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3
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1answer
70 views

Does this example contradict Earnshaw's theorem in one dimension?

This is basically a continuation of the post here. Consider electrostatics in $1$-dimension (say, the $x$-axis). Now consider a positive charge $+q$ located at $x=0$, and two equal negative charges $...
5
votes
2answers
1k views

Facing a paradox: Earnshaw's theorem in one dimension

Consider a one-dimensional situation on a straight line (say, $x$-axis). Let a charge of magnitude $q$ be located at $x=x_0$, the potential satisfies the Poisson's equation $$\frac{d^2V}{dx^2}=-\frac{\...
2
votes
1answer
186 views

Thermodynamic Stability - Convexity - Concavity of Thermodynamical potential

The thermodynamic stability principle requires convexity of the internal energy upon all of its independent variables. When we pass through the Legendre transforms to build all the other ...
1
vote
1answer
74 views

Equilibria and conserved quantities in a superposition of central potentials

I have a Lagrangian made up of a superposition of $k$ central repulsive potentials centered at $(a_k,b_k)$. Each potential is somewhat strange - they take the form $V(r)=-\ln\left(r\right)$. The whole ...
2
votes
1answer
111 views

Why physically do things in general tend to move toward a lower potential value in a potential field? [duplicate]

There are many answers on the site discussing motion of electrons in an electric potential field, See Why is voltage described as potential energy per charge? but also mass tends to move toward a ...
1
vote
1answer
314 views

Why is the helmholtz free energy minimized?

I don't quite understand the principle of minimum energy despite having read the derivation on Wikipedia. I think I got lost when the free energy was defined as $A= \max_S{\left(U-TS\right)}$, ...
5
votes
3answers
2k views

What is the state of the equilibrium for a second derivative equal to zero?

Considering a potential energy of $U$, and a displacement of $x$, the force is given by $F=-\frac{\partial U}{\partial x}$. Since equilibrium is defined as the point at which $F=0$, we can express ...
0
votes
3answers
491 views

Same Charge or Same Potential?

I was studying about sharing of charge between two charged conductors when I came across this statement - When two conductors are joined together through a conducting wire, charge begins to flow ...
0
votes
0answers
89 views

Deriving the Force on an ion in the Classical Harmonic Crystal

I was following Solid State Physics by Ashcroft and Memrmin and their introduction to the Classical Theory of the Harmonic Crystal. (Chapter 22) If there are a large number of ions in a straight line,...
-7
votes
1answer
3k views

Lennard-Jones potential - equilibrium [closed]

If I have Lennard-Jones potential, how can I calculate equilibrium? $$ U(r)=\epsilon \left ( \left (\frac{r_m}{r} \right )^{12}-2 \left (\frac{r_m}{r}\right )^6 \right ) \ $$