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Homework Statement
How do I find the total charge from a material with a charge density given by
[tex]\rho =10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m^3[/tex]
that exist between [itex]\frac{-\pi}{3}z_0<z<\frac{\pi}{3}z_0[/itex].
Homework Equations
None I can think of.
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 [itex]r[/itex]. However, the radius of the assumed shaped was not given, so let us assume that its radius is [itex]r_0[/itex]. We may solve this by:
[tex]q=\int_V \rho d\tau[/tex] [tex]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[/tex]
Everything will then be straight forward, but the issue is that
1. [itex]r_0[/itex] 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:
[tex]\rho =10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m^3[/tex]
must be equivalent to:
[tex]\rho =\delta(r,\phi - r',\phi')10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m[/tex]
then, things could now be easily solved in the integral:
[tex]q=\int_V \left (\delta(r,\phi - r',\phi')10^{-9} \text{cos}\left ( \frac{z}{z_0}\right ) C/m\right )d\tau[/tex]
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
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