Lorentz Force Equation – Is vxB Representable?

[Michael Lin]

Hi All,

Just a thought. The Lorentz force equation as we all know is: F = qE + qvxB. We know that Electrical Field can be written as del(Phi), where Phi is the electrical potential. Also, Force can be written as del(Energy) - correct me on this one. Hence is there a representative term for vxB. Can vxB be written as del(something) where something is a meaningful quantity?

Just curious,
Thanks - Michael

 

[quantumdude]

It sure can. Taking $\vec{F}=-\vec{\nabla}U$ and $\vec{E}=-\vec{\nabla}\Phi$, we have:

$-\vec{\nabla}U=-q\vec{\nabla}\Phi+q\vec{v}\times\vec{B}$.

Rearranging terms we get:

$q\vec{v}\times\vec{B}=q\vec{\nabla}\Phi-\vec{\nabla}U$.

Thanks to the linearity of the $\vec{\nabla}$ operator, we have:

$q\vec{v}\times\vec{B}=\vec{\nabla}(q\Phi-U)$
$\vec{v}\times\vec{B}=\vec{\nabla}(\Phi-\frac{U}{q})$.

 

[jtbell]

But a U that satisfies $\vec{F}=-\vec{\nabla}U$ exists only if $\vec F$ is a conservative force. The magnetic force isn't conservative.

 

[quantumdude]

Duh.

Well boys and girls, this is what happens when plug-n-chug runs amuck.

Is it 5:00 yet?

 

[Michael Lin]

Hi All,

I'm actually a little confused. Some guy in NASA said the magnetic field is not conservative http://pwg.gsfc.nasa.gov/istp/outreach/ed/back/1.html (he refers to magnetostatic field I think), some guy in madsci says it is conservative http://www.madsci.org/posts/archives/may98/895287804.Ph.r.html.

Actually, a little confused.

Thanks,
Michael

 

[jtbell]

If a force is conservative, you can define a potential energy function for it. The potential energy of a particle can depend only on its position, so a conservative force can depend only on position. But the magnetic force on a particle depends on the velocity (both magnitude and direction!) of the particle, not just on its position (which determines the magnetic field).

 

[dilloo]

how can we get potential from lorentz equations...?