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KayDee01
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1. The problem statement
I'm trying to show that the magnetic force is only conservative if dB/dt=0
F=q[E+(v[itex]\times[/itex]B)]
Conservative if ∇[itex]\times[/itex]F=0
∇[itex]\times[/itex](A[itex]\times[/itex]B)=A(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]A)+(B[itex]\cdot[/itex]∇)A-(A[itex]\cdot[/itex]∇)B
Maxwells equation: ∇[itex]\times[/itex]E=-∂B/∂t
So the magnetic force field is conservative if
∇[itex]\times[/itex][E+(v[itex]\times[/itex]B)]=0
=∇[itex]\times[/itex]E+∇[itex]\times[/itex](v[itex]\times[/itex]B)
=-∂B/∂t+v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B
So from here I know that I need to show:
v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)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 ∇[itex]\times[/itex](A[itex]\times[/itex]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.
I'm trying to show that the magnetic force is only conservative if dB/dt=0
Homework Equations
F=q[E+(v[itex]\times[/itex]B)]
Conservative if ∇[itex]\times[/itex]F=0
∇[itex]\times[/itex](A[itex]\times[/itex]B)=A(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]A)+(B[itex]\cdot[/itex]∇)A-(A[itex]\cdot[/itex]∇)B
Maxwells equation: ∇[itex]\times[/itex]E=-∂B/∂t
The Attempt at a Solution
So the magnetic force field is conservative if
∇[itex]\times[/itex][E+(v[itex]\times[/itex]B)]=0
=∇[itex]\times[/itex]E+∇[itex]\times[/itex](v[itex]\times[/itex]B)
=-∂B/∂t+v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B
So from here I know that I need to show:
v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)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 ∇[itex]\times[/itex](A[itex]\times[/itex]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.
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