Are Coefficients of Potential Non-Negative?

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SUMMARY

The discussion centers on proving that coefficients of potential, denoted as Pij, are non-negative (Pij ≥ 0) in a system of conductors. The relationship between the potential on the ith conductor (Φi) and the charges on conductors is established through the equation Φi = ∑ Pij Qj. The participants note that while Reitz and Milford indicate the difficulty of a rigorous proof, they suggest using the condition Pii ≥ Pij and consider specific arrangements of conductors to derive results, particularly when conductors are sufficiently distanced to minimize interaction effects.

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  • Understanding of electrostatics and conductors
  • Familiarity with the coefficients of potential method
  • Knowledge of potential theory in physics
  • Ability to interpret mathematical equations related to charge and potential
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  • Study the implications of conductor arrangements on potential coefficients
  • Explore the conditions under which Pij can be proven to be non-negative
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Homework Statement



I want to prove that coefficients of potential are non-negative. i.e. Pij ≥ 0

Homework Equations



By coefficients of potential method we know that in a system of conductors the potential on ith conductor is related to charges on conductors by:

\Phii = \sum Pij Qj

The Attempt at a Solution



It is written in Reitz and Milford that it is hard to prove in a rigorous way. I think that we should use Pii ≥ Pij.
 
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how is the system arranged?
 
This is a general statement. Of course we should set a specific arrangement to obtain a result. for example put Qi on ith conductor and assume jth conductor to be grounded and Q on other conductors to be zero.
But I haven't found the correct conditions.
 
in your scenario, should the conductors be sufficiently a long way away, their potentials will not effect each other (to a certain approximation) hence defeating the equation connecting the potential of one to the charge of another.
 

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