Фізика, збірник задач

24.9. Lhgdhkl•ggZ \hevnjZfh\Z dmey lh\sbghx d		ff sh gZ]j•lZ ^h
l_fi_jZlmjb L1 D hoheh^`m}lvky \gZke•^hd l_ieh\h]h \bijh-
f•gx\Zggy m \Zdmmf ^h l_fi_jZlmjb L2	D Mijh^h\` ydh]h qZkm
t \•^[m\Zehkv l_ieh\_ \bijhf•gx\Zggy • gZ kd•evdb af_grblvky
^h\`bgZ o\be• sh \•^ih\•^Z} fZdkbfmfm \bijhf•gx\Zevgh€ a^Zlghk-
l•" IblhfZ l_ieh}fg•klv \hevnjZfm k		>` d] D ]mklbgZ
\hevnjZfm ! 103 d] f3. k fdf

24.10. F_lZe_\Z dmey jZ^•mkhf R kf • l_ieh}fg•klx K >` D ijb

l_fi_jZlmj• Lh D i_j_[m\Z} m f•`ieZg_lghfm ijhklhj• Dh_n•p•}gl qhjghlb dme• ÂL = 0,8 Q_j_a ydbc qZk l_fi_jZlmjZ dme•

af_grblvky \^\•q•" k

24.11. <\Z`Zxqb sh ki_dljZevgbc jhaih^•e _g_j]•€ l_ieh\h]h \bijhf•gx- \Zggy hibkm}lvky nhjfmehx <•gZ r* L : 3 _-. /T ^_  = 7,5 nc K, agZclb ^ey l_fi_jZlmjb L D qZklhlm ydZ \•^ih\•^Z} fZdkb- fZevghfm agZq_ggx \bijhf•gx\Zevgh€ a^Zlghkl• (8 1014 k-1)

24.12. <bdhjbklh\mxqb nhjfmem IeZgdZ rν ,T	=	2πν 2		hν
		c2		hν	−1		h[qbkeblb
			e kT

klZem Kl_nZgZ – ;hevpfZgZ (5,67 10-8 <l f2 D4))

24.13. AZklhkh\mxqb nhjfmem IeZgdZ rλ,T =	2πc2 h	1
	λ5			hc			h[qbkeblb
			e kTλ			−1
klZem <•gZ (2,9 10-3 f D

D<:GLH<H-HILBQG1 Y<BS:

Hkgh\g• nhjfmeb

?g_j]•y fZkZ lZ •fimevk nhlhgZ

E = hν = hcλ ;

101

m =

E

=

hν

=

h

;

c2

c2

cλ

p =

E

=

hν

=

h

.

c

c

λ

J•\gyggy ?cgrl_cgZ ^ey nhlh_n_dlm

hν = A +	mvmax2	;	hν = A + eU a ,
2

^_ : – jh[hlZ \boh^m _e_dljhg•\	mvmax2	– fZdkbfZevgZ d•g_lbqgZ
2

_g_j]•y nhlh_e_dljhg•\ Ua – aZljbfmxqZ gZijm]Z _ – \_ebqbgZ aZjy^m _e_dljhgZ

Q_j\hgZ f_`Z nhlh_n_dlm

	νmin =	A	;	λmax	= ch .
		h
					A
	4. Lbkd k\•leZ

p = w(1 + ρ)cosi,

^_ w – h[¶}fgZ ]mklbgZ _g_j]•€ k\•leZ sh iZ^Z} ρ – dh_n•p•}gl \•^[b\Zggy k\•leZ \•^ ih\_jog• • – dml iZ^•ggy k\•leh\bo ijhf_g•\

Af•gZ ^h\`bgb o\be• λ jhak•ygbo j_gl]_g•\kvdbo ijhf_g•\ (γ-d\Zgl•\

λ = λ′− λ = 2λe sin 2 θ2 ,

^_ λ′ – ^h\`bgZ o\be• jhak•ygbo j_gl]_g•\kvdbo ijhf_g•\ λ – ^h\`bgZ o\be• j_gl]_g•\kvdbo ijhf_g•\ sh iZ^Zxlv λ_ – dhfilhg•\kvdZ ^h\`bgZ o\be• _e_dljhg•\ θ – dml jhak•yggy j_gl]_g•\kvdbo ijhf_g•\

