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I encountered a curious problem while trying to solve the case of a particle with v(v_x0, v_y0, 0), and B(0, 0, Bz); The elucidation of the differential equations obtained through the Lorentz force in this case, shouldcoincidewith those obtained through a simplification granted by using the Centripetal force, but here, instead of circular motion I get an unattractive ellipse :(.

Assuming the particle moves only on the x-y plane, and starts at (x0, y0, 0), the Lorentz force yields the following:

[itex]

\large

\vec{F} = q(\vec{v}\times\vec{B})

[/itex]

[itex]

ma_x(t) = -qv_yB

[/itex]

[itex]

ma_y(t) = qv_xB

[/itex]

Which in turn, with these initial conditions leads to:

With v0 = Sqrt(vx0^2+vy0^2);

Alpha derived from: Arctan[vy/vx] = alpha;

[itex]

x(t) = x_0+ \frac{v_0(-\sin(\alpha)+\sin(\omega t + \alpha))}{\omega}

[/itex]

[itex]

y(t) = y_0+ \frac{v_0(\cos(\alpha)-\cos(\omega t + \alpha))}{\omega}

[/itex]

[itex]

\omega = \frac{qB}{m}

[/itex]

Plugging in some random values, leads to the attached image, while we all know that motion in a magnetic field should be accompanied by uniform circular motion;

Where have I gone wrong?

Thanks,

Daniel

P.S

This is not related in anyway, to homework.

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# General Solutions for a particle in a magnetic field

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