# Problems in calculating potential of uniformly charged infinite plane or wire

From $$\int_Vd^3x \rho(\vec x)\mathrel{\mathop{=}\limits^!}Q_V$$ and by use of Dirac's delta distribution one finds that the charge density for the uniformly charged infinite plane is $$\sigma\cdot\delta(z)$$ and that the charge density for the uniformly charged infite wire is $$\kappa/\pi\cdot\delta(r)/r$$, where $$\sigma/\kappa$$ are some constants with dimension charge per area/lenght and $$r$$ denotes the orthogonal distance from the wire.

I want to calculate the electric potential associated to both densities, the general ansatz from coulomb superposition is:
$$\qquad\varphi(\vec x) = \frac1{4\pi\varepsilon_0}\int_{R^3}d^3r \frac{\rho(\vec r)}{\|\vec r-\vec x\|}.$$

Both times I try to do the integral and end up with $$\forall\vec x:\varphi(\vec x)=+\infty$$.

Plane:
Components: $$\vec x = (a,b,h)$$.
$$4\pi\varepsilon_0/\sigma\cdot\varphi(\vec x)= \int_{-\infty}^\infty dx\int_{-\infty}^\infty dy\int_{-\infty}^\infty dz \frac{\delta(z)}{ \sqrt{(x-a)^2+(y-b)^2+(z-h)^2}}$$
$$\quad= \int_{-\infty}^\infty dx\int_{-\infty}^\infty dy \sqrt{(x-a)^2+(y-b)^2+h^2}^{-1}$$
which can be reparametrised using polar coordinates and choosing new variables $$(x-a)/h$$ and $$(y-b)/h$$:
$$\quad=2\pi h\int_0^\infty dr\;r\sqrt{1+r^2}^{-1} = 2\pi h\left.\sqrt{1+r^2}\right|_{r=0}^\infty = \infty$$ while this result is independent of $$a,b,h$$.

Wire:
Components: $$\vec x = s\hat\rho(\alpha) +h\hat z$$.
$$4\pi^2\varepsilon_0/\kappa\cdot\varphi(\vec x)= \int_0^\infty r\;dr\int_0^{2\pi}d\phi\int_{-\infty}^\infty dz\; \delta(r)/\big(r\sqrt{\|\vec x-\vec r(r,\phi,z)\|}\big)$$
after the integral over $$r$$ the integrand is not anymore dependent on $$\phi$$ and note that $$\forall a>0,f:\int_0^adx\,f(x)\delta(x)=f(0)/2$$:
$$\quad = \pi\int_{-\infty}^\infty dz\; \sqrt{s^2+(z-h)^2}^{-1}=\left.\pi\, \mathrm{asinh}(w)\right|_{-\infty}^\infty=\infty$$,
again transforming $$w=(z-h)/s$$.

Literature results:
Plane: $$\varphi(\vec x) = \varphi(h) = C-|h|\sigma/(2\varepsilon_0)$$
Wire: $$\varphi(\vec x)=\varphi(s)=C-\kappa\ln(s)/(2\pi\varepsilon_0)$$