Фізика, збірник задач

13.14. >\• dhgp_gljbqg• ijh\•^g• kn_jbqg• ih\_jog• sh i_j_[m\Zxlv m \Zdmmf• aZjy^`_g• a ih\_jog_\hx ]mklbghx 1 fdDe f2 JZ-

^•mkb pbo ih\_johgv R1 f • R2	f <bagZqblb _g_j]•x
W _e_dljbqgh]h ihey f•` pbfb kn_jZfb f>`
13.15. Kmp•evgZ _[hg•lh\Z dmey jZ^•mkhf R	kf j•\ghf•jgh aZjy^`_gZ
aZjy^hf a h[¶}fghx ]mklbghx !	fdDe f3 >•_e_dljbqgZ

ijhgbdg•klv _[hg•lm ε = 3,0 H[qbkeblb _g_j]•x W1 _e_dljbqgh]h ihey ydZ ahk_j_^`_gZ \ kZf•c dme• • _g_j]•x W2 ihaZ g_x

fd>` fd>`

61

IV IHKL1CGBC KLJMF

14 HKGH<G1 A:DHGB IHKL1CGH=H KLJMFM

Hkgh\g• nhjfmeb

KbeZ \_ebqbgZ ihkl•cgh]h kljmfm

I = qt ,

^_ q – \_ebqbgZ aZjy^m ydbc i_j_ghkblvky kljmfhf q_j_a ^Zgbc i_j_j•aih\_jogx aZ ijhf•`hd qZkm t.

=mklbgZ _e_dljbqgh]h kljmfm j = SI ,

^_ S – iehsZ ihi_j_qgh]h i_j_j•am ijh\•^gbdZHi•j h^ghj•^gh]h ijh\•^gbdZ

R = ρ S ,

^_ ρ – iblhfbc hi•j fZl_j•Zem ijh\•^gbdZ • – ch]h ^h\`bgZ S – iehsZ ihi_j_qgh]h i_j_j•am ijh\•^gbdZ

4 ?e_dljhjmr•cgZ kbeZ

ε = Aq* ,

^_ : – jh[hlZ ydZ \bdhgZgZ klhjhgg•fb kbeZfb i•^ qZk i_j_f•s_ggy gZ ^Zg•c ^•eygp• m aZfdgmlhfm dhe• aZjy^m q.

AZdhg HfZ

^ey h^ghj•^gh€ ^•eygdb dheZ

, = ϕ −ϕ = 8 5 5

62

^ey g_h^ghj•^gh€ ^•eygdb dheZ

I = (ϕ1 −ϕ2 ) +ε12 = U ,

R R

^_ ε12 – ?JK sh ^•} gZ ^Zg•c g_h^ghj•^g•c ^•eygp• _e_dljbqgh]h dheZ

AZdhg HfZ ^ey aZfdgmlh]h _e_dljbqgh]h dheZ

ε, =

5 + U

^_ R – hi•j ah\g•rgvh€ qZklbgb dheZ r – \gmlj•rg•c hi•j ^`_j_- eZ kljmfm

Hi•j ihke•^h\gh a’}^gZgbo j_abklhj•\

R= R1 + R2 + .

Hi•j iZjZe_evgh a’}^gZgbo j_abklhj•\

1 = 1 + 1 + . R R1 R2

Jh[hlZ kbe _e_dljbqgh]h ihey gZ ^•eygp• dheZ ihkl•cgh]h kljm- fm aZ qZk t

A = IUt = I 2 Rt = U 2	t.
R

Ihlm`g•klv _e_dljbqgh]h kljmfm

P = IU = I 2 R = U 2 . R

Ih\gZ ihlm`g•klv ydZ \b^•ey}lvky \ aZfdgmlhfm dhe• ihkl•c-

gh]h kljmfm

3 = ε ,

I_jrbc aZdhg ijZ\beh D•jo]hnZ

Ze]_[jZ€qgZ kmfZ kljmf•\ yd• koh^ylvky m dh`ghfm \mae• jha]Zem`_gh]h _e_dljbqgh]h dheZ ^hj•\gx} gmex

n

∑ Ik = 0.

