Problem database

a. The formula for the transformation of the $\overrightarrow{E}$ and $\overrightarrow{B}$ fields when they move at the speed $\overrightarrow{\beta} c$ has the following form:

$\overrightarrow{E}' = \overrightarrow{E} {\parallel} + \gamma(\overrightarrow{E} {\perp} - [\overrightarrow{\beta} \times \overrightarrow{B} ]), \overrightarrow{B}' = \overrightarrow{B} {\parallel} + \gamma(\overrightarrow{B} {\perp} + (\overrightarrow{\beta} \times \overrightarrow{E} ]), \gamma = \frac{1}{\sqrt{1 - \beta^2}}$,

where $\overrightarrow{E}'$ and $\overrightarrow{B}'$ are the electric and magnetic fields in the drift; $\overrightarrow{E} {\parallel}$, $\overrightarrow{E} {\perp}$ and $\overrightarrow{B} {\parallel}$, $\overrightarrow{B} {\perp}$ — components electric and magnetic fields fields, parallel services and perpendicular lines $- c \overrightarrow{\beta}$ in the initial system. The movement of the $\overrightarrow{E}'$ and $\overrightarrow{B}'$ fields at a speed of $- c \beta$ returns the previous state. Check it out.

b. Using the field transformation formulas given in point $a$, solve the following problems: 14.3.1–14.3.3, 14.3.5.

c. Using the field transformation formulas given in point $a$, solve problems 14.3.6 a, b, and 14.3.7.

d. Prove that for $\beta \to 1$, the fields $\overrightarrow{E}'$ and $B'$ are perpendicular.