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

In summary: 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. 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?
  • #1
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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  • #2
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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  • #3
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?
 
  • #4
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?
 

1. What is the formula for calculating the electric field of a cylinder given the electric field of a ring?

The formula for calculating the electric field of a cylinder given the electric field of a ring is E = λ/2πε₀r, where λ is the charge per unit length, ε₀ is the permittivity of free space, and r is the distance from the center of the cylinder.

2. Can the electric field of a cylinder be calculated using the same formula as the electric field of a ring?

Yes, the formula for calculating the electric field of a cylinder is derived from the formula for the electric field of a ring. The only difference is the inclusion of the radius of the cylinder in the calculation.

3. How does the electric field of a cylinder compare to the electric field of a ring?

The electric field of a cylinder is similar to the electric field of a ring, but it is not the same. The electric field of a cylinder is stronger near the ends of the cylinder, while the electric field of a ring is uniform along its circumference.

4. What factors affect the electric field of a cylinder?

The electric field of a cylinder is affected by the charge per unit length, the radius of the cylinder, and the distance from the center of the cylinder. It is also affected by the surrounding medium and any other nearby charges.

5. Can the electric field of a cylinder be negative?

Yes, the electric field of a cylinder can be negative if the charge per unit length and the distance from the center of the cylinder are both negative. This indicates that the electric field is directed in the opposite direction of the positive charges.

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