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I'm trying to understand the use of conformal mapping to solve problems in electrostatics. To better understand the idea, I'm trying to learn how to solve this example (but you can propose any other example if you think it's better).

Consider conductor with cylindrical cross-section is joined to two semi-infinite planes that form an angle $\alpha$ between them like this figure. I want to find the potential between both planes.

enter image description here

The problem suggests to use the conformal mapping $z_2=z_1^\beta$, which I've never hear before to but I've already read about in a couple of sources (Zangwill and Wesley books on EM). Apparently this should allow me to transform this problem into an equivalent problem which is easier to work.

For example, if I only begin with Poisson's equation in polar coordinates:

$$\nabla^2 \phi=\rho / \epsilon$$

With the boundary conditions $\phi(\theta=0)=\phi(\theta=\alpha)=\phi(r=a)=0$. Assuming the charge density for a cylinder is constant, we can solve for the potential:

$$\frac{\partial^2 \phi}{\partial r^2}+\frac{1}{r}\frac{\partial \phi}{\partial r}+\frac{1}{r^2}\frac{\partial^2 \phi}{\partial \theta^2}=\rho / \epsilon$$

Which is very hard to solve (since I can't even use Green functions due to the boundary conditions).

Now, I'll try to use the conformal mapping suggested, I can start by setting:

$z_1=x+iy \ \ \ , \ \ \ z_2=u+iv$

$z_2=z_1^\beta \ \ \rightarrow \ \ \ u+iv=(x+iy)^\alpha$

Or even simpler, in polar coordinates:

$z_1=re^{i\theta} \ \ \ , \ \ \ z_2=\rho e^{o\phi}$

$z_2=z_1^\beta \ \ \rightarrow \ \ \ \rho e^{i\phi}=r^\beta e^{i\beta\phi}$

If $\beta<1$, the angle between the planes decreases and the radius of the cylinder shrinks.

If $\beta>1$, the angle between the planes increases and the radius of the cylinder is enlarged.

However, in both cases, I don't see how making such a conformal mapping makes the problem easier, as I'm only changing the angles and I'm still with the problem of finding the potential between two planes and a cylinder.

Update:

The only thing I can think would be better is choosing $\beta=\pi / \alpha$ such that the planes in the new mapping form a straight line like this:

enter image description here

Which I could potentially solve using the analogy to the case of a sphere in an uniform field, but I'm not sure if this is valid. Assuming it is valid, would that mean that, once I find the solution to this problem in terms of $\rho$ and $\phi$, I just work backwards the mapping to find the solution to the original problem in terms of $r$ and $\theta$?

The examples I read in the literature only dealt with mapping planes to planes or circles to circles and left the solutions only for the new mapping, so that's what I find this topic a bit confusing.

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  • $\begingroup$ If we take the potential at $\infty$ to be 0 (assuming all the charges are localized near the origin), then the potentials you have specified cannot be 0 (unless $\phi = 0$ everywhere). If the potential on the conductors is $V_0$ say, then you can tack on another mapping $w = z + 1/z$ to straighten out the semicircle and have the upper half plane as your region. See e.g. Spiegel "Schaum's Outline of Complex Variables", p. 206. $\endgroup$ – NickD Feb 10 at 2:50
  • $\begingroup$ ... or more accessibly perhaps, see No.13 here $\endgroup$ – NickD Feb 10 at 3:00
  • $\begingroup$ Sorry I forgot to include, the cylinder indeed has some charge density and thus a potential $V_0$. The surface of the cylinder as well as the intersecting planes are grounded ($\phi=0$). I see, so under that transformation I could transform the problem into one of a half plane, which is easier to transform. I'll check the link you gave me to understand how to apply the transformations and I'll reply back if I have a question. $\endgroup$ – Charlie Feb 10 at 3:19
  • $\begingroup$ From that link, using transformation no. 5, I could also map the intersecting planes into a single plane and solve for the upper half. In that case, how should I do the change of variables? Should I do something like $\rho e^{i\phi}=r^{\pi / \alpha}e^{i\pi\theta / \alpha}$ and then use Euler transform to find that how much is $\rho$ and $\phi$ in terms of $r$ and $\theta$? (Here I defined $r$ and $\theta$ before the mapping and $\rho$ and $\phi$ after the mapping). If that is true, then I could find the potential, express it in terms of $\rho$ and $\phi$ and then just substitute back. $\endgroup$ – Charlie Feb 10 at 3:23

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