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Методическое пособие по физике 2

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3.2.%2) 7 !!" 8 9 / )-& /"0" 2" .)-!1-2 2 )2"$-1-- 2" '.

") -& /-' *-!-* *

3 (1 * 2 &- ( — 2 ( % ' & ( -

&, & ( , " 0# " & + 1 * + -

& & + ' (", ) #@ + " 0 & (:

E =	F	.	(3.2.1)

= ' (" ( '+ * +).

( (, '" )- & & +) ' (" ) q (1 0-

& :

E = k	q		er ,	(3.2.2)

	r	2

-" er — " & + , + ' (" q " & (

( . 3.4).

A " & '" ( * ) & & +-

) ' (" ), ' * 0# (1 " ( ( $ -

) '$ :

E = Ei ,	(3.2.3)
i

-" Ei — (1 * (, '" )- i–) & & +) ' (" ) " & (3.2.2), . .

dq = σ dS

E = Ei =	1		qi	er .	(3.2.4)
	4πε0		2
i		i ri

A " & '" ( + "-

+) ' (" ), " ( *' ( $ '$ * B-

) V, ' ) )+ 2 ) ' (" ) ' ( & ) + B )+ dV.

1"+ & ) + B ) dV " 1 & ) + -

& & + ' (" dq. 3 (1 * ( " & & + ( ))-

) (1 , '" )+ 1"+) ) ' (" ):

E =	dE =	1			dq	er ,	(3.2.5)
E =	dE =	4πε	0	r 2		er ,	(3.2.5)
		4πε		r 2

-" ' (" dq ) 1 + * " & ' * - -

" (:

dq = ρdV , -" ρ – B@) ( * ' (";

, -" σ – ( * ' (";

dq = τ dL , -" τ – ( * ' (".

)-* )+ ) 3 !- $% %&

)-* ) 3. & & + 2 & ' (" q1 = 10–9 q2 = –2·10–9

( . 3.5) "( ( '" ( d = 10 ) "- "- . -

" * (1 * E (, '" )-

2 ) ' (" ) & , " ' ("				:6
2 ) ' (" ) & , " ' ("
q1 r1 = 9 ) ' (" q2 r2 = 7 ).			C
q1 r1 = 9 ) ' (" q2 r2 = 7 ).
3 !- . "( ' $ -		6		:
3 !- . "( ' $ -				:

'$ (3.2.4), & (1 * (				:7
				7
"+ ( ' (1 (

E1, '" )- ' (" ) q1 (-	6			7
1 E2, '" )- ' (" ) q2.		;<:;9

E = E1 + E2 ,

-" E1	=	q1			E2	=	q2				.	(3.2.6)
		4πε	0	r 2			4πε	0	r	2

				1					2

& ' * 0#- " ( ( ) #*0 )+ -

E = E 2	+ E 2	+ 2E E cosα ,		(3.2.7)
1	2	1	2

-" α — - ) 1" ) E1 E2. , - ) 1 + * " 1

) #*0 )+ , ) -* )

& " ' (":

d 2 = r12 + r22 + 2r1r2 cosα ,

	d 2	− r 2	− r	2
cosα =		1	2		= −0, 238 .	(3.2.8)
cosα =					= −0, 238 .	(3.2.8)
		2r1r2

= *, " (( (3.2.6) (3.2.7) (3.2.8), " ( (1 2 &-

- ( ! & )

								1			q2		q	2		q q
					E =						1	+		2	+ 2	1		2	cosα .
					E =							+			+ 2				cosα .
							4πε		0		r 4		r	4		r	2 r	2
									0		1		2			1	2
! " ) & + ' & (

		1				(1 0 − 9		) 2			( 2 1 0 − 9				) 2				1 0 − 9 2 1 0	− 9
E =										+						+ 2					( − 0 , 2 3 8 ) =
E =			− 1 2					4		+	( 0 , 0 7 )				4	+ 2		2		2	( − 0 , 2 3 8 ) =
	4 π 8 , 8 5 1 0				( 0 , 0 9 )						( 0 , 0 7 )							( 0 , 0 9 ) ( 0 , 0 7 )
3		H				"
= 3 , 5 8 1 0				=	3 , 5 8
= 3 , 5 8 1 0				=	3 , 5 8

% %&-

3.2.01. 3 (1 * 2 &- ( &, 1 # "

) 1" & & +) ' (" ) q1 = 8·10–9 q2 = – 86·10–9 . ( ) 1" ' (" ) r = 10 ).

