Earnshaw's Theorem Generalised

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In summary, the conversation discusses a system of N point charges in a box with electromagnetic interactions. Earnshaw's theorem states that the system cannot stay in equilibrium, and it is questioned if the system can have a periodic solution for a given initial configuration. The case of N=2 is also mentioned, with the possibility of the particles staying near opposite corners or approaching each other depending on the charges. The conversation also touches on the static version of Earnshaw's theorem and its limitations in determining the asymptotic behavior of the system. It is noted that the system could potentially have periodic solutions depending on the initial configuration.
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Eynstone
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Consider a system of N (>1) point charges inside a box with a totally reflective inner wall.Assume that there are no interactions other than electromagnetic interactions.
Earnshaw's theorem implies that the system can't stay in equilibrium. Can the system have a periodic solution for an ascribed initial configuration? If not, what's the asymptotic behaviour of the system?
Even the case when N=2 is not so easy . I guess that if the charges were like, the particles would stay near the opposite corners .If the charges were opposite, the particles
would approach . However, the system could be ergodic / chaotic in either cases.
The static version of Earnshaw's theorem rests on that potential has no extrema in free space. The difficulty is that this doesn't hold for solutions of hyperbolic equations (the
4-potential being of this type).
 
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Thus, the asymptotic behaviour of the system is not determined by Earnshaw's theorem. However, it is possible that the system could have periodic solutions for an ascribed initial configuration, depending on the details of the initial configuration.
 

1. What is Earnshaw's Theorem Generalised?

Earnshaw's Theorem Generalised, also known as Earnshaw's Theorem for Arbitrary Charge Distributions, is a mathematical theorem that states that it is impossible to achieve a stable equilibrium with only electric charges.

2. Who discovered Earnshaw's Theorem Generalised?

Earnshaw's Theorem Generalised was first proven by English physicist Samuel Earnshaw in 1842.

3. What is the significance of Earnshaw's Theorem Generalised?

Earnshaw's Theorem Generalised has important implications in electrostatics and magnetostatics, as it shows that it is not possible to have a stable equilibrium solely based on electric or magnetic forces. This theorem has also been applied in the study of molecular interactions and the stability of celestial bodies.

4. Are there any exceptions to Earnshaw's Theorem Generalised?

There are a few exceptions to Earnshaw's Theorem Generalised, such as the stability of atoms and molecules due to the presence of nuclear forces. Additionally, there are certain geometric configurations where stable electrostatic equilibrium can be achieved, but these are rare and require very specific conditions.

5. How is Earnshaw's Theorem Generalised related to Earnshaw's Theorem for Point Charges?

Earnshaw's Theorem Generalised is an extension of Earnshaw's Theorem for Point Charges, which states that it is impossible to achieve a stable equilibrium with only point charges. Earnshaw's Theorem Generalised expands this concept to arbitrary charge distributions, making it a more general and widely applicable theorem.

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