Calculate the charge of a density distributed along z axis?

flux!

1. The problem statement, all variables and given/known data
How do I find the total charge from a material with a charge density given by
$$\rho =10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m^3$$
that exist between $\frac{-\pi}{3}z_0<z<\frac{\pi}{3}z_0$.

2. Relevant equations
None I can think of.

3. The attempt at a solution

Attempt #1:

Since the charge density is a volume charge density we may assume that we are dealing with a cylindrical charge distribution that remains the same along $r$. However, the radius of the assumed shaped was not given, so let us assume that its radius is $r_0$. We may solve this by:

$$q=\int_V \rho d\tau$$ $$q=\int_0^{r_0} \int_0^{2\pi} \int_{\frac{-\pi}{3}z_0}^{\frac{\pi}{3}z_0} 10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m^3 rdrd\phi dz$$

Everything will then be straight forward, but the issue is that

1. $r_0$ is not given, and
2. the problem did not say it is a cylindrical charge density.

This is just, however, one way I could deal with a volume charge density that does not have a given volume, unusual huh.

Attempt #2:

Since the problem indicated that the volume charge density is distributed on a line, then it must be a line charge only. Thus it must have an infinitesimal radius that could be resolve by Dirac Delta function, so:

$$\rho =10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m^3$$

must be equivalent to:

$$\rho =\delta(r,\phi - r',\phi')10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m$$

then, things could now be easily solved in the integral:

$$q=\int_V \left (\delta(r,\phi - r',\phi')10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m\right )d\tau$$

4. My Question

Now, my question is, which two approaches is the right solution, Or if neither any of the two is correct, how should we solve the total charge of a density given above?

Cross-Link: Posted the Question In physics.exchange but, does not seem to get answered, Hope you guys help me out. http://physics.stackexchange.com/questions/240215/how-to-compute-the-charge-of-a-density-distributed-along-z-axis [Broken]

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Simon Bridge

Homework Helper
You are specifically given units of C/m3 ... so this is explicitly a volume charge density.

The only dpendence given is z, so a cylindrical symmetry is reasonable.
The question appears to call for the total charge between two planes so $r\in [0,\infty)$ ... but that gives you an infinite charge.
It follows that there is some information missing from the problem statement, or the statement is in error.

You need to use the context that the question is set in to further determine the limits for the volume.

flux!

Hi, that what all the problem given. No $r$ for a cylinder or so. The problem just tells that the charge is distributed on z.

Simon Bridge

Homework Helper
That's why I suggest going to the context - sometimes the needed information is not explicitly given.
Maybe the "material" is finite in a way mentioned elsewhere?

There are too many other possibilities to safely guess ... which is basically what you are doing above.
Have you asked the other students to see how they are handling it?

If you cannot ask anyone, and you must turn in something, then you will have to just leave $r_0$ as a variable.

haruspex

Homework Helper
Gold Member
2018 Award
I see no need to assume it is a cylinder. Just assume the cross-sectional area in the XY plane is constant, A.
However, this still leaves such a trivial problem that I suspect the missing information is crucial.

flux!

Thanks for the responses! I will just assume that there might be a missing information. In case there is none, is my solution involving dirac-delta correct?

"Calculate the charge of a density distributed along z axis?"

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