k =1

Kljmfb yd• \oh^ylv m \mahe \\Z`Zxlvky ^h^Zlgbfb Z yd• \boh^ylv a \maeZ – \•^¶}fgbfb Z[h gZ\iZdb

>jm]bc aZdhg ijZ\beh D•jo]hnZ

Ze]_[jZ€qgZ kmfZ kiZ^•\ gZijm] gZ \k•o ^•eygdZo aZfdgmlh]h dhg-

63

lmjm ^hj•\gx} Ze]_[jZ€qg•c kmf• ?JK yd• ^•xlv m pvhfm dhglmj•

∑n Ik Rk = ∑n εk . k =1 k =1

Ydsh gZijyf kljmfm a[•]Z}lvky a \b[jZgbf gZijyfhf h[oh^m dhglmjm lh \•^ih\•^gbc ^h[mlhd kbeb kljmfm gZ hi•j \oh^blv \ j•\gyggy •a agZdhf m ijhlbe_`ghfm \biZ^dm p_c ^h[mlhd \oh^blv •a agZdhf – ?JK [_jmlv •a agZdhf ydsh ijb

h[oh^• dhglmjm m \b[jZghfm gZijyfdm i_jrbc _e_dljh^ [m^_ g_]Zlb\gbf Z ^jm]bc – ihablb\gbf

AZdhg >`hmey – E_gpZ

Q = I 2 R t,

Z[h

t

Q = ∫ I 2 (t)Rd t,

0

^_ I(t) – fbll}\_ agZq_ggy kbeb kljmfm yd nmgdp•€ qZkm

14.1. GZijm]Z gZ d•gpyo ijh\•^gbdZ j•\ghf•jgh a[•evrm}lvky aZ qZk 2k \•^ U1 < ^h U2 < Hi•j ijh\•^gbdZ R Hf.

AgZclb aZjy^ q ydbc ijhcrh\ ih ijh\•^gbdm aZ qZk 2. De

14.2. GZ	h^ghfm d•gp• pbe•g^jbqgh]h aZe•agh]h kljb`gy hi•j ydh]h
R0	Hf ijb t = 0 hK i•^ljbfm}lvky l_fi_jZlmjZ t1 = 27 hK gZ
^jm]hfm t2 =533 hK L_fi_jZlmjgbc dh_n•p•}gl hihjm ^ey aZe•aZ

. = 6 10-3 ]jZ^-1 A [hdh\h€ ih\_jog• kljb`gy l_iehlZ g_ \•^\h- ^blvky <bagZqblb hi•j R ijh\•^gbdZ \\Z`Zxqb ]jZ^•}gl l_fi_-

jZlmjb \a^h\` ch]h hk• klZebf Hf

14.3.	=mklbgZ _e_dljbqgh]h kljmfm \ f•^ghfm ijh\•^gbdm j g: f2.
	Iblhfbc hi•j f•^• ! gHf f JhajZom\Zlb ]mklbgm l_ieh\h€
	ihlm`ghkl• kljmfm w. >` f3 k
14.4.	GZ d•gpyo ijh\•^gbdZ iblhfbc hi•j ydh]h !			fdHf f • ^h\`bgZ
	•	f i•^ljbfm}lvky gZijm]Z U		< YdZ ihlm`g•klv w

\b^•ey}lvky \ h^bgbp• h[¶}fm ijh\•^gbdZ" F<l f3)

64

<,

Hf,

Hf,

14.5. M _e_dljbqghfm dhe• yd_ ah[jZ-

`_g_ gZ jbkmgdm R1	Hf,	1
R2 Hf, 1 <, 2	<.

G_olmxqb \gmlj•rg•fb hihjZfb ^`_j_e \bagZqblb j•agbpx ih-

l_gp•Ze•\ 31 – 32. < )

14.6. M _e_dljbqghfm dhe• yd_ ah[jZ- `_g_ gZ jbkmgdm R1 = R2

R3 Hf, K	gN AZjy^
gZ dhg^_gkZlhj• q	fdDe.

