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.

 

[rwisz]

There are a few different ways you could approach this problem. One way would be to use the formula for current density, J = nev, where n is the number density of electrons and e is the charge of an electron. You could then integrate this over the length of the beam to find the total current, and divide by the cross-sectional area A to get the current density. From there, you could use the definition of charge density, ρ = Q/V, where Q is the total charge and V is the volume, to calculate the charge density as a function of x.

Another approach could be to use the continuity equation, which states that the rate of change of charge density in a given volume is equal to the negative divergence of the current density. In this case, you could set up a differential equation for the charge density as a function of x and solve for it using the given information about the beam's velocity and cross-sectional area.

It's also worth noting that the problem statement does not specify any information about the number density of electrons or the current, so it may be difficult to calculate the charge density without making some assumptions or obtaining more information. As a scientist, it's important to carefully consider the available data and make reasonable assumptions or approximations when necessary.