3.2.02. *- -* ) # 0 ( & & + ' ("+ " & + q. 3 (1 * ( ! $ -

-* , & ' ' (" ".

3.2.03. " + ' (" 10–7 "( ( '" ( 8 ) "- "- . " * (1 * !0 ( &, 1

( 5 ) ' (".

3.2.04. & & + ' (" q1 = 2q q2 = –q "( ( ( d "- "- . 3 1 & (), "(# & ' 2 ' ("+, - (1 * ! ( 0.

3.2.05. , & '" " )( & & +) ' (" ) q1 = 40 q2 = –10 , "(# ) ( ( d = 10 ) "- "- . " *

(1 * ! ( &, " - ' (" r1 = 12 )

- r2 = 6 ).

3.2.06. = *$ " ) R = 8 ) ' (", ) "-

+ *0 τ = 10 /). (1 * ! 2 &-

- ( &, " & *$ ( r = 10 )? ! * -% ' ) (1 ( E ' )

( ( " $ *$ .

3.2.07. ! % ' (", ) " +

*0 σ = 1 /)2. 3 (1 * ! 2 &- ( $ -

% +.

3.2.08. ! () " " L = 1 ) ' (", ) " +

- ". 3 & 2- ' (", (1 * E (

( = 0,5 ), 1 ) " * " ( " "- ' - $ , 200 /).

3.2.09.! () ( " L = 1 ) ' (" q = 40 ), )

" + ". 3 & (1 ( E &, 1 " 1 ( d = 1,5 )

$ .

3.2.10.! " " L = 40 ) - ) " ()+) -). ! - " ) " ' (" q = 10 ). " * (1 * (

E " &, 1 + -' &-( - - " - , ( d = 10 ) + 2- - .

3.3. ") *% % %.

") -& /-' *-!-* *

( 1" ( (1 2 &- ( " & + & ( " *' * ( ) .,

% ) ' & " 0# ) ' ): (1-

2 &- ( '* ' * 0 ' ) 0 *

)) 2 & ' (" 2 " 2 &-

0 ( 0 ε0.

	1	qi ,
(EdS) =		qi ,	(3.3.1)
s	ε0 i

-" dS = dS·n. ! 2 ) dS — 2 ) - , n —

) 2 ) dS, (EdS) = E·dS·cosα — E

'* 2 ) dS, α — - ) 1" ) E dS, qi —

)) ' (", "(# ( 2 . A ' ("

, 0. ( & ( +- "-

( ' (" ) 1 ' *:

qi	= ρdV , -" ρ – B@) ( * ' (";
i	V

qi = σ dS , -" σ – ( * ' (";

qi = τ dL , -" τ – ( * ' (".

-& (, "+ )+ *' )+ ., "-

0 ( 1" - ') 1 *0 +& ( - & -

(3.3.1). , ' ' & *- " ( ' (", '" 0#-

" & : " ( (- " ( ) 1 " *

*, '* 0 ) 1 " * & ) ".

3 (1 & + ( " ( ))-

& + " ' (" — % &, $" & .

)-* )+ ) 3 !- $% %&
)-* ) 4. 3 (1 * 2-
&- ( "	R,
' (1- B ) *0 ρ ( .
3.6). ! * -% ' ) (-
1 ( ! ( ( " $	-
.

3 !- . ! " * " ) (1 * ( !1 –

( r1 $ (1 * ( !2 – 1 -

( r2 $ .

+ ) % ) . . 0 & + )

" ( # ( +& ( - & (3.3.1)

" 1 + 0" * ( " 0# (:

. E = const 0 & " ( &, & '

+ 0;

. E dS E dS . +) ), - ( ) ' "

(3.3.1) " 1 + * α = 0°, α = 180°.

