Charge velocity and current density

[Defennder]

Homework Statement

Current density is given in cylindrical coordinates as $$\vec{J} = -10^6z^{1.5} \hat{z} Am^{-2} \ \mbox{in the region} \ 0 \leq r \leq 20\mu m , \mbox{and for} \ r \geq 20 \mu m, \ \vec{J} = 0$$

If the volume charge density at z=0.15 m is -2000C/m^3, find the charge velocity there.

Homework Equations

$$\nabla \cdot \vec{J} = -\frac{\partial \rho_v}{\partial t}$$

The Attempt at a Solution

Okay so this seems pretty straightforward. Given that J is only in the z-direction, then isn't it simply possible to find v_z by dividing J_z(0.15) by -2000 there? But this gives me 29 which isn't the answer. The answer is -2900.

Another method I tried was solving for $$\rho_v$$ using the relevant equations. This gives me $$\rho_v = 1.5x10^6 \int \sqrt{z} dt = 1.5x10^6\sqrt{z}t + g(z)$$. But how do I find what g(z) is? And I have to solve for t as well, since it's not stated what value of t I should evaluate the charge velocity for at z=0.15.

 

[Mindscrape]

I don't think that answer is right. You are looking for dQ/dt and you know that [itex]\rho=Q/V[/tex] for a uniform charge, so then you can solve for dQ/dt from the continuity equation.

 

[Defennder]

I don't think the charge is uniform. Why should it be?

 

[Mindscrape]

Oh, I guess I read it wrong. I suppose all the problem is really saying that only at the point z=.15 that it is uniform.

I mean, you have to know something about the charge density, or else you are shooting in the dark when it comes to finding the charge.

 

[Defennder]

The question is as stated. I didn't omit anything.