Average electric field over a spherical surface

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SUMMARY

The average electric field over a spherical surface is calculated using the integral formula \(\frac{1}{4\pi R^2}\oint\mathbf{E}\cdot d\mathbf{a}\). This formula results in a scalar value rather than a vector, which is a crucial distinction in understanding electric fields. The discussion references a previous thread on Physics Forums that addresses similar questions, providing additional context and insights into the topic.

PREREQUISITES
  • Understanding of electric fields and their properties
  • Familiarity with vector calculus and surface integrals
  • Knowledge of spherical coordinates
  • Basic principles of electromagnetism
NEXT STEPS
  • Study the concept of surface integrals in vector calculus
  • Explore the properties of electric fields in spherical symmetry
  • Learn about the divergence theorem and its applications in electromagnetism
  • Review related discussions on Physics Forums for deeper insights
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Students and professionals in physics, particularly those focusing on electromagnetism, as well as educators seeking to clarify concepts related to electric fields and integrals.

alexmahone
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Homework Statement
Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the center.
Relevant Equations
This is where I'm stuck.
I'm sure the average is going to be an integral, but \displaystyle\frac{1}{4\pi R^2}\oint\mathbf{E}\cdot d\mathbf{a} gives me a scalar, not a vector.
 
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