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where r is the distance from the dipole, 0 is the angle between the radius vector r and the axis of the dipole.
- Radiation power of an electric dipole with moment p (t) and of a charge q, moving with acceleration w:
P =^1 21 2^1 2q2w 4neo 3c3 '^ P =^ 4ns0 3c3 '
(4.4f)
4.189. An electromagnetic wave of frequency v = 3.0 MHz passes from vacuum into a non-magnetic medium with permittivity a = 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 1. The plate is made of non-magnetic substance whose permittivity decreases exponen- tially from a value alat the front surface down to a value a, 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 v = 10 MHz propagates in a poorly conducting medium with conductivity a =
= 10 mS/m and permittivity a = 9. Find the ratio of amplitudes
of conduction and displacement current densities. 4.192. A plane electromagnetic wave E = Emcos (cot — kr) propagates in vacuum. Assuming the vectors Emand k to be known, find the vector H as a function of time t at the point with radius vector r = 0. 4.193. A plane electromagnetic wave E = Emcos (cot — kr), where Em, = Eme y, k = kex, ex, e uare the unit vectors of the x, y axes, propagates in vacuum. Find the vector H at the point with radius vector r = xexat the moment (a) t = 0, (b) t = to. Consider the case when Em = 160 V/m, k = 0.51 m --1, x = 7.7 m, and to = = 33 ns. 4.194. A plane electromagnetic wave E = En, cos (cot — kx) propagating in vacuum induces the emf gin, in a square frame with side 1. The orientation of the frame is shown in Fig. 4.37. Find the amplitude va ue find , if (^) Ern = 0.50 mV/m, the frequency v =5.0 MHz and 1 =^ 50 cm. g
z
E
Fig. 4.37.
E
4.195. Proceeding from Maxwell's equations show that in the case of a plane electromagnetic wave (Fig. 4.38) propagating in
vacuum the following relations hold:
aE 2 OB OB aE
at ax at^ ar
4.196. Find the mean Poynting vector (8) of a plane electromag- netic wave E = Emcos (cot — kr) if the wave propagates in va- cuum. 4.197.. A plane harmonic electromagnetic wave with plane polari- zation propagates in vacuum. The electric component of the wave has a strength amplitude En, =^ 50 mV/m, the frequency is v 100 MHz. Find: (a) the efficient value of the displacement current density; (b) the mean energy flow density averaged over an oscillation period. 4.198. A ball of radius /I = 50 cm is located in a non-magnetic medium with permittivity a = 4.0. In that medium a plane electro- magnetic wave propagates,the strength amplitude of whose electric component is equal to Eni, =^ 200 Vim. What amount of energy reaches the ball during a time interval t = 1.0 min? 4.199. A standing electromagnetic wave with electric component E = Emcos kx•cos cot is sustained along the x axis in vacuum. Find the magnetic component of the wave B (x, t). Draw the approximate distribution pattern of the wave's electric and magnetic components (E and B) at the moments t = 0 and t = T/4, where T is the oscilla- tion period. 4.200. A standing electromagnetic wave E = Emcos kx• cos cot is sustained along the x axis in vacuum. Find the projection of the Poynting vector on the x axis (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 discs of radius R = 6.0 cm is connected to a source of an alternat- ing sinusoidal voltage with frequency co = 1000 s-1. Find the ratio of peak values of magnetic and electric energies within the capacitor. 4.202. An alternating sinusoidal current of frequency co 1000 s-1flows in the winding of a straight solenoid whose cross- sectional radius is equal to R = (^) 6.0 cm. Find the ratio of peak values of electric and magnetic energies within the solenoid. 4.203. A parallel-plate capacity whose electrodes are shaped as round discs is charged slowly. Demonstrate that the flux of the Poynting vector across the capacitor's lateral surface is equal to the increment of the capacitor's energy per unit time. The dissipation of field at the edge is to be neglected in calculations. 4.204. 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 diffe- rence U form a round beam with current I. Find the magnitude and
13* (^195)
Fig. 4.40.
equals B = 1.0 T. Find the ratio of the energy lost by the proton
due to radiation during its motion in the field to its initial kinetic energy. 4.216. A non-relativistic 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 kinetic 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 according to the
law y = a cos cot, and the point of observation P^ is located on the
axis at a distance 1 from the particle (1> a). Find the ratio of electro-
magnetic radiation flow densities S1/S 2 at the point P at the moments
when the coordinate of the particle yl= 0 and y2= a.. Calculate
that ratio if co = 3.3.106 s--1 and 1 = 190 m.
4.218. A charged particle moves uniformly with velocity v along
a circle of radius R in the plane xy (Fig. 4.40). An observer is located
on the x axis at a point P which is removed from the centre of the
circle by a distance much exceeding R. Find:
(a) the relationship between the observed values of the y projec- tion of the particle's acceleration and the y coordinate of the particle; (b) the ratio of electromagnetic radiation flow densities S1tS
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 Soat 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 Po. 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 electric dipole whose modulus is constant and whose moment is equal to p rotates with constant angular velocity w 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 electromagne- tic wave. Neglecting the magnetic component of the wave disturbing its motion, find the ratio of the mean energy radiated by the oscil- lating 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 co falls upon an elastically bonded electron whose natural frequency equals O. 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 ab- sorb 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 = 4.1026 W, and the density of the particle is p = 1.0 g/cms.
