# Lorentz Force Equation

1. Sep 30, 2005

### 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

2. Sep 30, 2005

### Tom Mattson

Staff Emeritus
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:

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

3. Sep 30, 2005

### Staff: Mentor

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.

4. Sep 30, 2005

### Tom Mattson

Staff Emeritus
Duh.

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

Is it 5:00 yet?

5. Sep 30, 2005

### Michael Lin

Last edited by a moderator: Apr 21, 2017
6. Oct 1, 2005

### Staff: Mentor

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).

7. May 9, 2011

### dilloo

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