where, is the distance from the dipele, $\theta$ is the angle between the radius vector $r$ and the axis of the dipole.
– Padiation power of an electrie dipole with moment $p(f)$ and of a charg 4. moving with acceleration $\mathrm{w}$ :
\[
P=\frac{1}{4 \pi \mu_{4}} \frac{\ddot{z p}^{3}}{3 p^{3}}, P=\frac{1}{4 \pi \mu_{4}} \frac{2 q^{2} w^{2}}{\mu^{3}} .
\]
4.189. An electromagnetic wave of frequency $\mathrm{v}=3.0 \mathrm{MHz}$ passes from vacuum into a non-magnetic medium with permittivity $\varepsilon=$ $=4.0$. Find the increment of its wavelength.
4.190. A plane electromagnetic wave falls at right angles to the surface of a plane-parallel plate of thickness $l$. The plate is made of non-magnetic substance whose permittivity decreases exponentially from a value $\varepsilon_{1}$ at the front surface down to a value $\varepsilon_{1}$ at the rear one. How long does it take a given wave phase to travel across this plate?
4.191. A plane electromagnetic wave of frequency $\mathrm{v}=10 \mathrm{MHz}$ propagates in a poorly conducting medium with conductivity $\sigma=$ $=10 \mathrm{mS} / \mathrm{m}$ and permittivity $\varepsilon=9$. Find the ratio of amplitudes of conduction and displacement current densities.
4.192. A plane electromagnetic wave $\mathbf{E}=\mathbf{E}_{m} \cos (\omega t-\mathbf{k r})$ propagates in vacuum. Assuming the vectors $\mathbf{E}_{m}$ and $\mathbf{k}$ to be known, find the vector $\mathbf{H}$ as a function of time $t$ at the point with radius vector $\mathbf{r}=0$.
4.193. A plane electromagnetic wave $\mathbf{E}-\mathrm{E}_{m} \cos (\omega t-\mathbf{k r})$, where $\mathrm{E}_{m}=E_{m} e_{n} \mathbf{k}=k_{n}, e_{n}, e_{n}$ are the unit vectors of the $x$, $y$ axes, propagates in vacuum. Find the vector $\mathbf{H}$ at the point with radius vector $\mathrm{r}=\mathrm{r}_{\mathrm{s}}$ at the moment (a) $t=0$, (b) $t=\mathrm{t}_{6}$. Consider the case when $E_{\mathrm{m}}=160 \mathrm{~V} / \mathrm{m}, k=0.51 \mathrm{~m}^{-1}, x=7.7 \mathrm{~m}$, and $t_{0}=$ – 33 ns.
4.194. A plane electromagnetic wave $\mathrm{E}=\mathrm{E}_{\mathrm{s}} \cos (\omega t-k x)$ propagating in vacuum induces the emf $\delta_{\text {isd }}$ in a square frame with side 1 . The orientation of the frame is shown in Fig. 4.37. Find the amplitude value $\epsilon_{m}$, if $E_{m}=0.50 \mathrm{mV} / \mathrm{m}$, the frequency $v=5.0 \mathrm{MHz}$ and $i$
Fig. 4.37.
Fia. $4.3 \mathrm{~s}$.
4.195. Proceeding from Maxwell’s equations show that in the case of a plane electromagnetic wave (Fig. 4.38) propagating in 194
vacuum the following relations hold:
\[
\frac{\partial E}{\hbar}=-c^{t} \frac{\partial B}{\partial x}, \quad \frac{\partial B}{\partial t}=-\frac{\partial E}{\partial x} .
\]
4.196. Find the mean Poynting vector (S) of a plane electromagnetic wave $\mathbf{E}=\mathbf{E}_{m} \cos (\omega t-\mathbf{k r})$ if the wave propagates in vacurm.
4.197. A plane harmonic electromagnetic wave with plane polarization propagates in vacuum. The electric component of the wave has a strength amplitude $E_{\mathrm{m}}=50 \mathrm{mV} / \mathrm{m}$, the frequency is $\mathrm{v}=$ $=100 \mathrm{MHz}$. Find:
(a) the effelent value of the displacement current density;
(b) the mean energy flow density averaged over an oscillation peried.
