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Question

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How do I eliminate the vector product to solve the differential equation as in the above question (Click the link to see the question)

Answer 1

$\vec r = (x,y,z)\\ \vec E = (E,0,0)\\ \vec H = (0,0,H)$

We have the diff eq.

$m\frac {d^2r}{dt^2} = e\vec E - \frac {e}{c}\left(\frac {dr}{dt}\times \vec H\right)$

Substitute with the vectors above.

$m(x'',y'', z'') = e(E,0,0) - \frac {e}{c}\left((x',y',z')\times (0,0,H)\right)$

Calculate the cross product.

$m(x'',y'', z'') = e(E,0,0) - \frac {e}{c}(Hy', -Hx', 0)$

Add the components together

$(mx'',my'', mz'') = (eE + \frac {e}{c}Hy', -\frac {e}{c}Hx', 0)$

If the LHS equals the RHS, each component of the respective vectors are equal

$mx'' = eE + \frac {e}{c}Hy'\\ my'' = -\frac {e}{c}Hx'\\ mz'' = 0$

Does that get you started?