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

so that $B= u_x$. Differentiating the above equation for $u_x$,

\begin{eqnarray*} \frac{dv_x}{dt} &=& A\omega_c\cos\omega_ct - B\omega_c\sin\omega_ct\\ &\equiv& \frac{qB}{m}v_y = \omega_cv_y \end{eqnarray*}

Since $v_y =0$at $t=0$, we have $A=0$. Thus, the velocity components at time $t$are given by

\begin{eqnarray*} v_x &=& u_x\cos\omega_ct\\ v_y &=& -u_x\sin\omega_ct = u_x\sin(\omega_ct + \frac{\pi}{2}) \end{eqnarray*}

which shows that $v_x$and $v_y$vary harmonically with time with the same amplitude but with a phase difference of $\pi/2$. Equation of the trajectory may be obtained by integrating the equations for velocity components

\begin{eqnarray*} x(t) &=& \frac{u_x}{\omega_c}\sin\omega_c t + x_0\\ y(t) &=& \frac{u_x}{\omega_c}\cos\omega_c t + y_0\\ z(t) = u_zt + z_0 \end{eqnarray*}

8
12
16
17
18
19