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

The first two equations may be solved by converting them into second order differential equatons. This is done by differentiating one of the equations with respect to time and substituting the other equation in the resulting second order equation. For instance, the equation for $v_x$is given by

\begin{displaymath}\frac{d^2v_x}{dt^2} = \frac{qB}{m}\frac{dv_y}{dt}= -\frac{q^2B^2}{m^2}v_x\end{displaymath}

The equation is familiar in the study of simple harmonic motion. The solutions are combination of sine and cosine functions.

\begin{displaymath}v_x = A\sin\omega_ct + B\cos\omega_ct\end{displaymath}

where

\begin{displaymath}\omega_c = \frac{qB}{m}\end{displaymath}

is called the cyclotron frequency and $A$and $B$are constants. These constants have to be determined from initial conditions. By our choice of x and y axes, we have

\begin{displaymath}v_x(t=0) = u_x \end{displaymath}

8
11
12
16
17
18
19