Motion In A Crossed Electric And Magnetic Fields

The equation to the trajectory is obtained by integrating the above equation and determining the constant of integration from the initial position (taken to be at the origin),

$\begin{eqnarray*} x &=& \frac{E}{B\omega_c}(1-\cos\omega_ct)\\ y &=& \frac{E}{B\omega_c}(\sin\omega_ct-\omega_ct) \end{eqnarray*}$

The equation to the trajectory is

which represents a circle of radius

whose centre travels along the negative y direction with a constant speed

The trajectory resembles that of a point on the circumference of a wheel of radius , rolling down the y-axis without slipping with a speed v₀. The trajectory is known as a cycloid.

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