25.1. >h\`bgZ o\be• sh \•^ih\•^Z} nhlhgm λ if <bagZqblb _g_j- ]•x ? fZkm m lZ •fimevk p nhlhgZ (13,24 10-14 >` 10-30 d]

4,4 10-22 d] f k

25.2.I•^ qZk hijhf•g_ggy ih\_jog• p_a•x n•he_lh\bf k\•lehf a ^h\`bghx

o\be• λ = 0,4 fdf fZdkbfZevgZ r\b^d•klv nhlh_e_dljhg•\ vmax = = 0, Ff k <bagZqblb q_j\hgm f_`m λmax nhlh_n_dlm fdf

102

25.3.Ih\_jogy kj•[eZ hk\•lex}lvky mevljZn•he_lh\bf \bijhf•gx\Zg- gyf a ^h\`bghx o\be• λ = 0,155 fdf Jh[hlZ \boh^m _e_dljhg•\ •a

kj•[eZ : _< H[qbkeblb fZdkbfZevgm r\b^d•klv vmax nhlh- _e_dljhg•\ Ff k

25.4.Ih\_jogy kj•[eZ hk\•lex}lvky γ-\bijhf•gx\Zggyf a ^h\`bghx o\be• λ = 2,47 if Jh[hlZ \boh^m _e_dljhg•\ •a kj•[eZ : _<. <bagZqblb fZdkbfZevgm r\b^d•klv vmax nhlh_e_dljhg•\ Ff k

25.5.Ihq_j]h\h hk\•lexxqb ih\_jogx ^_ydh]h f_lZem k\•lehf a ^h\- `bghx o\be• λ1 = 0,32 fdf • λ2 = 0,55 fdf \by\beb sh fZdkb- fZevgZ r\b^d•klv nhlh_e_dljhg•\ vmax m i_jrhfm \biZ^dm m n = 2 jZab [•evrZ g•` m ^jm]hfm <bagZqblb jh[hlm \boh^m _e_dljhg•\ a ih\_jog• f_lZem _<

25.6.GZ ih\_jogx dZe•x jh[hlZ \boh^m _e_dljhg•\ a ydh]h : _<, iZ^Z} k\•leh a ^h\`bghx o\be• λ = 0,2 fdf AgZclb gZcf_gr_ agZq_ggy aZljbfmxqh€ gZijm]b Ua ijb yd•c nhlhkljmf ijbib- gy}lvky <

25.7.I•^ qZk hk\•le_ggy ^_ydh]h f_lZem n•he_lh\bf k\•lehf a ^h\`b- ghx o\be• λ1 = 0,4 fdf \b[bl• k\•lehf _e_dljhgb ih\g•klx aZljb- fmxlvky gZijm]hx U31 = 2,50 < AgZclb aZljbfmxqm gZijm]m U32,

ydsh lhc kZfbc f_lZe hk\•lexxlv q_j\hgbf k\•lehf a ^h\`bghx o\be• λ2 = 0,75 fdf? <

25.8.I•^ qZk hk\•le_ggy \Zdmmfgh]h nhlh_e_f_glZ fhghojhfZlbqgbf k\•lehf a ^h\`bghx o\be• λ1 = 0,42 fdf \•g aZjy^`Z}lvky ^h ihl_gp•Zem 31 = 2,49 < <bagZqblb ^h ydh]h ihl_gp•Zem 32 aZ-

jy^blvky nhlh_e_f_gl ydsh ch]h hk\•lexxlv fhghojhfZlbqgbf k\•lehf a ^h\`bghx o\be• λ2 = 0,25 fdf. <

25.9.FhghojhfZlbqg_ k\•leh a ^h\`bghx o\be• λ = 0,331 fdf iZ^Z}

gZ p_a•}\bc dZlh^ nhlh_e_f_glZ Jh[hlZ \boh^m ^ey p_a•x

:_< <bagZqblb •fimevk p_ nhlh_e_dljhgZ lZ •fimevk pd,

sh hljbfm} dZlh^ ydsh \be•lZ} h^bg _e_dljhg (7,36 10-25 d] f k

7,38 10-25 d] f k

103

25.10. GZ ^a_jdZevgm ih\_jogx a dh_n•p•}glhf \•^[b\Zggy ! iZ^Z} k\•leh •gl_gkb\g•klv ydh]h I <l f2, Z dml iZ^•ggy  = 600.

