SLIDING RAIL
ε = BLv, and the ledger: Fv === I²R with residual 0 at every speed — work becomes heat, joule for joule.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A conducting rod slides along two rails a meter apart, closed through a resistor, with a magnetic field pointing into the page. The rod’s free charges are being carried through that field, and q v cross B pushes them along the rod. The loop has just become a battery with a moving part.
EMF and drag
The EMF is B L v — flux swept per second, nothing more — and it drives a current epsilon over R around the loop. But that current sits in the field too, so the rod feels B I L pointed against its own motion. Lenz’s law is not a decree from above; it is the sign falling out of two cross products.
The ledger
Mechanical power Fv and resistor heat I²R agree. With a hanging mass, the gravity-driven terminal branch solves F_drag = mg and reports v_t.
Sweeping the speed
The sweep runs the rod from half a meter per second to eight, quasi-static — every displayed frame is an exact solution. The two ledger bars rise together, pinned equal, growing as v squared: double the speed and both powers quadruple, bitwise.
Audit
Audited: the electrical/mechanical ledger and Lenz sign hold across the sweep; this selected hanging-mass branch also satisfies F_drag(v_t) = mg.