Question
A metro train enters a tunnel where Earth’s magnetic field has a vertical component T. The train’s two metal axles are m apart and the train moves at m/s. Find the EMF induced between the rails through the axle.
Solution — Step by Step
The axle (length ) moves through Earth’s vertical magnetic field with velocity perpendicular to both. The free electrons in the axle feel a force along the axle, which separates charges and produces an EMF — exactly the rod-on-rails setup from the textbook.
Only the vertical component of Earth’s field matters here, because the axle is horizontal and the train moves horizontally — only is perpendicular to both and .
Final Answer: mV.
Why This Works
The motional EMF comes from the magnetic Lorentz force on the free electrons in the moving conductor. Only the component of perpendicular to both and contributes — that’s why we used and not the total field.
This is real physics, not a textbook abstraction: aircraft wings tip-to-tip generate measurable EMFs while flying, and engineers must account for it in sensitive avionics.
Alternative Method
Use Faraday’s law on the rectangular loop formed by the axle, the two rails, and an imaginary closing wire at the back. In time , the loop area increases by , so flux change , giving EMF . Same answer.
Using the total magnetic field instead of just the perpendicular component is the most common error. The horizontal component of Earth’s field is parallel to the train’s velocity, so it produces zero EMF in the axle. Always identify which component is perpendicular to both and .
A useful sanity check: this EMF is in the millivolt range, way too small to electrocute a passenger. That’s why metro systems work fine despite this effect existing. If your answer for a similar problem comes out in volts, recheck the field magnitude and units.