S O V I E T P H Y S I C S J O U I { N A L 113

When the two sheets are of different thicknesses (~i ;" f~'), the thermal conduct ivi ty is

,~ c,.'~,.?,l. (?,~ i ; , ' ) .m (3)

17"~ ,,;c,z l/m ~.; I/;~-- :; c,~ l ,7 ;; �9 ~ / d (l

where c I is the :i>ecific heat of tide standard sheet, )'l is its specific gravi ty , and 81 is its thickness. The specific heat is de termined from the fami l ia r equation

a , (4)

6 1

80 ~ , 40 SCh �9 -o, , 4 0 SCh

Z5 --d, ~ . eT} sCh

65 0 gz7 40 80 80 t og

Fig. 2

It was shown in [31 that the :c la t ive error in measuring the t e m p e r a -

ture coeff icient ~a/,a due to failure to allow for heat loss through

the la tera l surfaces of the sheet is less than 4a,.;. fhe overall error

~ a / a -"-- 7%, w',tile &X. \ and &c ,c do not exceed 12% and 1 ~ . re-

spect ive ly .

The accompanying table gives exper imenta l results for different brands of ferrrte.

It Call De conchtded from the data in the table lizat the porosity of

polycrys ta l tme ferrites is one of the basic factors determining their

t he rma t conduct iv i ty . Lithium ferrt te, whose density is also low, has

the highest t he rma l conduct ivi ty .

The coeff ic ient of linear expansion o was rneast.red on a type

IZV-2 ver t ical opt ical measuring device over a t empera ture range

of 20~-100"( ". Re.d-shapod spec:mens 50 to 20(; m m long were used.

To ensure ulliform heating tile specimens were placed in a censtant-

temperat::r,z enclosure for at', hour al-d w~re t[ierl rapidly transferred

to the measur ing heat of tile nleatisring device; thus tile ti:ltiul

t empera ture (the t empera ture of the first reading} could be assumed

to be tile same as the ambient ter; :perature, rile second measure-

lnent was made after lO--ig n:in, ~ . c . , after the specimens had

comple te ly cooled. Tile error it! meas:lrmg ',( ,.'c was 1:'~-~ t.lun 1.55..

Fig:lrc '.2 ) shows tile e • ~ahles of the co-

eff icient of linear expansion for oOS(:h, 40sCh, and 40SChl ferrites. l 'he coeff icient is virtually : empera tu re - independen t and is equal

to 7 �9 !0 -6 per degree ( . For 4,O.<Ch and 40SChl ferritcs there is

a J J n l ? iu [.le coefficleP, t at the ( uric t empera ture , wi,ere there is a

scco:'d-,~rdcr p:;asc tr~nsiti::n.

Out reauits arc in agreenlent with the a ~ i l a b i e te:nperatuze data

[4] . "he accuracy m measunr:g a, c, arm \ us m~ our nlet!lod is

quite adequate for engineering calculat ions.

REFERt~NtlE8

1. L;. 8. Simonov, S t ro i te l ' aaya pronl)~hiennosI' , [i.,. ~, -'.ab2.

2. G. M. Kondrat 'ev , Regular I 'hermal ~ ouditicns [in Russiall], G I r l ' L , Moscow, 1954.

3. A. F. 3egunRova, Izvestiya VUL. l:riborostr~ci:ic, u.:. 2, i l/C3.

4. V. A. Khmara and K. it. :trandin, Voprosy radioclcl<trorliki, Seriy~ o~shchetekhnicheskya, uo. 20, :964.

7 .:an:.ary 196b K uznetsov >;iberian

Fh} sic a 1 - 'l'echl~ical ]:lstiturr

SOME EXPEi?.IMEN i'.<. WITH (:EN'I'IMf:I'ER WAVE.-.

