Problem — Earth's field and the train window

Problem

Estimate the average emf induced in the frame of a train window ($1{,}0\ \text{m}\times 0{,}8\ \text{m}$) travelling at $160\ \text{km/h}$ in a region where the vertical component of the Earth’s magnetic field is $B\approx 4\cdot 10^{-5}\ \text{T}$.

Solution

Convert the velocity: $v=160\ \text{km/h}=\frac{160}{3{,}6}\approx 44{,}4\ \text{m/s}$

The field is vertical and the motion is horizontal: the horizontal side of the frame (of length $\ell$) cuts the field lines and behaves like a moving rod. The motional emf is $E=B\,\ell\,v$. Taking the long side $\ell=1{,}0\ \text{m}$: $E=B\,\ell\,v=4\cdot 10^{-5}\cdot 1{,}0\cdot 44{,}4\approx 1{,}8\cdot 10^{-3}\ \text{V}$ (with the side $\ell=0{,}8\ \text{m}$ one would get $\approx 1{,}4\ \text{mV}$: the same order of magnitude).

$\ev{E\sim 1{,}8\ \text{mV}}$ A tiny voltage, entirely negligible: it shows why the Earth’s field produces no perceptible effects on vehicles.

Links

Topics: Induzione elettromagnetica Concepts: Legge di Faraday-Neumann-Lenz · Forza elettromotrice Skills: Stime di Fermi