<bagZqblb lbkd p k\•leZ gZ px ih\_jogx fdIZ

25.11. FhghojhfZlbqg_ k\•leh _g_j]•y ydh]h W >` ghjfZevgh

iZ^Z} gZ iehkdm ^a_jdZevgm ih\_jogx a dh_n•p•}glhf \•^[b\Zggy ! iehs_x S kf2 aZ qZk t o\ H[qbkeblb lbkd k\•leZ gZ

ih\_jogx fdIZ

25.12. Lbkd fhghojhfZlbqgh]h k\•leZ a ^h\`bghx o\be• λ = 0,6 fdf sh ghjfZevgh iZ^Z} gZ qhjgm ih\_jogx ! ^hj•\gx} J

fdIZ <bagZqblb d•evd•klv N nhlhg•\ yd• iZ^Zxlv aZ qZk t k gZ iehsm S kf2 p•}€ ih\_jog• (3 1019)

25.13.FhghojhfZlbqg_ k\•leh a ^h\`bghx o\be• λ = 0,662 fdf iZ^Z}

ghjfZevgh gZ ih\_jogx a dh_n•p•}glhf \•^[blly ! • qbgblv lbkd J fdIZ H[qbkeblb dhgp_gljZp•x n nhlhg•\ m imqdm

(5 1013 f-3)

25.14.< _n_dl• DhfilhgZ _g_j]•y nhlhgZ jhaih^•ey}lvky ihj•\gm f•` _e_dljhghf • jhak•ygbf nhlhghf Dml jhak•yggy θ = π/2 <bagZ-

qblb _g_j]•x ε′ lZ •fimevk pγ′ jhak•ygh]h nhlhgZ F_<

1,36 10-22 d] f k

25.15.Dhfilhg•\kvd_ af•s_ggy ^h\`bgb o\be• j_gl]_g•\kvdh]h d\ZglZ a ^h\`bghx o\be• λ = 5 nf ^hj•\gx} dhfilhg•\kvd•c ^h\`bg• o\be•

λ_ _e_dljhgZ AgZclb dml θ jhak•x\Zggy nhlhgZ • d•g_lbqgm _g_j]•x ?d _e_dljhgZ \•^^Zq• (900 F_<

25.16.<gZke•^hd _n_dlm DhfilhgZ nhlhg [m\ jhak•ygbc gZ \•evghfm _e_dljhg• gZ dml θ = 1800 ?g_j]•y jhak•ygh]h nhlhgZ ε′ = 0,2 F_%. <bagZqblb _g_j]•x ε nhlhgZ ^h jhak•yggy F_<

25.17.γ-d\Zgl a _g_j]•}x ? F_< jhak•x}lvky gZ \•evghfm _e_dljhg• I•key a•ldg_ggy _e_dljhg jmoZ}lvky i•^ dmlhf  = 450 ^h gZijyfdm

jmom d\ZglZ ^h a•ldg_ggy <bagZqblb dml θ jhak•x\Zggy γ-d\ZglZ

(230)

104

VIII. :LHFG: N1ABD:

L?HJ1Y ;HJ: >EY :LHF: <H>GX

Hkgh\g• nhjfmeb

Fhf_gl •fimevkm _e_dljhgZ gZ klZp•hgZjg•c hj[•l• mυn rn = n , n = 1,2, ,

^_ m – fZkZ _e_dljhgZ υn – r\b^d•klv _e_dljhgZ gZ n-c klZp•hgZjg•c hj[•l• rn – jZ^•mk _e_dljhgZ gZ n-c klZp•hgZjg•c hj[•l• n – ]heh\g_ d\Zglh\_ qbkeh = 2hπ – klZeZ IeZgdZ

?g_j]•y nhlhgZ sh \bijhf•gx}lvky Zlhfhf \h^gx i•^ qZk i_j_oh^m a h^gh]h klZp•hgZjgh]h klZgm \ •grbc

ε = hν = ω = Ek − En ε = ω = Ek − En ,

^_ ω – pbde•qgZ dheh\Z qZklhlZ \bijhf•gx\Zggy k i n – ]heh\g• d\Zglh\• qbkeZ klZp•hgZjgbo klZg•\ f•` ydbfb \•^[m\Z}lvky i_j_o•^ (k>n).