G_olmxqb \gmlj•rg•f hihjhf ^`_j_eZ H[qbkeblb ch]h ?JK

. <

14.7. >\Z	^`_j_eZ kljmfm a ?JK
1	< • 2 < lZ \gml-

j•rg•fb hihjZfb r1 Hf • r2 Hf i•^¶}^gZg• iZjZe_ev- gh ^h hihjm R Hf. <bagZqb-

lb kbeb kljmf•\ sh ijhl•dZxlv q_j_a ^`_j_eZ • hi•j R. :

: :

14.8.AgZclb kbem kljmfm ydbc ijhoh-

^blv q_j_a hihjb R1 = R4

R3 = R2 Hf m\•fdg_g• \ dheh yd ihdZaZgh gZ jbkmgdm ydsh 1 =

2 < <gmlj•rg•fb

hihjZfb ^`_j_e kljmfm fh`gZ ag_olm\Zlb : : :

14.9. <bagZqblb hi•j R f•` lhqdZfb : • A < ^ey dheZ yd_ ah[jZ`_g_ gZ jb- kmgdm Hihjb hdj_fbo ]•ehd ih- dZaZgh gZ jbkmgdm r Hf).

Hf

R1 1

2

2 R2

C

R1 R3 R2

1 r1

2 r2

R

R1 1

R2 2

R3

R4

2r			r	r	B
r
				2r

65

14.10. ;ZlZj_y •a N = 400

R1

1

	_e_f_gl•\ m\•fdg_gZ \ _e_dljbqg_ dheh •a
ah\g•rg•f hihjhf R	Hf	?JK dh`gh]h _e_f_glZ	<,
\gmlj•rg•c hi•j r	Hf	Lj_[Z kdeZklb [ZlZj_x •a	lZdh€

d•evdhkl• n1 iZjZe_evgbo ]jmi dh`gZ a ydbo f•klbeZ [ n2 ihke•-

^h\gh a¶}^gZgbo _e_f_gl•\ sh[ hljbfZlb fZdkbfZevgm kbem kljmfm <bagZqblb d•evdhkl• n1 • n2 kbem kljmfm 1 \ hihj• R • \ dh`ghfm _e_f_gl• 11. : :

14.11. M ko_f• ydZ ah[jZ`_gZ gZ jbkmgdm 1 =

R3 R2

2

<, 2 <, R1 Hf, R2 Hf, Z

\gmlj•rg•fb hihjZfb ^`_j_e fh`gZ ag_olm\Zlb <bagZqblb jh[hlm :1 • :2, \b-

dhgZgm ^`_j_eZfb ih\gm d•evd•klv l_iehlb Q sh \b^•ey}lvky \ dhe• aZ qZk ¨t k

ydsh R3	Hf Ijb ydhfm hihj• R3 ihlm`g•klv				sh \b^•ey}lvky
gZ gvhfm [m^_ fZdkbfZevghx" (– >` >` >` Hf
14.12. n = 6 ^`_j_e a ?JK		< • \gmlj•rg•f hihjhf r			Hf a¶}^-
gZg• ihke•^h\gh Ydbc hi•j R lj_[Z i•^¶}^gZlb ^h [ZlZj_€ sh[
dhjbkgZ ihlm`g•klv J [meZ fZdkbfZevghx • qhfm ^hj•\gx} JfZo?
Hf <l
14.13. n = 6 ^`_j_e a ?JK		< • \gmlj•rg•f hihjhf r				Hf
a¶}^gZg• iZjZe_evgh Ydbc hi•j R lj_[Z i•^¶}^gZlb ^h [ZlZj_€ sh[
dhjbkgZ ihlm`g•klv J [meZ fZdkbfZevghx • qhfm ^hj•\gx} JfZo?
Hf <l
14.14. M ijh\•^gbdm hi•j ydh]h R			Hf	aZ qZk 2	k j•\ghf•jgh
ajhkeZ kbeZ kljmfm \•^		11	: ^h 12	: <bagZqblb d•evd•klv
l_iehlb Q ydZ \b^•ebeZky m ijh\•^gbdm aZ qZk 2. >`
14.15. KbeZ kljmfm \ ijh\•^gbdm a qZkhf af•gx}lvky aZ aZdhghf 1						10 _- t ^_
10 : β	k-1 hi•j ijh\•^gbdZ R			Hf AgZclb d•evd•klv

