Фізика, збірник задач

A\¶yahd f•` fZ]g•lghx ijhgbdg•klx μ • fZ]g•lghx kijbcgyl- eb\•klx χ j_qh\bgb

μ = 1 + χ.

A\¶yahd				f•` gZijm`_g•klx		fZ]g•lgh]h			ihey H	fZ]g•lghx
•g^mdp•}x B lZ gZfZ]g•q_g•klx j_qh\bgb J
				H =
						B − J.
						μo
,g^mdlb\g•klv ^m`_ ^h\]h]h (• >> d) khe_gh€^Z
				L = μμ0 N 2 S = μμ0 n2V ,
^_ N – d•evd•klv				\bld•\ khe_gh€^Z	S – iehsZ				ihi_j_qgh]h i_j_j•am
	n =	N		– d•evd•klv \bld•\ gZ h^bgbpx ^h\`bgb • khe_gh€^Z
khe_gh€^Z
V – h[¶}f	khe_gh€^Z d – ^•Zf_lj khe_gh€^Z μ – fZ]g•lgZ									ijhgbdg•klv

k_j_^h\bsZ j_qh\bgb \k_j_^bg• khe_gh€^Z

?g_j]•y fZ]g•lgh]h ihey kljmfm sh l_q_ m dhglmj• a •g^md- lb\g•klx L

W= LI22 .

H[¶}fgZ ]mklbgZ _g_j]•€ fZ]g•lgh]h ihey

	w =	B2	=	μ0 μH 2	,
		2μ0 μ		2

^_ μ – fZ]g•lgZ ijhgbdg•klv k_j_^h\bsZ j_qh\bgb \ ydhfm •kgm} ihe_

19.1.Ih dheh\hfm dhglmjm jZ^•mkhf R = f ijhl•dZ} kljmf kbehx 1 : Dhglmj aZgmj_gbc \ j•^dbc dbk_gv fZ]g•lgZ kijbcgyleb- \•klv ydh]h χ = 0,0034 AgZclb gZfZ]g•q_g•klv J m p_glj• dhglmjm

f: f

19.2.H[fhldZ lhgdh€ lhjh€^Zevgh€ dhlmrdb •a aZe•agbf hk_j^yf kdeZ^Z}lvky

•a N = 628 \bld•\ K_j_^g•c jZ^•mk lhjZ R f Ih h[fhlp• ijhl•dZ} kljmf kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x ihey

81

\k_j_^bg• dhlmrdb gZfZ]g•q_g•klv • fZ]g•lgm ijhgbdg•klv hk_j^y

Le F: f

19.3. H[fhldZ lhgdh]h lhjh€^Z •a aZe•agbf hk_j^yf kdeZ^Z}lvky •a N = 1256 \bld•\ iehsZ ihi_j_qgh]h i_j_j•am hk_j^y S kf2 jZ^•mk hkvh\h€ e•g•€ hk_j^y R kf Ih h[fhlp• lhjh€^Z ijhl•dZ} kljmf kbehx

1 : AgZclb •g^mdlb\g•klv lhjh€^Z =g

19.4.Khe_gh€^ fZ} ^h\`bgm • kf iehsm ihi_j_qgh]h i_j_j•am

S kf2 • d•evd•klv \bld•\ N = 400 1g^mdlb\g•klv khe_gh€^Z L =f=g. Ih \bldZo khe_gh€^Z ijhl•dZ} kljmf kbehx 1 :.

