Calculating Electric Field at the Center of a Hemispherical Shell of Charge

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SUMMARY

The discussion focuses on calculating the electric field at the center of a hemispherical shell of charge Q, derived from a hollow sphere with outer radius R2 and inner radius R1. The problem requires integration using spherical coordinates due to the lack of symmetry, making Gauss's Law inapplicable. Participants emphasize the importance of setting up the integral correctly, considering charge density, and using Coulomb's law to find the electric field. A suggestion is made to approach the problem by summing the electric fields from infinitesimal charge elements, which can simplify the integration process.

PREREQUISITES
  • Understanding of electric fields and Coulomb's law
  • Familiarity with Gauss's Law and its limitations
  • Knowledge of spherical coordinates and their application in integration
  • Basic calculus skills, particularly in multivariable integration
NEXT STEPS
  • Learn how to set up integrals in spherical coordinates for electric field calculations
  • Study the derivation of electric fields from continuous charge distributions
  • Explore the concept of charge density and its role in electric field calculations
  • Investigate methods for summing electric fields from infinitesimal charge elements
USEFUL FOR

Students studying electromagnetism, physics educators, and anyone interested in advanced calculus applications in electric field calculations.

swervin09
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Homework Statement


A hollow sphere of outer radius R2 and inner radius of R1 carries a uniform charge 2Q. The sphere is then cut in half to create a hemispherical shell of charge Q. Calculate E at the center point (origin) P.


Homework Equations


equation of a hollow sphere = 2/3π(r2-r1)
Gauss' Law ∫E dot dA
surface area hemisphere = 2πr^2


The Attempt at a Solution


Well, I know this is an integration problem and that I am better off integrating with polar coordinates and that I will be integrating from 0-->π as my lower and upper integral bounds.
But in all honesty I haven't had much fortune setting the integral up. The set up is the help I am asking for.
 
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If I am not mistaken, you need to find the E field due to a HEMISPHERICAL shell

That involves some somewhat-complicated multiple integrals and E form Coulumb's law. Gauss's law will not work due to lack of symmetry
 
Assuming that that is your problem, here is my hint:

Use spherical coordinates. find the charge density. consider an infinitesimal piece of the shell and the Coulomb force on a test charge at the origin. Then choose appropriate limits for r, theta, and phi and integrate
 
Yes that is the problem. spherical coordinates make more sense. I will try that and post tomorrow what I have come up with. I don't have my calculus book nearby to refresh my memory of spherical coord. integration. This is one of those problems that has me intrigued and eager to "beat". It isn't worth a lot of points but it is due Thursday.
Thank you for the hint!
 
no problem!

I might add: depending on how comfortable you are with multivariable, you don't have to use a triple integral; just find the field of a ring, sum into a washer, sum the washers into a shell. The spherical coordinates are just a way of thinking, no need to get formal about it
 
I follow you up until you state sum the washers into a shell. Please provide a hint as to that specific.
 

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