danielakkerma
- 230
- 0
Hello everyone!
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, should coincide with 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:
<br /> \large<br /> \vec{F} = q(\vec{v}\times\vec{B})<br />
<br /> ma_x(t) = -qv_yB<br />
<br /> ma_y(t) = qv_xB<br />
Which in turn, with these initial conditions leads to:
With v0 = Sqrt(vx0^2+vy0^2);
Alpha derived from: Arctan[vy/vx] = alpha;
<br /> x(t) = x_0+ \frac{v_0(-\sin(\alpha)+\sin(\omega t + \alpha))}{\omega}<br />
<br /> y(t) = y_0+ \frac{v_0(\cos(\alpha)-\cos(\omega t + \alpha))}{\omega}<br />
<br /> \omega = \frac{qB}{m}<br />
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.
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, should coincide with 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:
<br /> \large<br /> \vec{F} = q(\vec{v}\times\vec{B})<br />
<br /> ma_x(t) = -qv_yB<br />
<br /> ma_y(t) = qv_xB<br />
Which in turn, with these initial conditions leads to:
With v0 = Sqrt(vx0^2+vy0^2);
Alpha derived from: Arctan[vy/vx] = alpha;
<br /> x(t) = x_0+ \frac{v_0(-\sin(\alpha)+\sin(\omega t + \alpha))}{\omega}<br />
<br /> y(t) = y_0+ \frac{v_0(\cos(\alpha)-\cos(\omega t + \alpha))}{\omega}<br />
<br /> \omega = \frac{qB}{m}<br />
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.