Induction and Energy Transfers — Question 3

Question 3

A falling conducting rod reaches a terminal speed set by electrical resistance

A horizontal rod of mass m slides downward on vertical rails in a perpendicular field B. A resistor R closes the rails above it. Rail spacing is ell. Start from rest; neglect friction and inductance.

1. Find terminal speed and the velocity as a function of time.

2. Find the current and heating rate at terminal speed.

Question 3 - Solution

A falling conducting rod reaches a terminal speed set by electrical resistance

(a) Method and calculation

$$m\dot v=mg-\frac{B^2\ell^2}{R}v,\quad\boxed{v_T=\frac{mgR}{B^2\ell^2}}.$$$$\boxed{v(t)=v_T(1-e^{-t/\tau}),\quad\tau=mR/(B^2\ell^2)}.$$

(b) Interpretation and further calculation

$$i_T=B\ell v_T/R=\boxed{mg/(B\ell)}.$$$$\boxed{P_R=i_T^2R=mgv_T}.$$

Physical check

At terminal speed, gravitational potential energy is converted to heat without further increase in kinetic energy.