B . sh. Perkal'skis and V . 1 , . Latin

lzvest iya VLJZ. Fizika, No. 2, pp.. �9 1966

1. rAUTO(.II[;:ONISh: OF A I.EN5

Image formation by a lens is an interference effect . All the rays

converging to a point in tire ima ge arrive in phase because their paths

are tautochtonous, wltich results ill mutual re inforcement , l'his can

be convenient ly demonstrated at mic rowave frequencies. At the

Physics Depar tment of Tomsk University we employed a 3.2 cm

klystron osczllator modula ted by a mul t iv ibra tor . The s i g n a l - w h i c h was radiated by a horn antenna--was picked up and fed to a l )K-l l

detector; the de tec ted low-frequency signal was fed through a [22-1A

(281M) ampl i f ie r to a Cl-1 fEd-;' ; scope.

The signal propagated by tile aorn was focused hy a plane-corl-

vex paraffin lens with a focal length of 50L; into and a d iameter of

300 ram. The detector was placed at a point where the rays converged;

the scope-screen (with tile sweet turned off) showed a received sig- nal of considerable ampl i tude .

Then a circular disk of paraffin or some other dielectr ic which

added a haLf-wave path difference was a t tached to the lens. The

center of the disk must be located on the main opt ical axis of the

lens; the radius was such that the disk covered half the area of the

wave front converging at the i m a g e point. Hence tile radius of tile

disk r = R/2~/2 , where R is the lens radius, since ,'rr z = rR 2 / 2 .

Addition of the disk (in front of or behind the lens) causes the

ampl i tude of the rece ived signal to drop to zero, since the rays con-

verged by the lens axe m antiphase.

The same result is obtained b)" placing a paraffin ring with inner

and outer radii of r and R respect ively neat the lens, but it is more

convemen t to use a rec tangular plate larger than tile lens but with

,J.

&

. / / ~ . - ~ / / | i ]

~4o

?ig. 1

an aperture of the same radius;

If ,~ rec tangular paraffin sheet causing a path difference ot L 2

is attaeiled to tile lens in such a way that its edge correqxmds to tile

" s imi la r results call he obtained ttl optics ny expc,il:g the tell., h)

coherent radlatiorl from a laser, if a r ing-shaped vc...sei ,:f radius r

between two plane paral lel p!cces ,;f bias., Is placed near {lie lc!:s;

a h a l f -wav e path differe::cc ,.~:1 be c r ra ted by tharw, iJ:g tac pressure

(which ,:an even be effeetcd :~) tht simple expedient of squeezing the vessel).

114 IZVESTIYA VUZ. FIZIKA

lens d iameter , the ampli tude of the rece ived signal wi l l again drop to zero. In this case, however, there wi l l be no signal even when the lens is removed, which can easi ly be seen even by using the Comu spiral method. When the detector is p laced facing the edge of the rec tangular sheet the result shown in Fig. 1 is obtained; if the probe is p laced on the other side of the sheet such that i t does not cover the two probes on the lef t -hand side of the plane front," the ampl i tude of the signal is 2 . 0 ~ , e tc . (Fig. 2):'0"

6/

A~

7/.~ 7 / X �9 I

, , \ \ , I / , , ! ! N r ~ 8t J , 1o 4, 4z

Fig. 2

To show that the cause is not due displacement of the focus, the wave front was covered up entirely by the sheet; the s ignal immedia re ly reappeared.

The thickness of the sheets is computed from the formula d(n - -- 1)= K/2, which at 1000 Mc yields d = 3.2 cm for paraffin and d = 4 - 5 cm for wood.

2. DEMONSTRATION OF LOSSES ON REFLECTION

The problem of phase shift when a wave is ref lected at the boundary of two media is t rea ted in mechanics and optics, but this

phenomenon is rarely demonstrated. In this connection the fol- lowing exper iment (which sirilultaneonsly sheds some l ight on Wiener's

**i. e . , the pro, he is placed at the point A 2. -*This is clea~ from the fact that, beginning with the third, the symmet r i ca l zones on both sides of the detector cancel each other

out. leaving only the double effect of the two nearest zones.

