What is the Electric Field at a Small Distance from a Charged Ring?

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SUMMARY

The electric field at a small distance \( r \) from the center of a charged ring with radius \( a \) is given by the formula \( \frac{Qr}{8\pi\epsilon_0 a^3} \). The discussion emphasizes that using Gauss's law is not appropriate in this scenario due to the non-uniformity of the electric field across the Gaussian surface. Instead, it is recommended to integrate the electric field contributions from each segment of the ring or to work with the electric potential for a more straightforward solution.

PREREQUISITES
  • Understanding of electric fields and potentials
  • Familiarity with Gaussian surfaces and Gauss's law
  • Knowledge of integration techniques in physics
  • Basic concepts of charge distribution
NEXT STEPS
  • Study the derivation of electric fields from charge distributions
  • Learn about the integration of electric field contributions from continuous charge distributions
  • Explore the concept of electric potential and its relationship to electric fields
  • Review examples of using Gauss's law in symmetric charge distributions
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Physics students, educators, and anyone studying electromagnetism, particularly those focusing on electric fields and charge distributions.

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Homework Statement



I have a ring, radius a, with a charge distributed evenly around it. Using a gaussian cylinder of radius r, r<<a (or otherwise). Find the electric field at at small distance r away from the centre of the ring, r is in the plane of the ring.

I know that the answer is

[tex]\frac{Qr}{8\pi\epsilon_0 a^3}[/tex]



Homework Equations



[tex]\int E\cdot ds = Q/\epsilon_0[/tex]


The Attempt at a Solution



I can't get started since I find that the charge enclosed is 0

Thanks
 
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Don't try to use Gauss's law. (What would be your Gaussian surface? Is the field uniform across it?) Instead, integrate the field contributions from each part of the ring. (It might prove easier to work with the potential instead of the field.)
 

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