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Math Help - Lorentz force law

  1. #1
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    Lorentz force law

    According to the Lorentz force law, an electron moving in a constant magnetic field follows a path  c(t) which satisfies  \ddot c = \dot c \times B , where  B is a constant vector on  \mathbb{R}^3 . Prove that this path is a helix.
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  2. #2
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    Quote Originally Posted by mathman88 View Post
    According to the Lorentz force law, an electron moving in a constant magnetic field follows a path  c(t) which satisfies  \ddot c = \dot c \times B , where  B is a constant vector on  \mathbb{R}^3 . Prove that this path is a helix.
    We can choose to align the z axis with the magnetic field B. Then:

    B=B_z, B_x=0, B_y=0

    So:

    \ddot c_x = \dot c_y B_z

    \ddot c_y = -\dot c_x B_z

    \ddot c_z = 0

    Solving the third equation by integrating twice with respect to t:  c_z = kt where k is a constant.

    Now if you differentiate the second equation and then substitute for \ddot c_x from the first then you will get:
     \ddot \dot c_y=-\dot c_yB_z^2

    which has the general solution:
     \dot c_y=-a sin(B_zt)+b cos(B_zt) = csin(B_zt+\phi)

    So
     c_y=\frac{-c cos(B_zt + \phi)}{B_z}

    Now using equation 1
     \ddot c_x=\dot c_y B_z= cB_zsin(B_zt+\phi)

    So
     c_x= \frac {-csin(B_zt+\phi)}{B_z}

    Now using c_x, c_y, c_z we can write:

    \bar c = \frac {-csin(B_zt+\phi)}{B_z} \hat e_x-\frac {ccos(B_zt+\phi)}{B_z} \hat e_y+kt\hat e_z

    This is the equation of a helix.
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