Electric field of a cylinder given the electric field of a ring

AI Thread Summary
The discussion centers on the necessity of introducing the variable z' for integrating the electric field of a cylinder, as z is fixed and cannot be varied. The integration with respect to z' allows for summing contributions from infinitesimal rings along the cylinder. There is a concern about a potential typo in the final solution regarding the missing R in the numerator, which affects dimensional correctness. Participants question the similarity between the integration for a disk and a cylinder, highlighting the differences in how each ring contributes to the overall electric field. Understanding these distinctions is crucial for correctly applying the integration process in electric field calculations.
notgoodatphysics
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Homework Statement
We’re given the equation for the electric field of a disk. From that the idea is to find the electric field of a cylinder.

I thought the best way would be to integrate the original equation over the surface area of a cylinder without the ends (2*pi*r*h). My attempt is similar to the solution except, the professor has introduced z’, and an R in the final solution has disappeared.

Why introduce z’ and have the dz’ above the origin instead of just using z like the original diagram?

And where did the R go in the numerator in final step of the solution?
Relevant Equations
The first pic is the question and my attempt, and the second pic is the solution.
 
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notgoodatphysics said:
Why introduce z’ and have the dz’ above the origin instead of just using z like the original diagram?
The symbol ##z## represents the position of the point on the ##z##-axis. It's fixed. You can't integrate over that variable. You need a different variable, namely ##z'##, which corresponds to the position of an infinitesimal ring. Then you integrate with respect to ##z'## to sum the contributions over the entire cylinder.

notgoodatphysics said:
And where did the R go in the numerator in final step of the solution?
I think this is just a typo, and the ##R## should still be there. The final answer in your professor's solution isn't dimensionally correct.
 
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vela said:
The symbol ##z## represents the position of the point on the ##z##-axis. It's fixed. You can't integrate over that variable. You need a different variable, namely ##z'##, which corresponds to the position of an infinitesimal ring. Then you integrate with respect to ##z'## to sum the contributions over the entire cylinder.I think this is just a typo, and the ##R## should still be there. The final answer in your professor's solution isn't dimensionally correct.
Thanks for the reply!

I’m not sure I’m totally understanding the need for z’ though. In the example of the electric field of due to a ring of charge with radius a, to find the electric charge due to a disk, the integral from 0 to R is calculated - isn’t this a similar case? Why isn’t it (a-R).

(From here: https://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html )

Also when we find the integral of (z-z’), why don’t we also take the direction of the electric field E(z) ##cos theta## because ##\cos \theta##is changing right?
 
notgoodatphysics said:
I’m not sure I’m totally understanding the need for z’ though. In the example of the electric field of due to a ring of charge with radius a, to find the electric charge due to a disk, the integral from 0 to R is calculated - isn’t this a similar case? Why isn’t it (a-R).
When you break up a disk into a collection of rings, what's different about each ring? Similarly, when you divide the cylinder up into a collection of rings, what's different about those rings?
 
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