experiments with Light rays) may be useful. When an e lec t romagnet ic wave is ref lec ted from a m e t a l surface

the ampl i tude of the ref lected wave is nearly the same as that of the incident wave, but the phase is shifted hy 180 ~ In our demonstra- t ions we employed a generator intended for school use which operated at a wavelength of 2 m. The signal was picked up by a ha l f -wave dipole connected to the detector. The low-frequency detected sig- na l was fed to a demonstration ga lvanometer . The distance between the receiving and t ransmit t ing dipoles was chosen such that the

ga lvanomete r deflect ion was fairly large (about 1 -1 .5 m). If a m e t a l p la te 1 m long and 6 - 8 m across was lowered into p lace close to the rece iv ing dipole, the ga lvanometer deflection was sharply

reduced, neat ly to zero when the plate was properly placed and the incident and ref lec ted waves were propagated in the same direct ion. This can be achieved by rotat ing the plate sLightly around the hori- zont',d axis. When the plate is raised the ga lvanometer deflect ion

increases sharply.

For the exper iment to be successful it is necessary that the path difference be small in comparison with the length of the standing

wave; hence the reflector must be placed as close to the dipole as possible. For this reason 3-cen t imete r waves are unsuited to this

exper iment , though dec imete r waves yield good results.

3. BREWSTER-ANGLE REFLECTION OF POLARIZED WAVES

Cemimete r waves can provide a c lea t demonstration of the dif- ference in Brewster-angle ref lect ion between waves polar ized in the p lane of inc idence and those polar ized perpendicularly to i t . The

s ignal from the horn antenena ks directed towards a d ie lec t r ic plate (paraffin. ebonite , e tc . ) at the Brewster angle (b7 ~ for paraffin). The ref lected signal is also picked up by a horn antenna, and after de- tec t ion is fed to a U2-1A ampli f ier and an EO-7 oscil loscope. In the

first part of the exper iment the two horns are oriented in such a way

tllat the wave is polarized perpendicularly to the plane of inc idence ; tile intensity of the ref lected signal is high. Then the receiving and

transmit t ing horns are rotated by 90' around their longitudinal axes without changing the angle of inc idence and ref lect ion. This causes

the ampl i tude of the received signal to drop to zero.

3 June 1965 Kuznetsov Siberian Physical-

Technica l Institute

A METHOD OF DERIVING THE LAW OF UNIVERSAL GRAVITATION FROM KEPLER'S LAWS

Yu. K. Gulak

lzvesuya VUZ, Fizika, No. 2, pp. 171-173, 1966

The programs of university courses on general physics, astronomy,

and somet imes theore t ica l mechanics include derivation of the Law of universal gravi ta t ion from Kepler 's laws as established expe t imen- ta.Uy. The t radi t ional procedure of obtaining an exact solution to

this lxoblem is to use the theorem of the k ine t ic energy differential , from wMeh the Binet theorem and then the Law of universal gravi- ta t ion are obta ined. This procedure involves a knowledge of higher ma themat i c s and mechanics and is therefore sometimes inadvisable.

To avoid the diff icult ies in this approach i t is necessary to Limit the

Ixoblem to the part icular case of c i rcular planetary orbits, which Urrdts the scope of the mater ia l and requires the students to accept

on fai th certain very basic ideas. In this paper we wi l l g ive a very s imple method of deriving the

l aw of universal gravi ta t ion from Kepler 's laws. This method has the following posit ive aspects:

a) eLlllXlCal planetary orbits are considered; b) different ia l ca lculus is not employed; c ) i t can be readi ly concluded that t h e f o r c e o f universal gravi-

tat ion, winch ks i n v e r ~ l y l~oportionat to the square of the distance, can give rise not only to eUlpt icaJ orbits but to any orbits which

/l" t //

can be represented by conic sections. Assume that the orbRal position of a planet with respect to the

sun is specif ied by the radius vector r . The planer 's motion is af- fected by the f o r c e e x e r t e d on i t by the sun. Taking the sun as the center of an Iner t ia l coordinate system, it Is easy to see that (figure)

~ l t l ~ . ,P, . - f ' s i n ( t o ) . ( 1 )