l_iehlb Q sh \b^•ey}lvky \ ijh\•^gbdm aZ qZk 2 k. >`

66

V ?E?DLJHF:=G?LBAF

F:=G1LG? IHE? KLJMFM M <:DMMF1 Hkgh\g• nhjfmeb

AZdhg ;•h – KZ\ZjZ – EZieZkZ

				μ0
							I d			r
	dB =										;
				4π				r3
	μ0		I sin
dB =							Â	d ;		B = ∫dB.
	4π			r	2
											L

^_ dB – \_ebqbgZ •g^mdp•€ fZ]g•lgh]h ihey fZ]g•lgh€ •g^mdp•€ kl\h-

j_gh]h _e_f_glhf	d•	ijh\•^gbdZ	a	kljmfhf I; μ0 – fZ]g•lgZ klZeZ
(μ0 = Œ˜10-7 =g f	U
		– jZ^•mk-\_dlhj ijh\_^_gbc \•^ _e_f_glZ			d ijh-
\•^gbdZ ^h lhqdb ^_ \bagZqZ}lvky dB ; Â – dml f•` \_dlhjZfb d • U .
A\¶yahd f•` fZ]g•lghx •g^mdp•}x B • gZijm`_g•klx fZ]g•lgh]h
ihey H m \Zdmmf•
			B	= μ0 H.
Ijbgpbi kmi_jihabp•€ fZ]g•lgbo ihe•\
		B =	B1	+ B2 + ,
		H =	H1 + H 2 + .

1g^mdp•y fZ]g•lgh]h ihey kl\hj_gh]h kljmfhf sh ijhl•dZ} ih g_kd•gq_ggh ^h\]hfm ijyfhfm ijh\•^gbdm

B = μ0 2πIR ,

^_ R – \•^klZgv \•^ ijh\•^gbdZ ^h lhqdb \ yd•c \bagZqZ}lvky <.

67

B = μ0

				1g^mdp•y fZ]g•lgh]h ihey kl\hj_-
	Â2			gh]h kljmfhf sh ijhl•dZ} m ijyfhfm ijh-
				\•^gbdm kd•gq_ggh€ ^h\`bgb
				B =	μ0		I	(cos 1 −cos 2 ) .
I	R		B		4π R
				1g^mdp•y fZ]g•lgh]h ihey \k_j_^bg•
				g_kd•gq_ggh ^h\]h]h khe_gh€^Z
	Â1						B = μ0 nI,

^_ n – d•evd•klv \bld•\ khe_gh€^Z gZ h^bgbp• ch]h ^h\`bgb

1g^mdp•y fZ]g•lgh]h ihey m p_glj• dheh\h]h kljmfm

2IR ,

^_ R – jZ^•mk dheh\h]h kljmfm

1g^mdp•y fZ]g•lgh]h ihey gZ hk• dheh\h]h kljmfm

B =	μ0 2IπR2	,
	4π (R2 + d 2 )3 2

^_ d – \•^klZgv \•^ p_gljZ dheh\h]h kljmfm ^h aZ^Zgh€ lhqdb gZ ch]h hk•1g^mdp•y fZ]g•lgh]h ihey \k_j_^bg• lhjh€^Z

B = μ0 In Rr ,

^_ R – jZ^•mk hkgh\gh€ e•g•€ lhjh€^Z r – \•^klZgv \•^ p_gljZ lhjh€^Z ^h aZ^Zgh€ lhqdb n – d•evd•klv \bld•\ gZ h^bgbpx ^h\`bgb hkgh\gh€ e•g•€ lhjh€^Z

