# Nabla calculus and conservative forces

1. May 1, 2013

### KayDee01

1. The problem statement

I'm trying to show that the magnetic force is only conservative if dB/dt=0

2. Relevant equations

F=q[E+(v$\times$B)]

Conservative if ∇$\times$F=0

∇$\times$(A$\times$B)=A(∇$\cdot$B)-B(∇$\cdot$A)+(B$\cdot$∇)A-(A$\cdot$∇)B

Maxwells equation: ∇$\times$E=-∂B/∂t

3. The attempt at a solution

So the magnetic force field is conservative if
∇$\times$[E+(v$\times$B)]=0
=∇$\times$E+∇$\times$(v$\times$B)
=-∂B/∂t+v(∇$\cdot$B)-B(∇$\cdot$v)+(B$\cdot$∇)v-(v$\cdot$∇)B

So from here I know that I need to show:
v(∇$\cdot$B)-B(∇$\cdot$v)+(B$\cdot$∇)v-(v$\cdot$∇)B=0

But when I write it out in all its components I get lost in the algebra. And then it got me thinking, if that equals zero, why doesn't ∇$\times$(A$\times$B) always equal zero.
I know there's a simple answer to this as we did it in lectures but I can't seem to find it in my notes anywhere.

Last edited: May 1, 2013
2. May 1, 2013

### Staff: Mentor

You cannot assign a scalar potential to the magnetic field. You can show that the magnetic field does not influence the speed (and therefore the energy) of the particle, however. This is related to $\vec{v} \cdot \vec{F}$.