Lenz's Law — Question 10

Question 10

The sign of induction is required for a passive moving circuit to conserve energy

A freely sliding rod on closed resistive rails moves right in a uniform perpendicular field. There is no battery or external driving force. The emf magnitude is B ell v and resistance is R; neglect inductance.

1. Using the Lenz-law force, compare kinetic-energy loss with Joule power.

2. If the current were assigned the opposite direction while retaining its magnitude, show the resulting inconsistency.

Question 10 - Solution

The sign of induction is required for a passive moving circuit to conserve energy

(a) Method and calculation

$$F_x=-B^2\ell^2v/R\Rightarrow\frac{dK}{dt}=F_xv=-B^2\ell^2v^2/R.$$$$P_R=i^2R=B^2\ell^2v^2/R\Rightarrow\boxed{dK/dt+P_R=0}.$$

(b) Interpretation and further calculation

$$\text{opposite current:}\quad F_x=+B^2\ell^2v/R\Rightarrow dK/dt=+P_R.$$$$\boxed{d(K+H)/dt=2P_R>0}\quad\text{with no energy source}.$$

Physical check

The incorrect sign would create kinetic energy and heat simultaneously in an isolated passive system.