	Khe_gh€^ agZoh^blvky \ ^•ZfZ]g•lghfm k_j_^h\bs• <bagZqblb
	fZ]g•lgm •g^mdp•x < •	\_dlhj gZfZ]g•q_ghkl• J \k_j_^bg• kh-
	e_gh€^Z (0 Le : f
19.5.	M khe_gh€^ ^h\`bghx • f sh fZ} N = 400 \bld•\ \\_^_gh
	aZe•ag_ hk_j^y Ih khe_gh€^m l_q_ kljmf kbehx 1 : AgZclb
	\_ebqbgm gZfZ]g•q_ghkl• aZe•aZ J \k_j_^bg• khe_gh€^Z <\Z`Zlb
	fZ]g•lg_ ihe_ \k_j_^bg• khe_gh€^Z h^ghj•^gbf (1, F: f
19.6.	GZ qZ\mgg_ hk_j^y m \b]ey^• lhjZ a ^h\`bghx hkvh\h€ e•g•€ • f
	gZfhlZgZ h[fhldZ a d•evd•klx \bld•\ N = 800 < hk_j^• ajh[e_gZ
	\mavdZ ihi_j_qgZ s•ebgZ rbjbghx •0 ff FZ]g•lgZ •g^mdp•y m
	ih\•ljyg•c s•ebg• <0	Le Jhak•yggyf fZ]g•lgh]h ihlhdm m
	ih\•ljyg•c s•ebg• fh`gZ ag_olm\Zlb AgZclb kbem kljmfm 1 \
	h[fhlp• :
19.7.	Ih h[fhlp• khe_gh€^Z [_a hk_j^y sh f•klblv N = 1000 \bld•\
	ijhl•dZ} kljmf kbehx 1 : FZ]g•lgbc ihl•d q_j_a ihi_j_qgbc
	i_j_j•a khe_gh€^Z N<	f<[ <bagZqblb _g_j]•x W fZ]g•lgh]h

ihey \ khe_gh€^• >`

19.8. Khe_gh€^ [_a hk_j^y a s•evgh gZfhlZghx h^ghrZjh\hx h[fhldhx

•a ^jhlm ^•Zf_ljhf d	ff fZ} ^h\`bgm • f • iehsm
ihi_j_qgh]h i_j_j•am S	kf2 Ydsh gZijm]Z gZ d•gpyo h[fhldb

U < ih h[fhlp• ijhl•dZ} kljmf kbehx 1 : AZ ydbc qZk \

h[fhlp• \b^•eblvky d•evd•klv l_iehlb ydZ ^hj•\gx} _g_j]•€ ihey \k_j_^bg• khe_gh€^Z" FZ]g•lg_ ihe_ \k_j_^bg• khe_gh€^Z \\Z`Zlb h^ghj•^gbf fdk

82

f d•evd•klv \bld•\ N = 340 iehsZkf2 <bagZqblb _g_j]•x fZ]g•lgh]h ihey

19.9. >h h[fhldb khe_gh€^Z hi•j ydh€ R Hf ijbdeZ^_gZ ihkl•cgZ gZijm]Z AZ qZk t k \ h[fhlp• \b^•eblvky d•evd•klv l_iehlb Q ydZ ^hj•\gx} _g_j]•€ fZ]g•lgh]h ihey W khe_gh€^Z <bagZqblb •g^mdlb\g•klv L khe_gh€^Z =g

19.10. Khe_gh€^ ^h\`bghx • f [_a hk_j^y f•klblv N = 100 \bld•\ Ih khe_gh€^m l_q_ kljmf kbehx 1 : Ydhx [m^_ h[¶}fgZ ]mklbgZ w

_g_j]•€ fZ]g•lgh]h ihey \k_j_^bg• khe_gh€^Z" <\Z`Zlb fZ]g•lg_ ihe_

\k_j_^bg• khe_gh€^Z h^ghj•^gbf • ehdZe•ah\Zgbf ijZdlbqgh \k_j_^bg• khe_gh€^Z >` f3)

19.11. Ih \bldZo khe_gh€^Z •a aZe•agbf hk_j^yf l_q_ kljmf kbehx 1 :. >h\`bgZ khe_gh€^Z •

ihi_j_qgh]h i_j_j•am S

khe_gh€^Z <\Z`Zlb fZ]g•lg_ ihe_ \k_j_^bg• khe_gh€^Z h^ghj•^gbf • ehdZe•ah\Zgbf ijZdlbqgh \k_j_^bg• khe_gh€^Z >`