4.198. A ball of radlus $R=50 \mathrm{~cm}$ is located in a non-magnetic medium with permittivity $\varepsilon=4.0$. In that medium a plane electromagnetic wave propagates, the strength amplitude of whose electric component is equal to $E_{m}=200 \mathrm{~V} / \mathrm{m}$. What amount of energy reaches the ball during a time interval $t=1.0 \mathrm{~min}$ ?
4.199. A standing electromagnetic wave with electric component $\mathbf{Z}=\mathbf{E}_{\mathrm{m}} \cos \mathrm{kx}$-cos $\mathrm{t}$ is sustained along the $x$ axis in vacuum. Find the magnetic component of the wave $\mathbf{B}(x, t)$. Draw the approximate distribution pattern of the wave’s electric and magnetic components ( $\mathbf{Z}$ and $\mathbf{B}$ ) at the moments $t=0$ and $t=T / 4$, where $T$ is the oscillation period.
4.200. A standing electromagnetic wave $\mathbf{E}=\mathrm{E}_{-} \cos k x \cdot \cos$ at Is sustained along the $x$ axis in vacuum. Find the projection of the Poynting vector on the $x$ axis $S_{s}(x, t)$ and the mean value of that projection averaged over an oscillation period.
4.201. A parallel-plate air capacitor whose electrodes are shaped as dises of radius $R=6.0 \mathrm{~cm}$ is connected to a source of an alternating sinusoidal voltage with frequency $a=1000 \mathrm{~s}=1$. Find the ratio of peak values of magnetic and electric energies within the eapacitor.\”
4.202. An alternating sinusoidal current of frequency $-=$ $-1000 \mathrm{~s}^{-1}$ flows in the winding of a straight solenoid whose crosssectional radius is equal to $\boldsymbol{R}=6.0 \mathrm{~cm}$. Find the ratio of peak values of electric and magnetie energies within the solenold.
4.203. A parallel-plate capacity whose electrodes are shaped as round dises is charged slowly. Demonstrate that the flux of the Poynting vector scross the capacitor’s lateral surface is equal to the increment of the capacitor’s energy per unlt tlme. The dissipation of field at the edge is to be neglected in calculations.
4.205. A current $I$ flows along a straight conductor with round cross-section. Find the flux of the Poynting vector across the lateral surface of the conductor’s segment with resistance $R$.
4.205. Non-relativistic protons accelerated by a potential diflerence $U$ form a round beam with current $I$. Find the magnitude and
direction of the Poynting vecter outside the bean at a distance $r$ from its axis.
4.206. A current flowing in the winding of a long straight solenoid is increased at a sulficiently slow rate. Demonstrate that the rate at which the energy of the magnetic field in the solenoid increases is equal to the flux of the Poynting vector across the lateral surface of the solenoid.
4.207. Fig. 4.39 illustrates a segment of a double line carrying direct current whose direction is indicated by the arrows. Taking into account that the potential $\varphi_{1}>\varphi_{1}$, and making use of the Poynting vector, establish on which side (left or right) the source of the current is located.
Fie. 4.39.
4.208. The energy is transferred from a source of constant voltage $V$ to a consumer by means of a long straight cosxial cable with negligible active resistance. The consumed current is $I$. Find the energy flux across the cross-section of the cable. The conductive sheath is supposed to be thin.
4.209. A source of ac voltage $\boldsymbol{V}=V$, cos cot delivers enercy to a consumer by means of a long straight coaxial cable with negligible active resistance. The current in the circuit varies as $I=$ $=I_{0} \cos \omega t-q$ ). Find the time-averaged energy flux through the cross-section of the cable. The sheath is thin.
4.210. Demenstrate that at the boundary between two media the normal components of the Poynting vector are continuous, i.e. $s_{\text {in }}=s_{\text {s. }}$.
4.211. Demonstrate that a closed system of charged non-relativistic particles with identical specific eharges emits ao dipole radiation.
