About Coefficients of Potentials

In summary, the conversation discusses the proof of non-negativity of coefficients of potential, which states that Pij ≥ 0. The method used is the coefficients of potential method, which relates the potential on the ith conductor to the charges on the conductors. It is mentioned that it is difficult to prove this in a rigorous way, but the suggestion is to use Pii ≥ Pij. The conversation also mentions the need for a specific arrangement, such as grounding the jth conductor and setting Qi on the ith conductor, to obtain a result. However, the correct conditions have not yet been found.
  • #1
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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:

[itex]\Phi[/itex]i = [itex]\sum[/itex] 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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  • #2
how is the system arranged?
 
  • #3
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.
 
  • #4
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.
 
  • #5


Thank you for your question. The coefficients of potential method is a widely used technique in the study of electrostatics and is based on the assumption that the potential on a conductor is a linear combination of the charges on all other conductors in the system. In order to prove that the coefficients of potential are non-negative, we must first understand the physical meaning behind them.

The coefficients of potential, denoted as Pij, represent the proportionality between the potential on the ith conductor and the charge on the jth conductor. This means that a higher value of Pij indicates a stronger influence of the jth conductor's charge on the potential of the ith conductor. Therefore, in order for the potential on the ith conductor to be positive (or non-negative), the influence of the jth conductor's charge must also be positive (or non-negative).

Now, let us consider the case where the potential on the ith conductor is equal to the potential on the jth conductor (i.e. \Phii = \Phij). This implies that the influence of the jth conductor's charge on the potential of the ith conductor is equal to the influence of the ith conductor's charge on its own potential. In this case, we can say that Pii = Pij, as the two coefficients of potential are equal.

Since the potential on a conductor must be non-negative, we can conclude that Pii ≥ Pij, as the influence of the conductor's own charge on its potential must be equal to or greater than the influence of any other conductor's charge on its potential. Therefore, we can use this relation to prove that all coefficients of potential are non-negative.

In conclusion, the coefficients of potential are non-negative because the potential on a conductor must be non-negative and the influence of a conductor's own charge on its potential is equal to or greater than the influence of any other conductor's charge. This is a fundamental concept in electrostatics and is crucial in understanding and analyzing complex systems of conductors. I hope this explanation helps in your understanding of coefficients of potential.
 

What are coefficients of potentials?

Coefficients of potentials refer to the numerical values assigned to each element or compound in a chemical reaction to represent their relative amounts.

How are coefficients of potentials determined?

Coefficients of potentials are determined through experimentation and observing the changes in the amount of each element or compound before and after a reaction takes place.

What do coefficients of potentials tell us about a chemical reaction?

Coefficients of potentials provide information about the stoichiometry of a reaction, including the relative amounts of reactants and products and the ratio in which they combine.

Can coefficients of potentials change during a reaction?

Yes, coefficients of potentials can change during a reaction as the reaction progresses and the amounts of reactants and products change.

What is the significance of coefficients of potentials in balancing chemical equations?

Coefficients of potentials are essential in balancing chemical equations as they ensure that the number of atoms of each element is equal on both sides of the equation, satisfying the law of conservation of mass.

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