19.12. D•evd•klv \bld•\ gZ dh`ghfm kZglbf_lj• ^h\`bgb khe_gh€^Z •a aZe•agbf hk_j^yf n kf-1 Ih h[fhlp• khe_gh€^Z ijhl•dZ} kljmf

kbehx 1 : <bagZqblb h[¶}fgm ]mklbgm w _g_j]•€ fZ]g•lgh]h ihey \ hk_j^• <\Z`Zlb fZ]g•lg_ ihe_ \k_j_^bg• khe_gh€^Z h^gh- j•^gbf >` f3)

?E?DLJHF:=G1LG1 DHEB<:GGY L: O<BE1

Hkgh\g• nhjfmeb

I_j•h^ \eZkgbo _e_dljhfZ]g•lgbo dheb\Zgv \ •^_Zevghfm dheb- \Zevghfm dhglmj• nhjfmeZ LhfkhgZ

T = 2π LC ,

^_ L – •g^mdlb\g•klv dhlmrdb K – }fg•klv dhg^_gkZlhjZA]ZkZxq• dheb\Zggy \ j_Zevghfm dhglmj• (R •

q = q0 e−δ t cos(ωt + ϕ),

83

^_

ω =

, ω

0

–

\eZkgZ

qZklhlZ

dhglmjm ω

=

); δ –

ω −δ

/&

dh_n•p•}gl a]ZkZggy δ =

5

).

/

I_j•h^ \eZkgbo _e_dljhfZ]g•lgbo dheb\Zgv \ j_Zevghfm dheb-

\Zevghfm dhglmj• hf•qgbc Zdlb\gbc hi•j ydh]h R

T = 2π

=

2π

.

ω

1

R 2

−

LC

2L

>h\`bgZ _e_dljhfZ]g•lgh€ o\be• m \Zdmmf• ydZ \bijhf•gx}lvky dheb\Zevgbf dhglmjhf

λ = cT = 2πc LC ,

^_ k – r\b^d•klv ihrbj_ggy _e_dljhfZ]g•lgbo o\bev m \Zdmmf•

k ˜108 f k

NZah\Z r\b^d•klv ihrbj_ggy _e_dljhfZ]g•lgbo o\bev

υ =		1	=	c .
	ε0	εμ0 μ		εμ

J•\gyggy iehkdh€ _e_dljhfZ]g•lgh€ o\be• ydZ ihrbjx}lvky \a^h\` ^h^Zlgh]h gZijyfm hk• x

E= E0 cos(ωt − kx +ϕ0 )

iH = H0 cos(ωt − kx +ϕ0 ),

^_ E0 • H0 – Zfie•lm^g• agZq_ggy \•^ih\•^gh gZijm`_ghkl_c _e_dljbqgh]h

• fZ]g•lgh]h ihe•\ _e_dljhfZ]g•lgh€ o\be• ω – pbde•qgZ qZklhlZ dheb\Zgv k – o\bevh\_ qbkeh k = ωυ = 2λπ ).

A\¶yahd f•` Zfie•lm^gbfb E0 • H0

ε0εE0 = μμ0 H 0.

=mklbgZ ihlhdm _g_j]•€ _e_dljhfZ]g•lgh€ o\be• \_dlhj Ihcgl•g]Z

			,
P = wυ		= E H

P = EH ,

^_ w – h[¶}fgZ ]mklbgZ _g_j]•€ _e_dljhfZ]g•lgh€ o\be•

84

< <bagZqblb fZdkbfZevgm kbem kljmfm 10 \

,gl_gkb\g•klv fhghojhfZlbqgh€ [•`mqh€ _e_dljhfZ]g•lgh€ o\be•

I = P = w υ.

20.1. Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhg^_gkZlhjZ _e_dljh}fg•klx K fdN • dhlmrdb •g^mdlb\g•klx L f=g :dlb\gbc hi•j

dhglmjm ^m`_ fZebc FZdkbfZevgZ gZijm]Z gZ h[deZ^dZo dhg^_gkZlhjZ U0

dhglmj• :

20.2. Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhlmrdb •g^mdlb\g•klx L f=g • dhg^_gkZlhjZ _e_dljh}fg•klx K iN :dlb\gbc hi•j R dhglmjm ^m`_ fZebc FZdkbfZevgZ kbeZ kljmfm \ dhglmj•

