Module 3 : Electromagnetism
Lecture 13 : Electric Current and Current Density

The constant $A$may be determined by substituting the solutions in eqn. (1) which gives

\begin{displaymath}mA\omega_c\cos\omega_ct = qE + qBA(\cos\omega_ct-1)\end{displaymath}

Since the equation above is valid for all times, the constant terms on the right must cancel, which gives $A= E/B$. Thus we have

\begin{eqnarray*} v_x &=& \frac{E}{B}\sin\omega_ct\\ v_y &=& \frac{E}{B}(\cos\omega_ct-1) \end{eqnarray*}

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*}

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