How Do You Calculate Volume Charge Density in a Varying Electron Beam?

[ajith.mk91]

Homework Statement

There is a beam of electrons in the space starting at x=0 and ending at x=20cm. The cross section of the beam is A.The velocity of the electrons is a function of position x and let it be v(x).Now i need to calculate the volume charge density as a function of x.

Homework Equations

The Attempt at a Solution

What i did is this:
I took an imaginary plane perpendicular to the beam.In a small time interval dt the amount of charge crossing the plane of area A is J.So dividing this J with v(x) should give the density. But J is not specified.Is there any other way to solve this?

 

[vela]

I don't think there's enough information to solve the problem. Was that the complete problem as given to you?

 

[ajith.mk91]

The question has an additional info which i did not mention in my previous post.The current density is given at x=20cm as 10000A/m2(Actually a cathode is placed at x=0 and an anode is placed at x=20cm).But i was asked to find both the volume charge density and current density as a function of x.

 

[vela]

The equation of continuity

$$\nabla\cdot\vec{J}+\frac{\partial\rho}{\partial t} = 0$$

should help you. ρ is the charge density.