobility)

3. In the atomic view, the relaxation time τ is:
   A. the time an electron takes to cross the whole wire
   B. the time for the field to reach the far end of the wire
   C. the time for which the electron stays at rest
   D. the average time between collisions of an electron with lattice ions/impurities
   Answer: D. After each collision the electron loses memory of its motion; τ is the mean time between collisions, over which it gains drift velocity. (Key point: Relaxation time)

4. Which sequence gives ρ = m/(ne²τ)?
   A. v_d = mEτ/e, so J = (nm τ/e)E, then ρ = σ
   B. v_d = eEτ/m, so J = (ne²τ/m)E = σE, then ρ = 1/σ
   C. v_d = eE/(mτ), so J = (ne²/mτ)E, then ρ = σ
   D. v_d = eEτ/m, so J = n e v_d²/E, then ρ = 1/σ
   Answer: B. Drift velocity is aτ = eEτ/m; J = n e v_d gives σ = ne²τ/m and ρ = 1/σ. (Key point: Derivation of ρ)

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