Conducting bar in a magnetic field: induction, dynamics and energy

Problem

A conducting bar of mass $m$, length $\ell$, resistance $R$ slides without friction on two parallel horizontal rails, immersed in a uniform magnetic field $\pq{B}$ perpendicular to the plane of the rails. The bar starts from rest with a constant external force $\pq{F}_0$ applied. Determine: (a) the velocity as a function of time, (b) the terminal velocity, (c) the power dissipated in the resistance at steady state.

Solution

Setup (staged approach). The problem interweaves three laws: Newton’s second law (dynamics), the Faraday-Neumann-Lenz law (induction) and the energy balance. We put them in sequence.

Stage 1 — induction. Moving with velocity $v$, the bar generates an induced electromotive force (Faraday’s law, changing area): $\Gamma_E = -B\ell v$ The current in the circuit, of resistance $R$, has magnitude: $i = \frac{B\ell v}{R}$

Stage 2 — dynamics. The current in the field produces a Laplace force $F_\text{ind} = Bi\ell$ which, by Lenz’s law, opposes the motion. Newton’s second law along the rails is: $m\,\ddt{v} = F_0 - Bi\ell = F_0 - \frac{B^2\ell^2}{R}\,v$ This is a first-order linear differential equation in $v$.

(a) Velocity as a function of time. Setting $\tau = \dfrac{mR}{B^2\ell^2}$ and integrating with the condition $v(0)=0$: $\ev{v(t) = \frac{F_0 R}{B^2\ell^2}\left(1 - e^{-t/\tau}\right)}$ The velocity grows exponentially and saturates: the same form as the charging of a capacitor.

(b) Terminal velocity. At steady state $\ddt{v}=0$, hence $F_0 = \dfrac{B^2\ell^2}{R}v_\infty$: $\ev{v_\infty = \frac{F_0 R}{B^2\ell^2}}$

(c) Energy balance at steady state. At constant velocity the acceleration is zero: all the power supplied by the external force is dissipated by the Joule effect in $R$. $P_\text{diss} = P_\text{esterna} = F_0\, v_\infty = \frac{F_0^2 R}{B^2\ell^2}$ $\ev{P_\text{diss} = \frac{F_0^2 R}{B^2\ell^2}}$

The three areas interlock: induction (ch. 18) supplies the current, which feeds into the dynamics (variable force, ch. 2) and determines the velocity as a function of time; conservation of energy (ch. 4) closes the account at steady state. The mathematical structure is the same differential equation $\dot v = a - bv$ that also governs falling with viscous drag.

Links

Topics: Electromagnetic induction · Dynamics · Work and energy Concepts: Faraday-Neumann-Lenz law · Laplace force · Joule effect Skills: Conservation of energy