10 f: AgZclb fZdkbfZevgm gZijm]m U0 gZ h[deZ^dZo dhg- ^_gkZlhjZ <

20.3. H[qbkeblb \•^ghr_ggy _g_j]•€ _e_dljbqgh]h ihey We ^h _g_j]•€ fZ]g•lgh]h ihey Wf dheb\Zevgh]h dhglmjm ^ey fhf_glm qZkm

	t = 1/8 L. (1)
20.4.	?e_dljh}fg•klv dhg^_gkZlhjZ dheb\Zevgh]h dhglmjm K					fdN
	GZijm]Z gZ h[deZ^dZo dhg^_gkZlhjZ a qZkhf af•gx}lvky aZ aZdhghf
	uK = 50 cos 103π t <. <bagZqblb •g^mdlb\g•klv L dhlmrdb • aZdhg af•gb a
	qZkhf kbeb kljmfm \ dhe• =g 1			cos (103πt + π :
20.5.	1g^mdlb\g•klv dheb\Zevgh]h dhglmjm L				=g KbeZ kljmfm \
	dhglmj• a qZkhf af•gx}lvky aZ aZdhghf i = –0,02 sin 500πt : AgZc-
	lb }fg•klv dhglmjm K fZdkbfZevg• _g_j]•€ fZ]g•lgh]h lZ _e_dljbq-
	gh]h ihe•\ fdN f>`
20.6.	M dheb\Zevghfm dhglmj• a fZebf Zdlb\gbf hihjhf sh kdeZ^Z}lvky a
	dhg^_gkZlhjZ }fg•klx K		fdN •		dhlmrdb •g^mdlb\g•klx
	L f=g kbeZ kljmfm af•gx}lvky aZ aZdhghf i = –0,02 sin ω t : H[-
	qbkeblb fbll}\• agZq_ggy gZijm]b uK gZ dhg^_gkZlhj• • gZijm]b uL
	gZ dhlmrp• \ fhf_gl qZkm t			L \•^ ihqZldm \bgbdg_ggy
	dheb\Zgv. < <
20.7.	?e_dljh}fg•klv dhg^_gkZlhjZ dheb\Zevgh]h dhglmjm K					iN Z
	•g^mdlb\g•klv dhlmrdb L		f=g GZijm]Z gZ dhg^_gkZlhj•

85

af•gx}lvky aZ aZdhghf uK = 0,01 cos ωt. AgZclb fbll}\_ agZq_ggy kbeb kljmfm i • fbll}\_ agZq_ggy gZijm]b uL gZ dhlmrp• q_j_a qZk

t L fd: f<

20.8.Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhg^_gkZlhjZ _e_dljh}fg•klx

K iN • dhlmrdb •g^mdlb\g•klx L fd=g	:dlb\gbc hi•j
dhglmjm R Hf <bagZqblb i_j•h^ L \•evgbo dheb\Zgv dhglmjm
dh_n•p•}gl a]ZkZggy δ • eh]Zjbnf•qgbc ^_dj_f_gl a]ZkZggy æ.
fdk 106 k-1; 3,6)
20.9. M dheb\Zevghfm dhglmj• •g^mdlb\g•klv ydh]h L	f=g aZjy^

dhg^_gkZlhjZ af_grm}lvky \ N = 10 jZa•\ aZ i_j•h^ L fk. H[qbkeblb hi•j dhglmjm R. Hf

20.10. Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhg^_gkZlhjZ _e_dljh}fg•klx K fdN • dhlmrdb •g^mdlb\g•klx L =g :dlb\gbc hi•j dhglmjm R Hf FZdkbfZevgbc aZjy^ h[deZ^hd dhg^_gkZlhjZ q0 fDe <bagZqblb eh]Zjbnf•qgbc ^_dj_f_gl a]ZkZggy æ •

gZijm]m gZ h[deZ^dZo dhg^_gkZlhjZ \ fhf_gl qZkm t	L.
<

20.11. Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhlmrdb •g^mdlb\g•klx L =

