Drift of Electrons & the Origin of Resistivity 🔌⚡

1  How Electrons Move

Without an electric field, free electrons in a metal zip around randomly, so their average velocity is zero — the random directions cancel out ✨ :contentReference[oaicite:16]{index=16}.

The moment we switch on an electric field E, each electron feels a steady acceleration \(a=-\dfrac{eE}{m}\) 🔋 . Between collisions with heavy ions, an electron picks up extra speed; after many electrons are considered together, they share a tiny average drift speed \(v_d\) opposite to E: \(v_d=-\dfrac{eE\tau}{m}\) (τ = relaxation time) 🚶‍♂️💨 .

2  Link to Current & Ohm’s Law

Every electron that drifts carries charge, so a net current appears. For number density n and cross-sectional area A: \(I = neA|v_d|\) :contentReference[oaicite:17]{index=17}. Combining this with the expression for \(v_d\) gives the current density: \(\mathbf{j}= \dfrac{ne^{2}\tau}{m}\,\mathbf{E}\) .

Compare this with the familiar form \(\mathbf{j}= \sigma \mathbf{E}\); the conductivity is \(\sigma=\dfrac{ne^{2}\tau}{m}\) 🧲 . Its reciprocal \(\rho=1/\sigma\) is the resistivity.

3  Mobility µ

Mobility tells us how easily carriers respond to E: \(\mu=\dfrac{|v_d|}{E}=\dfrac{e\tau}{m}\) 🚀 . In metals, only electrons contribute; in electrolytes or ionised gases, both positive and negative ions jump in.

4  Numerical Feel 🔢

For a 1.0 × 10-7 m2 copper wire carrying 1.5 A, the drift speed is a leisurely \(v_d\approx1.1\times10^{-3}\,\text{m s}^{-1}\) (≈ 1 mm s-1) 🐢 . That’s trillions of times slower than the electric field itself, which races along the wire nearly at light-speed!

5  Why Current Starts Instantly 🤔

  • Field spreads fast: the electric field fills the circuit almost at once, nudging electrons everywhere at the same time. :contentReference[oaicite:18]{index=18}
  • Steady speed: each electron gains speed, collides, slows, and repeats. The average settles at \(v_d\). :contentReference[oaicite:19]{index=19}
  • Huge numbers: even tiny \(v_d\) moves an immense crowd (~1029 m-3) of electrons, so the total current can be large. :contentReference[oaicite:20]{index=20}

6  When Ohm’s Law Breaks Down ⚠️

Some materials refuse to keep \(V\) proportional to \(I\):

  1. Non-linear behaviour — current rises unevenly with voltage.
  2. Direction matters — reversing V changes the slope (think diodes).
  3. Multiple V for one I — some semiconductors (e.g., GaAs) show looping curves.

7  Resistivity & Material Types 🏅

Classify substances by resistivity (ρ) in increasing order: conductors → semiconductors → insulators. Lower ρ means easier current flow 💡 .

High-Yield NEET Pointers 🎯

  • Microscopic Ohm’s law: \(\mathbf{j}= \dfrac{ne^{2}\tau}{m}\,\mathbf{E}\) and \(\sigma=ne^{2}\tau/m\).
  • Drift velocity expression \(v_d=-eE\tau/m\) and its tiny magnitude in metals.
  • Mobility relation \(\mu = e\tau/m\) and its connection \( \mathbf{j}=ne\mu\mathbf{E}\).
  • Relaxation time (τ) concept and its role in electron collisions.
  • Recognising non-Ohmic devices (diodes, GaAs) from their V-I graphs.