PX157 - D1b - a moving crossbar

revisiting force on a moveable bar, but now, there is no power source

the crossbar is forced externally to move at velocity, $\vec{v} = v \hat{x}$
in the time-interval, $Δ t$ , the crossbar will have moved by a distance, $Δ x = v Δ t$
area enclosed by the crossbar increases:

Δ A = Δ x L = v Δ t L

the magnetic flux increases:

Δ ϕ_{B} = B Δ A = B v L Δ t

the rate of change in magnetic flux:

\frac{Δ ϕ_{B}}{Δ t} = B v L

from faraday's law, and using the right-hand rule (emf points counter-clockwise):

ϵ = - B v L

emf drives a current in the counter-clockwise direction:

I = \frac{| ϵ |}{R} = \frac{B v L}{R}

force on a section $d \vec{l}$ of the crossbar due to the induced current:

\begin{aligned} d \vec{F} & = I d \vec{l} \times \vec{B} \\ = I d l B \hat{y} \times (- \hat{z}) \\ = - I d l B \hat{x} \end{aligned}

total force on the crossbar:

\vec{F} = - I B \int d l \hat{x} = - I B L \hat{x}

if there is no external forcing:

\begin{aligned} m \frac{d v}{d t} & = - I B L \\ = - \frac{B^{2} L^{2}}{R} v \\ ⟹ \frac{d v}{d t} & = - \frac{v}{τ} \end{aligned}

where, $τ = {(\frac{B^{2} L^{2}}{R m})}^{- 1}$

the solution for the 2nd order homogeneous ODE:

v (t) = v (0) \exp (- \frac{t}{τ})

- $τ :$ typically slowing down (damping) time

if there is external forcing, to maintain constant speed, $v :$

{\vec{F}}_{e x t} = + I B L \hat{x}

work done by the external force (per unit time), ie: power input:

P = F_{e x t} v = I B L v = \frac{B^{2} L^{2}}{R} v^{2}

power dissipated due to ohmic heating:

P_{l o s s} = R I^{2} = R \frac{B^{2} L^{2}}{R^{2}} v^{2} = \frac{B^{2} L^{2}}{R} v^{2}

P_{i n p u t} = P_{l o s s}