Torque on a rotating solid conducting cylinder in B field

[merrypark3]

Homework Statement

panofsky 10.3

Find the torque on a solid conducting cylinder rotating slowly in a uniform magnetic field perpendicular to the axis of the cylinder.

The Attempt at a Solution

let the radius of cylinder r, and the conductivity is σ, the rotating angular velocity is $\stackrel{\rightarrow}{ω}$

j=σ(u×B)=ρv=σ((ω×r)×B)
$\stackrel{\rightarrow}{j}=σ((\stackrel{\rightarrow}{ω}×\stackrel{\rightarrow}{r}) ×\stackrel{\rightarrow}{B})=ρ\stackrel{\rightarrow}{v}$


$\stackrel{\rightarrow}{τ}=∫(ρ\stackrel{\rightarrow}{v}×\stackrel{\rightarrow}{B} )dV=0$

Homework Statement

Is it right?

 

[TSny]

j=σ(u×B)=ρv=σ((ω×r)×B)
$\stackrel{\rightarrow}{j}=σ((\stackrel{\rightarrow}{ω}×\stackrel{\rightarrow}{r}) ×\stackrel{\rightarrow}{B})=ρ\stackrel{\rightarrow}{v}$$\stackrel{\rightarrow}{τ}=∫(ρ\stackrel{\rightarrow}{v}×\stackrel{\rightarrow}{B} )dV=0$

The net torque will not be zero. Your last integral looks like the net force rather than the net torque. Otherwise your expressions look ok to me if $\vec{v}$ denotes the drift velocity of charge carriers. I don't think using $\rho \vec{v}$ for $\vec{j}$will help much.