# Infinite wire, vector potential

If I want to calculate the magnetic vector potential $\vec{A}$ of a infinite wire using the formula $$A(\textbf r)=\frac{\mu_0}{4\pi}\space \int \frac{\textbf j(\textbf r')\space d\textbf r'^3}{|\textbf r-\textbf r'|}$$ at a distance $a$ from the wire I get a integral that does not converge: $$A(| \textbf{r}|=a)=\frac{\mu_0 I}{4\pi}\ \int_{-\infty}^{\infty}\frac{dy}{\sqrt{(x+a)^2+y^2}}$$. Question: Is this a consequence of the fact that $\nabla\space \textbf j=0$, because of charge conservation, and an infinite wire wouldn't conserve charge, since the both ends never meet to form a "loop "?

The energy of something that is constant and infinite is (usually) divergent. Loosely speaking, we are in a situation where $$A=\rho_0 L+\text{finite}$$ where $\rho_0$ is a certain constant and $L$ is the length of the wire, which is formally infinite. This term - in more complex situations - is sometimes called a zero-point energy: a constant shift in the energy which is not measurable and doesn't affect physical predictions.
On the other hand, the magnetic field $\boldsymbol B$ is the derivative of $\boldsymbol A$ with respect to $a$, which is finite, because the $\rho_0 L$ term doesn't depend on $a$ (its derivative vanishes). The magnetic field is finite, as it should be: this object is measurable, unlike $\boldsymbol A$.
The same situation appears when one considers an infinite sheet of charge: there the electric field is finite, but the scalar potential diverges: $$\phi=\rho_0 S+\text{finite}$$ where $S$ is the area of the sheet, formally infinite $S\to\infty$. Again, this is a consequence of the fact that the potential is an extensive magnitude, and it scales with the volume of space.