Is Electric Potential the Key to Understanding Earnshaw's Theorem?

[M-Speezy]

I was working on a problem out of Griffiths, and have become a bit confused. The problem is regarding to Earnshaw's theorem, which states that a particle cannot be held in stable equilibrium by electrostatic forces. (3.2 for anyone with the text). He suggests a cube with a positive charge on each of the eight corners, and asks what would happen to a positive charge placed in the center. It seems to me that the charge could stay at rest if it were somehow perfectly placed between all the corners, but that's not really possible so it is not worth considering. Earnshaw's theorem state that it could not be contained and will be removed. However, what if the corners became really close, or there were more positive charges... Wouldn't it simply be contained due to repulsion from the sides? That may just be the answer, but I think I've become a little bit confused. There's also the matter of the electric potential. The middle would be the point of highest electric potential. He warns against thinking of electric potential as being 'potential energy' but is it correct to assume the particle would move to a point of lower electric potential?

Thanks for your time, and sorry if this is a bit scatter-brained. I found I was unclear about several things when writing this.

 

[Simon Bridge]

...is it correct to assume the particle would move to a point of lower electric potential?

By convention, a positive test charge will seek the lower potential and a negative test charge will seek the higher potential.

In the example, the potential at the center of the cube is a local saddle point - bringing the corners closer just squeezes the saddle so the point becomes even more unstable.

 

[M-Speezy]

How literally should we take Earnshaw's theorem then? What if instead of a cube it were a spherical array of positive charges. Would you still have the same result, of it being mathematically impossible to contain the charge using electrostatic repulsion? I can see why this would be the case, knowing about Laplace's equation, but it also seems to defies intuition.

 

[Simon Bridge]

What if instead of a cube it were a spherical array of positive charges. Would you still have the same result, of it being mathematically impossible to contain the charge using electrostatic repulsion?

Yes - the resulting potential cannot contain a local minima.

I can see why this would be the case, knowing about Laplace's equation, but it also seems to defies intuition.

Well - take the extreme example - let's add charges to the sphere until they touch each other!
i.e. Imagine you have a uniform spherical shell of positive charge instead of a collection of discrete charges.
Now there are no gaps - intuitively all that charge would push a positive charge to the center perhaps?

But: the field everywhere inside the shell is zero, a result you should already be familiar with.

A small positive charge in the center, given a slight nudge, will therefore drift right to the shell. Once it enters the shell, it experiences a net force pointing radially away from the center. Hence it is inevitable that the charge should escape.

A distribution of discrete charges will just have favored escape routes built in.

We can electromagnetically trap charges, it is just not done statically.

 

[M-Speezy]

Oh, now that is an interesting thought. Thanks for your help, you've done wonders to clear this up for me. Have a good day!

 

[Simon Bridge]

No worries.

There are lots of surprising and counter-intuitive things in electromagnetism, and in physics for that matter.
It turns out that this is because our intuition is not that great-a tool for understanding things we don't regularly experience directly. We have to think about things more carefully. Remember - intuition is what tells you the Earth is flat.

Enjoy.