JZ^•mkb klZp•hgZjgbo hj[•l _e_dljhgZ \ Zlhf• \h^gx

	rn =	4πε0 2	n2 , n = 1,2, ,
		e2 m

^_ ε0 – _e_dljbqgZ klZeZ _ – \_ebqbgZ aZjy^m _e_dljhgZ?g_j]•y ZlhfZ \h^gx \ n-fm klZp•hgZjghfm klZg•

En = −

e4 m

1

= −

e4 m

1

.

32π 2ε02 2

n2

8ε02 h2

n2

MaZ]Zevg_gZ nhjfmeZ ;Zevf_jZ

1

1

ν = R

−

,

n2

k 2

105

	^_ ν – qZklhlb ki_dljZevgbo e•g•c ZlhfZ \h^gx R ( R =	e4 m	) – klZeZ
		8ε02 h3

J•^[_j]Z k (k = 1,2,3,4,5,6) – ]heh\g_ d\Zglh\_ qbkeh yd_ \bagZqZ} k_j•x

^h ydh€ gZe_`blv ki_dljZevgZ e•g•y ghf_j _g_j]_lbqgh]h j•\gy gZ ydbc i_j_oh^blv _e_dljhg

k = 1 – k_j•y EZcfZgZ k = 2 – k_j•y ;Zevf_jZ k = 3 – k_j•y IZr_gZ k = 4 – k_j•y ;j_d_lZ k = 5 – k_j•y Inmg^Z k = 6 – k_j•y O_fnj•

n (n = k + 1, k + 2, … , ∞) – ]heh\g_ d\Zglh\_ qbkeh sh \•^ih\•- ^Z} _g_j]_lbqghfm j•\gx a ydh]h i_j_oh^blv _e_dljhg

26.1.AgZclb jZ^•mkb rn ljvho i_jrbo [hj•\kvdbo _e_dljhggbo hj[•l \ Zlhf• \h^gx • r\b^dhkl• vn _e_dljhg•\ gZ gbo (0,53 10-10 f

21,18 10-10 f 10-10 f 106 f k 106 f k 106 f k

26.2.<bagZqblb d•g_lbqgm ?d ihl_gp•Zevgm ?i • ih\gm ?1 _g_j]•x _e_dljhgZ gZ i_jr•c [hj•\kvd•c hj[•l• ZlhfZ \h^gx _<

– _< – _<

26.3. :lhf \h^gx \bijhf•gx} nhlhg a ^h\`bghx o\be• fdf.

<bagZqblb gZ kd•evdb \h^ghqZk af•gbeZkv d•g_lbqgZ _g_j]•y _e_dljhgZ _<

26.4.<bagZqblb gZcf_grm min • gZc[•evrm max ^h\`bgb o\bev ki_dljZev- gbo e•g•c \h^gx m \b^bf•c ^•eygp• ki_dljZ fdf fdf

26.5.AgZclb i_jrbc ihl_gp•Ze a[m^`_ggy U1 ZlhfZ \h^gx <

26.6.H[qbkeblb ihl_gp•Ze •hg•aZp•€ Ui ZlhfZ \h^gx <

26.7.I•key i_j_oh^m _e_dljhgZ gZ j•\_gv a ]heh\gbf d\Zglh\bf qbkehf n = 2 jZ^•mk hj[•lb _e_dljhgZ \ Zlhf• \h^gx af•gb\ky m jZa•\

<bagZqblb qZklhlm k\•leZ sh \bijhf•gx}lvky Zlhfhf \h^gx

(0,731· 1015 =p

106

26.8.:lhf \h^gx ydbc i_j_[m\Z} m a[m^`_ghfm klZg• fh`_ ih\_jlZx- qbkv \ hkgh\gbc klZg \bijhf•gblb N = 6 e•g•c <bagZqblb ghf_j n a[m^`_gh]h klZgm (4)

