- #1
etotheipi
If a particle is in a magnetic field ##\vec{B} = B\hat{z}## with velocity ##\vec{v} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}##, then in Cartesian coordinates we can obtain the pair of differential equations $$\ddot{x} = \frac{qB}{m}\dot{y}$$$$\ddot{y} = -\frac{qB}{m}\dot{x}$$which give the solution$$\vec{r}(t) = \begin{pmatrix}r_L \cos{\omega t}\\r_L \sin{\omega t}\\v_z t\end{pmatrix}$$where ##\omega = \frac{qB}{m}## and ##r_L = \frac{mv_{\bot}}{qB}##, i.e. a helix.
Furthermore, we see that the acceleration vector has magnitude ##\frac{v_{\bot}^2}{r_L}## in a direction perpendicular to the velocity (and the magnetic field).
However, I'm having some trouble relating this to the acceleration in intrinsic coordinates. The radius of curvature of the helix given by ##\vec{r}(t)## is ##\rho = \frac{r_L^2 + v_z^2}{r_L}## and the normal component of acceleration in intrinsic coordinates is ##\frac{v^2}{\rho} = \frac{v^2 r_L}{r_L^2 + v_z^2}##, with ##v^2 = v_z^2 + v_{\bot}^2##.
The normal component of acceleration in intrinsic coordinates should equal ##\frac{v_{\bot}^2}{r_L}##, though, since both are the components of acceleration perpendicular to the velocity. I wondered what I've done wrong?
Furthermore, we see that the acceleration vector has magnitude ##\frac{v_{\bot}^2}{r_L}## in a direction perpendicular to the velocity (and the magnetic field).
However, I'm having some trouble relating this to the acceleration in intrinsic coordinates. The radius of curvature of the helix given by ##\vec{r}(t)## is ##\rho = \frac{r_L^2 + v_z^2}{r_L}## and the normal component of acceleration in intrinsic coordinates is ##\frac{v^2}{\rho} = \frac{v^2 r_L}{r_L^2 + v_z^2}##, with ##v^2 = v_z^2 + v_{\bot}^2##.
The normal component of acceleration in intrinsic coordinates should equal ##\frac{v_{\bot}^2}{r_L}##, though, since both are the components of acceleration perpendicular to the velocity. I wondered what I've done wrong?