F07
Quiz P260 #7
Name_______________________
A bit more on Potential Theory. We saw for electrostatics that, in the spirit of Helmholtz's theorem (a vector field is completely specified by its divergence and curl), E and E 0. The zero-curl equation is important in that it tells us that
0
the electric field vector can be written as the gradient of a scalar potential function of the position coordinates, V(r). Referring back to quiz #3 you verified the vector identity V 0 and thus telling us that we can write E V . Now, if we our consider equations for magnetism the curl of B is not zero (Ampere's law-Monday), but it is Gauss's law that tells us that the divergence of B is zero, B 0. Show the vector identity ( A) 0 and hence conclude that B can be written as the curl of a magnetic vector potential A(r ), where B A. The physical significance of this magnetic vector potential is not as straightforward to understand as the electric scalar potential, but you will see this again in more advanced E&M courses so it helps to be aware of this concept.