Question

Magnetic scalar potential is defined as ${\displaystyle U\left({\overrightarrow{r}}_2\right)-U\left({\overrightarrow{r}}_1\right)=-{\int }_{\overrightarrow{r_1}}^{\overrightarrow{r_2}}}$ ${\displaystyle \overrightarrow{B}.d\overrightarrow{l}}$

Apply this equation to a closed curve enclosing a long straight wire. The RHS of the above equation is then ${\displaystyle -u_{0}i}$ by Ampere's law. We see that ${\displaystyle U\left(\overrightarrow{r_2}\right)\ne U\left(\overrightarrow{r_1}\right)}$ even when ${\displaystyle {\overrightarrow{r}}_2={\overrightarrow{r}}_1}$ .Can we have a magnetic scalar potential in this case?

Answer 1

No, we cannot have a magnetic scalar potential here.

Ampere's law is a method of calculating magnetic field due to current distribution. On the other hand, magnetic scalar potential requires a magnetic field due to pole strength m.
Potential at a distance r is given by ${\displaystyle \frac{u_{0}m}{4\pi r}}$

As there is no current distribution, no magnetic field due to poles or the pole strength is present. That is why we cannot have a magnetic scalar potential in this case.