Uniqueness: The Physics Problem That Shouldn’t Be Solved

Uniqueness: The Physics Problem That Shouldn’t Be Solved

The Uniqueness Theorem can PROVE that this problem only has one possible solution… so however we can find it (e.g. guessing), we know we’ve got the right one!

In this video, we’ll be taking a look at how this uniqueness theorem is derived for the Poisson and Laplace equations in electrostatics.

To begin with, we’ll recall what an electric field and potential field are in electrostatics. The electric field is a vector field generated by charged objects. It tells us about the forces experienced by other charged objects placed in the field. A potential field is a scalar field that also gives us similar information. In fact, the gradient operator applied to the potential field gives us (the negative of) the electric field!

Both the electric and potential fields follow Poisson’s equation for electrostatics, if they are to obey the laws of physics. This equation relates the divergence of the gradient of the potential field, to the charge density in any region of space we want to study.

If we consider a particular region of space, such as a sphere, we can simplify our discussion if we assume there are no charges in this region. With this assumption, the charge density becomes zero and the Poisson Equation becomes the Laplace Equation. Also, we can assume we already know the potential values at the surface of our region of space since we measured them. (The potential field is the same as the voltage that we use when talking about electric circuits). The measured potential values make up our « boundary conditions ».

At this point, we want to calculate what the potential field is everywhere in our region of space, and we know it must satisfy the boundary conditions (i.e. be the same as our measured values on the boundary).

We can now assume that there are two solutions that are different within our region of space, but the same on the boundary. And then we can subtract one from another to study a new quantity, which also happens to obey the Laplace equation. Additionally, this new quantity is zero on the boundary since our two assumed solutions matched our measured values on the boundary.

Then we can apply some general vector calculus identities to show that the value of our new quantity must be a constant. But since this value is also zero on the boundary, it therefore must be zero everywhere. In other words, our two assumed solutions must be the same as each other if they are to subtract to be equal to zero. We’ve just found that it is impossible for there to be two different solutions to our problem.

This theorem is the uniqueness theorem, and shows that there is only one possible solution to this problem. This is useful because it means we don’t need to solve the problem exactly to figure out the right solution for our scenario. Even if we can GUESS the solution, and then plug it back in to our equations to check that it works, this is ok. The reason is that if we somehow find the solution, we know that it’s the only possible one – so we’ve not accidentally found a solution that works, but doesn’t apply to our scenario!

The discussion in this video limits itself to « Dirichlet » boundary conditions (potential known on the boundary). Read this page for more information on « Neumann » boundary conditions (electric field known on boundary) and mixed boundary conditions: https://en.wikipedia.org/wiki/Uniquen…
For a more detailed breakdown of the Poisson and Laplace Equations, check out this video: https://www.youtube.com/watch?v=k91KD…

Timestamps:
0:00 – The Problem That Only Has One (Guessable) Solution
0:45 – The Electric Field Generated by Charged Objects
2:04 – The Potential Field Generated by Charged Objects
3:05 – The Relationship Between the Potential and Electric Fields
3:45 – The Poisson Equation, Charge Density, and Boundary Conditions
6:05 – The Laplace Equation for Electrostatics (Charge Density = 0)
6:17 – Assuming There Are Two Possible Solutions (V1 and V2)
8:01 – Proving There is Only ONE Possible Solution (V1 = V2)
9:03 – The Uniqueness Theorem, and Why One Solution is Useful – Guesswork!
10:32 – Dirichlet, Neumann, and Mixed Boundary Conditions
10:56 – Summary