Homework Help: Irrotational and divergenceless?

1. Feb 3, 2004

jlmac2001

I'm not really sure what I'm being asked to do with the following questions. Will someone help me?

How can you show that E(r)=-gradV(r) has zero curl an is irrotational, i.e. the quantity (grad x E) =0

How can you show that the magnetic field ,B(r)=grad x A(r), is divergenceless i.e. that the quantity grad dot B(r) =0?

2. Feb 3, 2004

HallsofIvy

Basically, you just go ahead and do it!

That is, start by assuming that you have V(x,y,z) so that E(r)=-gradV(r)= -Vxi- Vy-Vz and calculate curl(E)= curl(grad V). See what happens!

3. Feb 4, 2004

Norman

if B=curl A, then div B = div (curl A), the divergence of a curl is always zero. Actually this proof is usually done in the opposite order since the div B=0 is one of Maxwell's equations, valid even for time dependent phenomenon (the old there are no magnetic monopoles theorem). If div B=0 then B can be expressed as the curl of another vector, which gets labeled A and we call it the magentic vector potential.
Hope this helps.
Cheers,
Norm