Is Earnshaw’s Theorem wrong?
Physics Journal: Think outside the Square
Earnshaw’s Theorem states that no particle can be in stable equilibrium under the action of electrostatic forces alone.
But consider a point P at the centre of a square of four equal positive charges. If you put a positive test charge at P, it seems to be in stable equilibrium because every one of the four external charges pushes it towards P.
Has Earnshaw’s Theorem been disproved?
What makes an equilibrium ‘stable’?
For a stable equilibrium, even if the object is slightly displaced, it will return to equilibrium. The surrounding forces are pointing towards the equilibrium position.
So let’s start by looking at the forces around P.
Forces around P
If the test charge is brought a bit closer to the purple charge, that charge will repel it more strongly than the others.
The resultant force will point towards the equilibrium position. So the equilibrium seems to be stable in this case.
Can we say the same at every point around P?
Here, I must admit I cheated by using an electric field simulation made by Wolfgang Christian.
An electric field is the electric force on a unit charge. So for our purposes, we just need to use the fact that the field and the force are in the same direction.
Yes! All the forces around P are pointing towards it.
We can get a stable equilibrium from electrostatic forces alone and Earnshaw was completely wrong!
But wait! have we considered all the directions the test charge could be displaced by a tiny bit?
All this is happening, or assumed to be happening, in our world, which is 3D. What if we pulled it out of the square?
3D diagram
To answer that, we have to move our diagram up a dimension.
Then we add to the diagram the forces exerted on the test charge by each of the four charges in the square.
By symmetry, the x and y components of the vectors cancel so the resultant force will point upwards along the z axis.
Because the resultant force is pointing away from P, the equilibrium is actually unstable.
One little push out of the plane of the square and the delicate balance of forces would break down.
It turns out Earnshaw’s Theorem has survived the 200-year test of time.
I started keeping a ‘Physics Journal’ in the hopes of solving physics problems regularly and sharing my thought process with anyone interested. Who knows, maybe you might be inspired to delve deeper into the depths of physics?
Aphysicist’s approximation of a cow. In an alternate universe, I am a college student studying maths and physics. If you enjoyed this story, don’t hesitate to follow and give it a clap!
