Mean Value Theorem and electrostatic potential

[Hypnotoad]

Prove that for charge-free two-dimensional space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any circle centered on that point. Do this by considering the electrostatic potential as the real part of an analytic function.

I have no idea how to start this problem and am not sure what to do with the analytic function information. Any hints on how to start this would be really appreciated.

 

[Wong]

In fact what you said is a general theorem regarding harmonic functions in any dimensions. By definition a harmonic function is a function f satisfying $$\sum_{i} {\partial_{i}^{2} f}= 0$$.

To prove the assertion in two dimension, you may like to recall that in a source free region, the potential satisfies the laplace equation and may be regarded as the real part of an analytic function (because of the riemann condition on analyticity). Then you may like to recall which theorem in complex analysis allows you to express the value of an analytic function at a point as an integral over a contour?

 

[M98Ranger]

The Mean Value Theorem is a fundamental concept in calculus that states that for a continuous function f on a closed interval [a,b], there exists a point c in the interval such that the average value of the function over that interval is equal to the value of the function at c. In other words, if we take the total change in the function over the interval [a,b] and divide it by the length of the interval, we will get the value of the function at some point c in the interval.

In the context of electrostatics, the electrostatic potential is a function that describes the electric potential energy per unit charge at a given point in space. It is a continuous function and satisfies the Laplace's equation, which is a partial differential equation that relates the potential to the charge distribution in the space.

Now, let's consider a charge-free two-dimensional space, which means that there are no charges present in the space. In this case, the electrostatic potential is a harmonic function, which means that it satisfies the Laplace's equation and can be written as the real part of an analytic function. This means that we can write the electrostatic potential as the real part of a complex function f(z) = u(x,y) + iv(x,y), where z = x + iy is a complex variable and u(x,y) and v(x,y) are real-valued functions.

Using the Mean Value Theorem, we can say that for any circle centered at a point z0 = x0 + iy0, the average value of the potential over the surface of the circle is equal to the potential at some point z1 = x1 + iy1 on the circle. In other words, we can write:

1/2π ∫0^2π u(x0 + rcosθ, y0 + rsinθ) dθ = u(x1,y1)

where r is the radius of the circle and θ is the angle around the circle. This means that the potential at any point z0 is equal to the average potential over the circle centered at that point.

To prove this, we can use Cauchy's Integral Formula, which states that for an analytic function f(z) = u(x,y) + iv(x,y) and a simple closed curve C, we have:

f(z0) = 1/2π i ∫C f(z)/ (z-z0) dz

Applying this formula to our