15.1.M ijyfhe•g•cghfm g_kd•gq_ggh ^h\]hfm ijh\•^gbdm l_q_ kljmf kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < m lhqp• sh jhaf•- s_gZ gZ \•^klZg• R f \•^ ijh\•^gbdZ fdLe

15.2.M ^\ho g_kd•gq_ggh ^h\]bo ijyfhe•g•cgbo iZjZe_evgbo ijh\•^gbdZo \

h^ghfm gZijyfdm l_qmlv kljmfb kbeZfb 11 : • 12 : <•^klZgv f•` ijh\•^gbdZfb d f <bagZqblb fZ]g•lgm •g^mdp•x < m lhqp•

sh jhaf•s_gZ ihk_j_^bg• f•` ijh\•^gbdZfb fdLe

68

15.3. >\Z g_kd•gq_ggh ^h\]• ijyfhe•g•cg• iZjZe_evg• ijh\•^gbdb \ ydbo

ijhl•dZxlv m ijhlbe_`gbo gZijyfdZo kljmfb kbeZfb 11 : • 12 = = : agZoh^ylvky gZ \•^klZg• d f AgZclb fZ]g•lgm •g^md-

p•x < m lhqp• sh jhaf•s_gZ gZ \•^klZg• R f \•^ i_jrh]h ijh\•^gbdZ gZ ijh^h\`_gg• \•^j•adZ ijyfh€ sh a¶}^gm} ijh\•^- gbdb fdLe

15.4. >\Z iZjZe_evg• g_kd•gq_ggh ^h\]• ijyfhe•g•cg• ijh\•^gbdb \ ydbo

ijhl•dZxlv \ h^ghfm gZijyfdm kljmfb 11 12 : jhaf•s_g• gZ \•^klZg• d f h^bg \•^ h^gh]h. AgZclb fZ]g•lgm •g^mdp•x < m

lhqp• sh jhaf•s_gZ gZ \•^klZg• R1 f \•^ h^gh]h ijh\•^gbdZ

• gZ R2 f \•^ ^jm]h]h fdLe

15.5. >\Z g_kd•gq_ggh ^h\]• ijyfhe•g•cg• ijh\•^gbdb koj_s_g• i•^ ijyfbf

dmlhf M ijh\•^gbdZo l_qmlv kljmfb kbeZfb 11 : • 12 :. <•^klZgv f•` ijh\•^gbdZfb d f AgZclb fZ]g•lgm •g^mdp•x < m

lhqp• sh h^gZdh\h \•^^Ze_gZ \•^ h[ho ijh\•^gbd•\ fdLe

15.6. M \•^j•adm ijyfhe•g•cgh]h ijh\•^gbdZ aZ\^h\`db L f ijhoh-

^blv kljmf kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < m lhqp• sh e_`blv gZ i_ji_g^bdmeyj• ^h k_j_^bgb \•^j•adZ gZ \•^klZg•

R1 f \•^ gvh]h (2 fdLe

15.7.1a ^jhlm ^h\`bgZ ydh]h l f ajh[e_gh d\Z^jZlgm jZfdm Ih

g•c ijhoh^blv kljmf kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < m p_glj• p•}€ jZfdb fdLe

15.8.M ijh\•^gbdm a•]gmlhfm m \b]ey^• d\Z^jZlgh€ jZfdb ^h\`bgZ

klhjhgb ydh€ Z f ijhl•dZ} kljmf kbehx 1 : H[qbkeblb fZ]g•lgm •g^mdp•x < ihey \ lhqp• ydZ j•\gh\•^^Ze_gZ \•^ \_jrbg d\Z^jZlZ gZ \•^klZgv sh ^hj•\gx} ^h\`bg• ch]h klhjhgb fdLe