4.212. Find the mean radiation power of se electron performing harmonic oscillations with amplitude $a=0.10 \mathrm{~nm}$ and frequen cy $\omega=6.5 \cdot 10^{14} \mathrm{~s}^{-1}$
4.213. Find the radiation power developed by a non-relativistic particle with charge $e$ and mass $m$, moving along a circular orbit of radius $B$ in the field of a stationary point charge $q$.
4.214. A particle with charge $e$ and mass $m$ flies with non-relativistic velocity $o$ at a distance b past a statienary particle with charge q. Neglecting the bending of the trajectory of the moving particle, find the energy lost by this partiele due to radiation during the total night time.
4.215. A non-relativistie proton enters a half-space along the normal to the transverse uniform magnetic field whose induction 196
equals $B=1.0 \mathrm{~T}$. Find the ratio of the energy lest by the proton due to radiation during its motion in the field to its initial kinetic energy.
4.216. A non-relativistie charged particle moves in a transverse uniform magnetic field with induction $B$. Find the time dependence of the particle’s kinetic energy diminishing due to radiation. How soon will its kinetie energy decrease e-fold? Calculate this time interval for the case (a) of an electron, (b) of a proton.
4.217. A charged particle moves along the $y$ axis accorting to the law $y=a$ cos et, and the point of observation $P$ is located on the $x$ axis at a distance $l$ from the partiele $(l>a)$. Find the ratio of electromagnetic radiation flow densities $S_{1} / S_{4}$ at the point $P$ at the moments when the coordinate of the particle $y_{1}=0$ and $y_{1}=a$. Caleulate that ratio if $e=3.3 \cdot 10^{\circ} \mathrm{s}-1$ and $i=190 \mathrm{~m}$.
4.218. A charged particle moves uniformly with velocity $v$ along a circle of radius $R$ in the plane $x y$ (Fig. 4.40). An observer is located
Fie. 4.40.
on the $x$ axis at a point $P$ which is removed from the centre of the circle by a distance much exceeding $\boldsymbol{R}$. Find:
(a) the relationship between the observed values of the $y$ projection of the particle’s acceleration and the $y$ coordinate of the particle;
(b) the ratio of electromagnetic radiation flow densities $s_{1} / s_{3}$ at the point $P$ at the moments of time when the particle moves, from the standpoint of the observer $P$, toward him and away from him, as shown in the figure.
4.219. An electromagnetic wave emitted by an elementary dipole propagates in vacuum so that in the far field zone the mean value of the energy flow density is equal to $S_{\text {, at }}$ at the point removed from the dipole by a distance $r$ along the perpendicular drawn to the dipole’s axis. Find the mean radiation power of the dipole.
4.220. The mean power radiated by an elementary dipole is equal to $P_{0}$. Find the mean space density of energy of the electromagnetic field in vacuum in the far field zone at the point removed from the dipole by a distance $r$ along the perpendicular drawn to the dipole’s axis.
4.221. An electrie dipole whose modulus is constant and whose moment is equal to $p$ rotates with constant angular velocity 0 about the axis drawn at right angles to the axis of the dipole and passing through its midpoint. Find the power radiated by such a dipole.
4.222. A free electron is located in the field of a plane electromagnetic wave. Neglecting the magnetic component of the wave disturbing its motion, find the ratio of the mean energy radiated by the oscillating electron per unit time to the mean value of the energy flow density of the incident wave.
4.223. A plane electromagnetic wave with frequency $\omega$ falls upon an elastically bonded electron whose natural frequency equals $\omega_{0}$. Neglecting the damping of oscillations, find the ratio of the mean energy dissipated by the electron per unit time to the mean value of the energy flow density of the incident wave.
4.224. Assuming a particle to have the form of a ball and to absorb all incident light, find the radius of a particle for which its gravitational attraction to the Sun is counterbalanced by the force that light exerts on it. The power of light radiated by the Sun equals $P=4 \cdot 10^{26} \mathrm{~W}$, and the density of the particle is $\rho=1.0 \mathrm{~g} / \mathrm{cm}^{3}$.