Question

Magnetic scalar potential is defined as `U(vec r_2) - U(vec r_1) = - ∫_vec(r_1)^vec(r_2)` `vec (B) . dvec(l)`

Apply this equation to a closed curve enclosing a long straight wire. The RHS of the above equation is then `-u_0 i` by Ampere's law. We see that `U(vec(r_2)) ≠ U(vec(r_1))` even when `vec r_2 =vec 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 `(u_0m)/(4pir)`

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.