f=g • dhg^_gkZlhjZ _e_dljh}fg•klx K		fdN AZ i	ih\g•
dheb\Zggy gZijm]Z gZ h[deZ^dZo	dhg^_gkZlhjZ af_grm}lvky \
N = 5 jZa•\ <bagZqblb eh]Zjbnf•qgbc ^_dj_f_gl a]ZkZggy æ •
Zdlb\gbc hi•j R dhglmjm Hf
20.12. Dheb\Zevgbc dhglmj kdeZ^Z}lvky a dhlmrdb •g^mdlb\g•klx L			f=g •
dhg^_gkZlhjZ _e_dljh}fg•klx K fdN Hi•j dhglmjm R			Hf. M
dhglmj• i•^ljbfmxlvky g_a]ZkZxq• dheb\Zggy a Zfie•lm^gbf
agZq_ggyf gZijm]b gZ dhg^_gkZlhj• U0		< Ydm	k_j_^gx
ihlm`g•klv ih\bg_g kih`b\Zlb dhglmj" <l
20.13. ?e_dljhfZ]g•lgZ o\bey a qZklhlhx ν	F=p i_j_oh^blv a \Zdmmfm

\ k_j_^h\bs_ a ^•_e_dljbqghx ijhgbdg•klx ε = 4 • fZ]g•lghx ijh- gbdg•klx μ = 1 <bagZqblb af•gm Δλ ^h\`bgb o\be• f)

20.14. JZ^•hehdZlhj \by\b\ m fhj• i•^\h^gbc qh\_g <•^[blbc \•^ qh\gZ kb]gZe jZ^•hehdZlhjZ ih\_jgm\ky ^h gvh]h aZ qZk t fdk.

86

< f.

: f.

>•_e_dljbqgZ ijhgbdg•klv \h^b ε = 81 AgZclb \•^klZgv \•^ ehdZ- lhjZ ^h i•^\h^gh]h qh\gZ f

20.15.IehkdZ _e_dljhfZ]g•lgZ o\bey ihrbjx}lvky \ h^ghj•^ghfm •ah- ljhighfm k_j_^h\bs• a ε = 4 • μ = 1 :fie•lm^Z gZijm`_ghkl•

_e_dljbqgh]h ihey o\be• ?0	< f H[qbkeblb nZah\m r\b^d•klv

o\be• • Zfie•lm^m gZijm`_ghkl• fZ]g•lgh]h ihey o\be• G0.

(1,5 108 f k f: f

20.16.< h^ghj•^ghfm •ahljhighfm k_j_^h\bs• a ε = 9 • μ = 1 ihrb- jx}lvky iehkdZ _e_dljhfZ]g•lgZ o\bey :fie•lm^Z •g^mdp•€ fZ]g•l-

gh]h ihey o\be• <0 = 1 10-3 Le H[qbkeblb nZah\m r\b^d•klv o\be• • Zfie•lm^m gZijm`_ghkl• _e_dljbqgh]h ihey (1 108 f k 5 < f

20.17.1gl_gkb\g•klv iehkdh€ _e_dljhfZ]g•lgh€ o\be• sh ihrbjx}lvky m \Zdmmf• ^hj•\gx} 1 f<l f2 <bagZqblb Zfie•lm^m gZijm`_- ghkl• _e_dljbqgh]h ihey o\be• < f

20.18.IehkdZ _e_dljhfZ]g•lgZ o\bey ? VLQ,28 108t – 4,55x) < f ih- rbjx}lvky \ j_qh\bg• a μ = 1 AgZclb ^•_e_dljbqgm ijhgbdg•klv j_qh\bgb lZ •gl_gkb\g•klv _e_dljhfZ]g•lgh€ o\be• <l f2)

20.19.M k_j_^h\bs• a ε = 3 • μ = 1 ihrbjx}lvky iehkdZ _e_dljhfZ]g•lgZ

o\bey :fie•lm^Z gZijm`_ghkl• fZ]g•lgh]h ihey o\be• G0

<bagZqblb _g_j]•x sh i_j_ghkblvky o\be_x aZ qZk W k q_j_a ih\_jogx iehs_x 6 kf2 sh jhalZrh\ZgZ i_ji_g^bdmeyjgh

^h gZijyfdm ihrbj_ggy o\be• I_j•h^ dheb\Zgv o\be• L << t.

