Calculating Current Density Vector in Solid Cylinder

[1q2w3e]

Hi this is the first time I've posted something don't know how long it usually take for replies but any help would be muchly apprecitated.

This question asks: A long solid cylinder, radius R, carries uniform charge density p. the cylinder is rotating around the cetral axis with constant angular velocity omega(vector) = omega * z (unit vector)

the first qustion asks to calculate the current density vector in the cylinder...

heres what I have done,

I know that the equation to use is J(vector) = p * Vd, where Vd is the drift velocity and it can defines by several ways, Vd= (eE(tal)/m) or Vd = I/(NAe) where N is the number of free electrons per unit volume in conductor, A is the cross section area and e is simply the charge of the electron, so I don't know if I should or how to interpert the angular velocity into the drift velocity and then to actually "calculate" the current density vector. Just need some guardance thanks

 

[Sojourner01]

I'm not sure what you're asking. You've given all the formulae you need to calculate the current density, so what's the problem?

Edit: Ah, I see. I had the same quandry with this kind of question once. The formula you have there assumes it's a straight wire.

In your problem, the drift velocity is a function of position with respect to the central axis. Find the current density at position r from the central axis, then average the current density from 0 to r(edge) to find a single current density vector to use in your calculations.

 

[1q2w3e]

well in the formula you need to use the drift velocity yeah? but then in the original quesiton, you are also given the angular velocity, I am just not too sure if I need to somehow combine this angular velocity into the drift velocity and if so how?

 

[Sojourner01]

OK, the charge in the rod has no intrinsic motion relative to the medium it's sitting in, it's the rotation of the rod itself that is in effect a current. The charges in the rod are rotating around the central axis at precisely the same speed as the point on the rod itself that each charge occupies.

 

[1q2w3e]

I understand the last part about the rod haveing no intrinsic motion, but I don't get what you mean by "Find the current density at position r from the central axis" to do this, i'd need to use the current density formula again which lead me back to the problem of what I should use for the drift velocity in that equation. So at point r (the edge), the motion would be rotation with the max angular velocity. So each point, depending on the distance away from the central axis, it's velocity is going to vary... Sorry is that even relavent?

 

[Meir Achuz]

Just use j=rho v=omegaXr rho.

 

[1q2w3e]

so you are saying substitude rho for j(current density) in the equation, and that the drift velocity Vd= omega(x)*rho... is it omega or omega(vector)? then that means the entire equation becomes

rho=rho*omega(s)*rho
and the omega(x)=1

I don't get how that works, can you please explain with just a little bit more detail? thanks

 

[1q2w3e]

it's all good I've worked it out thanks to all for help