26.9.A[m^`_gbc Zlhf \h^gx i_j_crh\rb \ hkgh\gbc klZg \bimklb\

ihke•^h\gh ^\Z d\Zglb k\•leZ a ^h\`bgZfb o\bev 1 fdf •2 = 0,09725 fdf <bagZqblb _g_j]•x ?n ihqZldh\h]h klZgm ZlhfZ • \•^ih\•^g_ chfm d\Zglh\_ qbkeh n. (– _<

26.10. >\hf e•g•yf k_j•€ ;Zevf_jZ ZlhfZ \h^gx \•^ih\•^Zxlv ^h\`bgb

o\bev 1 fdf • 2 fdf <bagZqblb ^h ydh€ k_j•€ gZe_`blv ki_dljZevgZ e•g•y o\bevh\_ qbkeh ydh€ ^hj•\gx} j•agbp• o\bevh\bo qbk_e pbo e•g•c k_j•€ IZr_gZ

26.11.IhdZaZlb sh qZklhlZ k\•leZ yd_ \bijhf•gx}lvky i•^ qZk i_j_oh^m _e_dljhgZ a (n+1)-€ gZ n-lm hj[•lm ijb n:’ gZ[eb`Z}lvky ^h qZklhlb h[_jlZggy _e_dljhgZ gZ\dheh y^jZ

O<BE1 >? ;JHCEY Hkgh\g• nhjfmeb

>h\`bgZ o\be• ^_ ;jhcey ^ey f•djhqZklbgdb a •fimevkhf P = mυ λ = Ph = mhυ ,

^_ h – klZeZ IeZgdZ

27.1.	?e_dljhg ijhcrh\ j•agbpx ihl_gp•Ze•\ U	< <bagZqblb
	^h\`bgm o\be• ^_ ;jhcey λ _e_dljhgZ ( if)
27.2.	< h^ghj•^ghfm fZ]g•lghfm ihe• a •g^mdp•}x <	fLe jmoZ}lvky

_e_dljhg ih dhem jZ^•mkhf R kf AgZclb ^h\`bgm o\be• ^_ ;jhcey λ _e_dljhgZ (129,3 if)

107

27.3.Fhe_dmeZ Zahlm jmoZ}lvky •a k_j_^gvhx d\Z^jZlbqghx r\b^d•klx ijb l_fi_jZlmj• L D <bagZqblb ^h\`bgm o\be• ^_ ;jhcey λ fhe_dmeb if)

27.4.?e_dljhg jmoZ}lvky ih ^jm]•c hj[•l• ZlhfZ \h^gx H[qbkeblb ^h\`bgm o\be• ^_ ;jhcey λ _e_dljhgZ if)

27.5.GZ \mavdm s•ebgm rbjbghx Z fdf kijyfh\Zgh iZjZe_evgbc imqhd _e_dljhg•\ yd• fZxlv r\b^d•klv v = 3,6 106 f k <jZoh-

\mxqb o\bevh\• \eZklb\hkl• _e_dljhg•\ \bagZqblb \•^klZgv f•` ^\hfZ fZdkbfmfZfb •gl_gkb\ghkl• i_jrh]h ihjy^dm \ ^bnjZd-

p•cg•c dZjlbg• gZ _djZg• ydbc \•^^Ze_gbc gZ L	f \•^ s•ebgb
( fdf)
27.6. Dmey fZkhx m ] jmoZ}lvky •a r\b^d•klx v	f k <bagZ-

qblb ^h\`bgm o\be• ^_ ;jhcey λ dme• (2,76 10-34 f)

27.7.Imqhd _e_dljhg•\ iZ^Z} gZ iehsbgm i•^ dmlhf dh\aZggy × = 300,

_e_dljhgb \•^[b\Zxlvky i•^ dmlhf sh ^hj•\gx} dmlm iZ^•ggy KlZeZ djbklZe•qgh€ ‰jZldb d gf AgZclb agZq_ggy i_jrh€ ijbkdhjxxqh€ j•agbp• ihl_gp•Ze•\ U ijb yd•c kihkl_j•]Z}lvky fZdkbfZevg_ \•^[b\Zggy (26,1 <)

