- #1
PFStudent
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Homework Statement
Our professor gave this as sort of a challenge to find on our own; I've been stumped for a while and just can't seem to figure it out.
Find the electric field at a point [itex]{P}[/itex] a distance [itex]{Z_{0}}[/itex] along the z-axis due to a charged cylinder of radius [itex]{R_{0}}[/itex], length [itex]{L}[/itex], and uniformly distributed charge [itex]{q}[/itex]. Find the electric field for a
(a) Hollow Cylinder and for a
(b) Solid Cylinder
http://aycu26.webshots.com/image/29425/2005877188898265207_rs.jpg
Yea,...my MS Paint skills could be better.
Homework Equations
Electric Field Equation
[tex]
{\vec{E}_{P1}} = {\frac{{k_{e}}{q_{1}}}{{\left(r_{_{1P}}\right)}^{2}}{\hat{r}_{1P}}
[/tex]
[tex]
{{E}_{P1}} = \frac{{k_{e}}{|q_{1}|}}{{\left(r_{_{1P}}\right)}^{2}}
[/tex]
[tex]
{{dE}_{P1}} = \frac{{k_{e}}{dq}}{{\left(r_{_{1P}}\right)}^{2}}
[/tex]
Charge Density Equations
[tex]
{\lambda} = \frac{q}{L}
[/tex]
[tex]
{\sigma} = \frac{q}{A}
[/tex]
[tex]
{\rho_{q}} = \frac{q}{V}
[/tex]
Electric Field due to a Charged Ring (C.R.)
[tex]
{E_{P1_{z}}} = \frac{{k_{e}}{q}{Z_{0}}}{\left({{R_{0}}^{2}}+{{Z_{0}}^{2}}\right)^{3/2}}
[/tex]
Electric Field due to a Charged Disk (C.D.)
[tex]
{E_{P1_{z}}} = \frac{{2}{k_{e}}{q}{Z_{0}}}{{{R}_{0}}^{2}}{\left[{1}-{\frac{{{Z}_{0}}}{\sqrt{\left({{R_{0}}^{2}}+{{Z_{0}}^{2}}\right)}}}\right]}
[/tex]
The Attempt at a Solution
For easier notation let,
[tex]
{q} = {q_{1}}
[/tex]
(a) Hollow Cylinder
For the hollow charged cylinder, I treat the cylinder as the sum of rings of differential length [itex]{ds}[/itex] and charge [itex]{dq}[/itex]. That is, a differential segment of the ring has a length [itex]{ds}[/itex] and charge [itex]{dq}[/itex].
I recognize that,
[tex]
{{dE}_{net}} = {dE}_{P1_{z}}
[/tex]
Where,
[tex]
{dE}_{P1_{z}} = {dE}{{cos}{\beta}}
[/tex]
Now, because I am using a ring I refer to [itex]{\lambda}[/itex], so that,
[tex]
{\lambda} = {\frac{dq}{ds}}
[/tex]
However, I can already recognize that this approach is folly because I have already indicated (in the figure) [itex]{ds}[/itex] as a differential width of the cylinder as opposed to a differential length segment.
So already I am stuck, any ideas?
Particularly, I know how I want to break up the hollow cylinder (in to a sum of differential rings) however, I am not sure how to put it together. Specifically in terms of what constant: [itex]{\lambda}[/itex], [itex]{\sigma}[/itex], or [itex]{{\rho}_{q}}[/itex]? So, that I only have to integrate once to find the electric field.
Any help is appreciated.
Thanks,
-PFStudent