>`

20.20.M k_j_^h\bs• a ε = 4 • μ = 1 ihrbjx}lvky iehkdZ _e_dljhfZ]g•lgZ

o\bey Zfie•lm^Z gZijm`_ghkl• _e_dljbqgh]h ihey ydh€ ?0

GZ reyom o\be• i_ji_g^bdmeyjgh ^h gZijyfdm €€ ihrbj_ggy jhalZrh\ZgZ ih]ebgZevgZ ih\_jogy sh fZ} nhjfm djm]Z jZ^•mkhf R f Ydm _g_j]•x ih]ebgZ} py ih\_jogy aZ qZk W k? I_j•h^

dheb\Zgv o\be• L << t. >`

20.21.M \Zdmmf• ihrbjx}lvky iehkdZ _e_dljhfZ]g•lgZ o\bey a pbde•q- ghx qZklhlhx ω = 1010 jZ^ k :fie•lm^Z gZijm`_ghkl• _e_dljbq-

87

gh]h ihey o\be• ?0 < f GZ reyom o\be• i_ji_g^bdmeyjgh ^h gZijyfdm €€ ihrbj_ggy jhalZrh\ZgZ ih]ebgZevgZ ih\_jogy sh fZ} nhjfm i•\kn_jb a jZ^•mkhf R f ydZ \_jrbghx ih\_j-

gmlZ \ gZijyfdm ihrbj_ggy o\be• Ydm _g_j]•x ih]ebgZ} py ih\_jogy aZ qZk W k? >`

88

VI O<BEVH<: HILBD:

1GL?JN?J?GP1Y K<1LE:

Hkgh\g• nhjfmeb

Hilbqgbc reyo ijhf_gy

L = nS,

^_ S – ]_hf_ljbqgbc reyo ijhf_gy m k_j_^h\bs• a ihdZagbdhf aZehfe_ggy n.

HilbqgZ j•agbpy oh^m ^\ho ijhf_g•\ sh ihrbjxxlvky m j•agbo k_j_^h\bsZo \•^ih\•^gh a ihdZagbdZfb aZehfe_ggy n1 • n2

=n1S1 − n2 S2 .

A\¶yahd f•` j•agbp_x nZa Δϕ k\•leh\bo dheb\Zgv yd• ^h^Zxlvky

lZ hilbqghx j•agbp_x oh^m \•^ih\•^gbo ijhf_g•\

ϕ= 2π ,

λ0

^_ λ0 – ^h\`bgZ k\•leh\bo o\bev m \Zdmmf•

Mfh\Z •gl_jn_j_gp•cgbo fZdkbfmf•\

ϕ= ±2mπ,

= ±mλ0 , m = 0,1,2,...,

^_ m – ghf_j ihjy^hd •gl_jn_j_gp•cgh]h fZdkbfmfm

Mfh\Z •gl_jn_j_gp•cgbo f•g•fmf•\

ϕ= ±(2m +1)π,

= ±(2m +1)λ0 , m = 0,1,2,...,
			2
^_ m – ghf_j ihjy^hd •gl_jn_j_gp•cgh]h f•g•fmfm
JZ^•mkb k\•lebo d•e_pv GvxlhgZ m \•^[blhfm k\•le•
r m =		1
	m −	2	Rλ, m = 1,2, ,

89

^_ R – jZ^•mk kn_jbqgh€ ih\_jog• e•gab λ – ^h\`bgZ k\•leh\bo o\bev m ijhf•`dm f•` e•gahx lZ kdeyghx i•^klZ\dhx m – ghf_j k\•leh]h d•evpy GvxlhgZ

JZ^•mkb l_fgbo d•e_pv GvxlhgZ m \•^[blhfm k\•le• r m = mRλ, m = 1,2, ,

^_ m – ghf_j l_fgh]h d•evpy GvxlhgZ

21.1. M ijbkljh€ Xg‰Z \•^klZgv f•` s•ebgZfb G ff Z \•^klZgv \•^

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