28 KI1<<1>GHR?GGY G?<BAG:Q?GHKL?C

Hkgh\g• nhjfmeb

1.Ki•\\•^ghr_ggy g_\bagZq_ghkl_c =_ca_g[_j]Z

xPx ≥ ,

yPy ≥ ,

zPz ≥ ,

^_ x, y, z – g_\bagZq_ghkl• dhhj^bgZl x, y, z qZklbgdb Px, Py, Pz – g_\bagZq_ghkl• \•^ih\•^gbo ijh_dp•c •fimevkm qZklbgdb

Ki•\\•^ghr_ggy g_\bagZq_ghkl_c ^ey _g_j]•€ qZklbgdb

E t ≥ ,

108

^_ E – g_\bagZq_g•klv _g_j]•€ qZklbgdb	t – qZk ljb\Ze•klv `blly
qZklbgdb m klZg• a ^Zgbf agZq_ggyf _g_j]•€

28.1.Imqhd _e_dljhg•\ jmoZ}lvky \a^h\` hk• 0X a• r\b^d•klx v = 106 f k, ydZ \bagZqZ}lvky a lhqg•klx ^h 0,01 % \•^ €€ qbkeh\h]h agZq_ggy AgZclb g_\bagZq_g•klv x dhhj^bgZlb _e_dljhgZ (1,16 10-6 f

28.2.IbebgdZ fZkhx m = 10-12 d] fZ} e•g•cg• jhaf•jb ihjy^dm 10-6 f. Dhhj^bgZlm ibebgdb fh`gZ \bagZqblb a lhqg•klx ^h 0,01 €€ jhaf•j•\

YdZ g_\bagZq_g•klv r\b^dhkl• vx ibebgdb" (1,05 10-14 f c)

28.3. ?e_dljhggbc imqhd ijbkdhjx}lvky \ _e_dljhggh-ijhf_g_\•c ljm[p• j•agbp_x ihl_gp•Ze•\ U < <\Z`Zxqb sh g_\bagZ- q_g•klv •fimevkm ^hj•\gx} 0,1 % \•^ ch]h qbkeh\h]h agZq_ggy agZclb g_\bagZq_g•klv dhhj^bgZlb x _e_dljhgZ gf

28.4.?e_dljhg jmoZ}lvky \ Zlhf• \h^gx ih i_jr•c [hj•\kvd•c hj[•l• a• r\b^d•klx v = 2,18 106 f k <\Z`Zxqb sh g_\bagZq_g•klv r\b^dhkl•

^hj•\gx} 10 % \•^ €€ qbkeh\h]h agZq_ggy agZclb g_\bagZq_g•klv x dhhj^bgZlb _e_dljhgZ (5,31 10-10 f

28.5.?e_dljhg jmoZ}lvky \ Zlhf• \h^gx ih i_jr•c [hj•\kvd•c hj[•l• G_oZc g_\bagZq_g•klv dhhj^bgZlb _e_dljhgZ } ki•\f•jgZ a jhaf•-

jZfb kZfh]h ZlhfZ lh[lh x = 10-10 f AgZclb g_\bagZq_g•klv r\b^dhkl• v jmom _e_dljhgZ ih hj[•l• (1,16 106 f k

28.6. D•g_lbqgZ _g_j]•y _e_dljhgZ \ Zlhf• \h^gx ?d _< <bdhjbk-

lh\mxqb ki•\\•^ghr_ggy g_\bagZq_ghkl_c hp•gblb f•g•fZevg• e•g•cg• jhaf•jb ZlhfZ (1,23 10-10 f

28.7.?e_dljhg i_j_[m\Z} \ g_kd•gq_ggh ]eb[hd•c ³ihl_gp•Zevg•c yf•´ rbjbghx • if AZklhkh\mxqb ki•\\•^ghr_ggy g_\bagZq_- ghkl_c agZclb f•g•fZevgm _g_j]•x ?min _e_dljhgZ _<

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109

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