Question

ELECTRODYNAMICS;

A compact area C around the origin contains (time-dependent) sources rho and j. The fields E, B are generated by those sources. a> 0 is a number. Justify if the following statements are true or false (even if it is true you have to justifywhy) shortly 1) IfE0 or B-0, no radiationis emitted. (ap, aj) ikefwe changerto, and jlike: (ρ, j) then the radiated power changes like: P- ->a 2 P then the product E B does not 3) If we change rho and j like: change. (Po +p,j) with ρ0 time independent (p,j) → 4) If we change rho and j like:PiJ the powerradiated doesn't change 5) The electric field cannot decay more slowly (in the distance r to C) than 1/ra2.

Answer 1

1) True, because electromagnetic radiations are synchronized oscillations of electric and magnetic fields. If anyone of the component like E or B is zero then no radiation is emitted.

P=_ lim φ (E × B).nds

2) True , because $E\propto \rho$ and  $B\propto J$ therefore, when ,

(p, J) → ( J) ap. a

$P_{new} \propto (E\times B) \propto a^{2}P$

3) True, because $E\propto \frac{\rho}{a}$ and $B\propto aJ$ ,that implies, . The product E.B remains the same.

Pneu, x (E × B) x-P = P

4) True, because the radiation waves are due to accelerating charges and changing currents. According to the continuity equation.
at
which implies that the time independent $\rho_{0}$ will not contribute to the accelerating charge or changing currents.

5)False, because according to Coulomb's law, electrostatic fields fall off like 1/r ² and the Biot-Savart law says that magnetostatic fields go like 1/r ² , which leads to  S ~ 1/r ⁴ , for static configurations. Such static sources do not emit radiation. However according to Jefimenko's equations, the time-dependent fields include terms that go like 1/r. It  is these terms that are responsible for electromagnetic radiation. Therefore the electric field can decay more slowly that 1/r².