So I am constructing an analogy between the self replicating fracturing effect on thin films and the path of a charged particle. (Qualitatively, several cracks have similar shapes to charged particle motion) I won't go into the details of the fracture mechanics, so I will only use E+M nomenclature.(adsbygoogle = window.adsbygoogle || []).push({});

My problem: Start with a parameterized path function, s, of a charged particle from which the velocity and acceleration functions can be easily derived. (a and R are constants)

$$\vec{s} = ( tcos(at) , tsin(at),(x^{2} + y^{2}) R )$$

I want to use this in a modified Lorentz Force equation where the mass-charge ratio is set to one (m/q=1)In the equation, I ultimately want to find a way to solve for all x, y, z components of E and B without resorting to Maxwell's equations.

$$ dv/dt = E + v × B $$

Current attempt: In order to reduce the amount of unknowns from 6 to 3, (i.e. from E_{x}, E_{y}, E_{z}, B_{x}, B_{y}, B_{z}to just B_{x}, B_{y}, B_{z}) I thought I can use the relation between kinetic energy and the electric field. I am not sure if this is the right thing to do, so feel free to suggest a different method for minimizing unknowns. I used the following relation for this (m/q=1 again):

$$Work = qEs = \frac{1}{2}mv^{2} $$

So now s, v, dv/dt, and each component of E is known thus leaving the components of B left to be solved. The following vector equation is the result of taking the cross product:

$$(a_{x},a_y,a_z )= (E_x,E_y,E_z)+(v_yB_z-v_zB_y,v_zB_x-v_xB_z, v_xB_y-v_yB_x)$$

I think this then simplifies to:

$$

\begin{pmatrix}

0& -v_{z} &v_{y} \\

v_{z}&0 &-v_{x} \\

-v_{y}& v_{x}& 0

\end{pmatrix}\begin{pmatrix}

B_{x}\\

B_{y}\\

B_{z}

\end{pmatrix} =\begin{pmatrix}

a_{x}-E_{x}\\a_{y}-E_{y}

\\ a_{z}-E_{z}

\end{pmatrix}

$$

Which doesn't seem to have a solution. What identities can I implement to yield solutions for this equation? Do I need to set one of the B components to 0? Is my equation for E useful in this case? Can I solve for E

_{x}, E_{y}, E_{z}, B_{x}, B_{y}, B_{z}numerically- what numerical method would be useful?

tldr: I just want to know how to take a particle path and derive the electric and magnetic field components

Feel free to ask any questions for clarification!

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# I Use Lorentz Force to Find Magnetic Field Components

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