15.9. M	ijh\•^gbdm a•]gmlhfm m \b]ey^• ijyfhdmlgbdZ a• klhjhgZfb
Z	f • \ f l_q_ kljmf kbehx 1 : <bagZqblb

fZ]g•lgm •g^mdp•x < ihey \ lhqp• i_j_lbgm ^•Z]hgZe_c ijyfhdml- gbdZ fLe

15.10.1a ^jhlm ^h\`bgZ ydh]h • f \b]hlh\e_gh jZfdm m \b]ey^• jhf[Z a ]hkljbf dmlhf 3 0 Ih p•c jZfp• ijhoh^blv kljmf kbehx 1 : AgZclb fZ]g•lgm •g^mdp•x < m p_glj• jhf[Z fLe

69

: l_q_ ih g_kd•gq_ggh ^h\]hfm ijh\•^-

15.11. Kljmf kbeZ ydh]h 11

gbdm sh a•]gmlbc i•^ ijyfbf dmlhf <bagZqblb fZ]g•lgm •g^mdp•x < ihey gZ \•^klZg• d f \•^ \_jrbgb dmlZ \ lhqp• gZ

ijh^h\`_gg• h^g•}€ a• klhj•g fdLe

15.12.G_kd•gq_ggh ^h\]bc ijh\•^gbd ih ydhfm ijhl•dZ} kljmf kbehx 11 =

: a•]gmlbc i•^ ijyfbf dmlhf <bagZqblb fZ]g•lgm •g^mdp•x <

ihey gZ \•^klZg• d f \•^ \_jrbgb dmlZ \ lhqp• sh e_`blv gZ [•k_dljbk• ijyfh]h dmlZ ( fdLe)

15.13. Ih ijh\•^gbdm m \b]ey^• j•\ghklhjhggvh]h ljbdmlgbdZ a ^h\`b-

ghx klhjhgb Z f ijhl•dZ} kljmf kbehx 1 : AgZclb fZ]g•lgm •g^mdp•x < ihey \ lhqp• i_j_lbgm \bkhl fdLe

15.14. Ih lhgdhfm ijh\•^gbdm a•]gmlhfm m \b]ey^• ijZ\bevgh]h r_klb-

dmlgbdZ a ^h\`bghx klhjhgb Z kf l_q_ kljmf kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < ihey \ p_glj• r_klbdmlgbdZ

fdLe

15.15.H[qbkeblb fZ]g•lgm •g^mdp•x < ihey \ p_glj• dheh\h]h ijh\•^-

gbdZ jZ^•mkhf R f ih ydhfm l_q_ kljmf kbehx 1 :

fdLe

15.16. Ih dheh\hfm \bldm jZ^•mk ydh]h R f ijhoh^blv kljmf

kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < ihey m lhqp• sh e_`blv gZ i_ji_g^bdmeyj• ijh\_^_ghfm a p_gljZ \bldZ ^h ch]h iehsbgb gZ \•^klZg• d f \•^ p_gljZ fdLe

15.17. Ih dheh\hfm \bldm jZ^•mk ydh]h R f ijhoh^blv kljmf

kbehx 1 : <bagZqblb fZ]g•lgm •g^mdp•x < ihey m lhqp• sh j•\gh\•^^Ze_gZ \•^ mk•o lhqhd \bldZ gZ \•^klZgv r f. fd Le

15.18. M p_glj• dheh\h]h ^jhlygh]h \bldZ \bgbdZ} fZ]g•lg_ ihe_ a •g^mdp•}x < ijb j•agbp• ihl_gp•Ze•\ U1 < gZ d•gpyo \bldZ Ydm gZijm]m U2 ihlj•[gh ijbdeZklb sh[ hljbfZlb lZdm kZfm fZ]g•lgm •g^mdp•x ihey \ p_glj• \bldZ \^\•q• [•evrh]h jZ^•mkZ \b]hlh\e_gh]h a lZdh]h kZfh]h ^jhlm" <

15.19. M khe_gh€^• sh fZ} n = 1500 \bld•\ gZ f ^h\`bgb ijhoh^blv kljmf kbehx 1 : >h\`bgZ khe_gh€^Z L f ^•Zf_lj D f. <bagZqblb fZ]g•lgm •g^mdp•x < ihey gZ hk• khe_gh€^Z fdLe

70