April 15, 1998

PHY 712 - Problem Set # 16

The Liénard-Wiechert fields for a point charge are summarized in Jackson in Eqs. 14.13 and 14.14. These are equations are consistent with the equations derived in class, where ${\bf{r}}_q(t_{ret})$ describes the particle trajectory evaluated at the retarded time:

t_ret = t - R(t_ret)/c,

and

$$R(t_{ret}) \equiv \vert{\bf{r}} - {\bf{r}}_q(t_{ret})\vert.$$

Other variables are defined by:

$${\bf{\hat{n}}} \equiv {\bf{R}}(t_{ret})/R(t_{ret}),$$

$${\bf{\beta}} \equiv \frac{1}{c} \frac{\partial{\bf{r}}_q(t_{ret})}{\partial t_{ret}},$$

and

$${\bf{\dot{\beta}}} \equiv \frac{\partial{\bf{\beta}}(t_{ret})}{\partial t_{ret}}.$$

Using these equations, directly calculate the electric and magnetic fields of the constant velocity particle of charge q shown in Figure 11.8 in the K frame of reference.

1.

Show that ${\bf{r}}_q(t_{ret}) = v t_{ret}{\bf{\hat{x}_1}}$.

2.

Write an expression for R(t_ret) which eliminates t_ret and includes t, b, v, $\beta$, and $\gamma$.

3.

Show that your results for ${\bf{E}}$ and ${\bf{B}}$ are consistent with Eqs. 11.152.