So, I've been trying to come up with an idea that seems plausible to me for the last half week or so, but I'm stuck and I can't seem to get anywhere with this problem.
The problem is as follows (probably crude translation of the task):
Consider two uniformly charged wires of a certain line charge density $\lambda$ that intersect at a 90 degree angle. Compute the electric field $\vec{E}(\vec{r})$ and the electric scalar potential $\phi(\vec{r})$.
Now, here's my progress (if you want to call it that) so far:
From Gauss' Law we know that the electric field of a single charged wire should be
$$\vec{E}_1(\vec{r})=\frac{Q}{2\pi\epsilon_0r_1l}\hat{r}_1$$
Using the line charge density $\lambda=\frac{Q}{l}$, I know that $Q=\lambda l$. Once I insert that into the equation for my electric field and cross out the $l$s I get
$$\vec{E}_1(\vec{r})=\frac{\lambda}{2\pi\epsilon_0r_1}\hat{r}_1$$
This should be my electric field for just one charged wire, right? Now if I add another wire that is exactly the same I could make use of the superposition principle and add them together. Then my total electric field should look something like this:
$$\vec{E}_{total}(\vec{r})=\frac{\lambda}{2\pi\epsilon_0}(\frac{1}{r_1}\hat{r_1}+\frac{1}{r_2}\hat{r_2})$$
With $\hat{r_1}$ and $\hat{r_2}$ being the unit vectors into the respective direction of a test charge and $r_1$ and $r_2$ being the distances from that test charge to my wires.
Here's my first big problem: I do not know whether this is actually the correct answer to the problem and I lack the confidence to just assume that it is. I can't seem to find any means of checking my result either.
As the second part of the question relies heavily on the first any further results may already be wrong from the get-go. Here's my attempt at finding the electric scalar potential:
Instead of using the equation for $\vec{E}_{total}$ I tried to compute the two potentials individually and add them together afterwards relying on the superposition principle once more.
$$\phi_1(\vec{r})=-\int_\infty^r{\vec{E}_1(\vec{r})dr'}=-\int_\infty^r{\frac{\lambda}{2\pi\epsilon_0r'_1}dr'}=-\frac{\lambda}{2\pi\epsilon_0}\int_\infty^r{\frac{1}{r'_1}dr'}$$
So, here's my next big problem. If I compute this integral and form the anti-derivative of $\frac{1}{r'_1}dr'$ I end up with:
$$\phi_1(\vec{r})=-\frac{\lambda}{2\pi\epsilon_0}[\ln{r'_1}]_\infty^r=-\frac{\lambda}{2\pi\epsilon_0}(\ln{r}-\ln{\infty})=-\frac{\lambda}{2\pi\epsilon_0}\ln{\frac{r}{\infty}}$$
Now I have no idea where to go from there … if I continue and plug in the integration limits I'd end up with some infinite potential so I tried a different approach and used a different equation for the scalar potential:
$$\phi_1(\vec{r})=\frac{1}{2\pi\epsilon_0}\frac{Q}{r_1}=\frac{1}{2\pi\epsilon_0}\frac{\lambda l}{r_1}$$
In one way or another this does resemble the above result, which makes it seem pretty plausible to me, so I continued with this and after creating a "second" potential like this one I added them together relying on the superposition principle:
$$\phi_{total}(\vec{r})=\frac{\lambda l}{2\pi\epsilon_0}(\frac{1}{r_1}+\frac{1}{r_2})$$
Again, I stand (sit) here with little to no confidence in my result.
Best Answer
When dealing with infinitely long line charges (basically a cylindrical geometry) calculating the potential relative to infinity becomes a problem. You have to establish a reference (ground/earth) at a finite location. So, your result of an infinite potential difference is not incorrect, although it is confusing the first time you see it.
This site provides a description of why this happens. You are effectively grounding one end of your distribution, then stacking an infinite amount of charge down the line.