Sliding Copper Rod

A copper rod slides on frictionless rails in the presence of a constant magnetic field B = B₀ẑ. At t = 0, the rod is moving in the y direction with velocity (see Figure 1.1).

Figure 1.1

a) What is the subsequent velocity of the rod if is its conductivity and ρ_m the mass density of copper.

b) For copper, σ = 5 × 10¹⁷s^-1 and ρ_m = 8.9 g/cm³. If B₀ = 1 gauss, estimate the time it takes the rod to stop.

c) Show that the rate of decrease of the kinetic energy of the rod per unit volume is equal to the ohmic heating rate per unit volume.

SOLUTION

a) In this problem, B = B₀ẑ, so the magnetic flux through the surface limited by the rod and the rails changes as a result of the change of the surface area S (since the rod is moving). This gives rise to an electromotive force ε_e:

where  is the magnetic flux, so

In its turn, ε_e produces the current through the rod:

where R is the resistance of the rod, A is its cross section, and σ is the conductivity of the rod. The force acting on the rod is

On the other hand,  So we have

b) For an estimate we can take that the rod practically stopped when t/τ = 10. (It is good enough for an estimate, since for t/τ = 10 the final velocity is v_f = 5 × 10^-5 v₀ and for t/τ = 10, v_f = 2 × 10^-9 v₀) So for t/τ = 10,

c) The kinetic energy of the rod is T = Mv²/2 = (Mv²₀/2)e^-2t/τ, where M is the total mass of the rod. We can simply take the derivative of this kinetic energy per unit volume:

where V is the volume of the rod. On the other hand, the Joule heating per unit volume

so