' 1 )) (, & * . " ) &

" + * " % + " r, $& ) R. ! 2 )

0 & + (0 ( ' 1 + + (: E = const

E dS 0 & ). . 3.6.

) " * 0 0 & )+ . (3.3.1) " ( -

1 # ' (1- :

1.		(E dS) =		E dS cos 0 = E			dS = E 4π r 2
		1		1	1		1 1
	S1		S1			S1
					16

1	ρdV =	1	ρ dV =	1	ρ	4	π r13 .	(3.3.3)

ε0 V		ε0 V		ε0 3
1		1

! ( ) & + (3.3.2) (3.3.3) ' *:


E 4π r 2		=	1		ρ		4	π r3	,
E 4π r 2		=			ρ			π r3	,
1	1		ε0				3	1
			ε0				3
" # ( ": E =				ρr1		. ", & (1 *
" # ( ": E =						. ", & (1 *
	1			3ε0
				3ε0

) ' (1- ' $ % .

2.		(E	dS) =		E dS cos 0 = E		dS = E	4π r 2
		2			2	2	2	2
	S2			S2		S2

ρdV =

ρ dV =

π R3 .

ε0 V

ε0 3

-& .1 (

(3.3.4)

(3.3.5), & ): E =

ρ R3

3ε

r 2

. % & ' ) * A(r) "-

. 3.7.

(3.3.5)

% %&-

3.3.01.*' ( ) ., (1 * (, '"- '-

(1 & *0. ( * ' (" λ .

3.3.02.3 (1 * ( % &- ", ' (-

1- ' (" ) q. " " r, — R.

3.3.03.3 (1 * ( % + " R, ' (1-

*0 σ . ! * -% ' ) ! -

( ( " $ % +.

3.3.04.3 (1 * 2 &- ( "

R, ' (1- B ) *0 ρ = ρ0(1 – r/R), -" r — (

$ , ρ0 — * ' (" $ . ! * -% '-

) ! ( ( " $ % +.

3.3.05.3 (1 * (, '"- & , ' (-

1 *0 σ .

3.3.06. 3 (1 * ( - ", +

- & + ' (1 + *0 +σ .

3.3.07.3 (1 * ( - &- $ -

" " R, ' (1- B@) *0 ρ . ! * -%

' ) ! ( ( " $".

3.3.08.3 (1 * ( - &- $ -

" " R, ' (1- *0 σ . ! * -% ' ) ! ( ( " $".

3.3.09. " + + * + " ) R1 = 2 )R2 = 4 ) ' ("+, ) " + " +)

() τ1 = l /) τ2 = –0,5 /). " * (1 * ! (

&, "(# ( ( ( r1 = 1 ), r2 = 3 ), r3 = 5 ) .

! * -% ' ) ! r.

3.3.10. * ( (, # d = 1 ) ' (", )

" +: B ) B ) *0 ρ =100 /)3. 3 -

(1 * E 2 &- (: ' $* & +

, ) ) ( .

3.4." !1-% . %," % 2" .

") -& /-' *-!-* *

,-& 2 &- ( ( ( ( $ -

:
	ϕ =		W			,			(3.4.1)

				q
'" * W – 2-(, " & ( " ' (" q.
!$ ( )+ & & + ' (" qi:
ϕ =	1						qi	.	(3.4.2)
		4πε0
					i		ri

2 &- ( 0 ' (" q ' & 1 &

2:
A12 = q (ϕ1 − ϕ2 ) = qU ,	(3.4.3)

'" * U = ϕ1 − ϕ2 – ' * $ .

" ' " ( (- 2 &- (

* - $*. !$* * 2 &- -

( ' 0& ( ), & , ' " ( ) " ' (" ) ' )-

) , 0:
E dl = 0 .	(3.4.5)
L

" * = E dl — $($( (1 2 &- (

' ) ) . " * ) " ) '+ * -

+) ) " ( 2 &- ( ( — )

.).

('* ) 1" (1 *0 $) 2 &- ( " @ (

E = −gradϕ ,

(3.4.6)

			∂ϕ			∂ϕ			∂ϕ
		gradϕ =				;			;	.	(3.4.7)
			∂			∂			∂z
	A ( r ' (" q (1 * !, ' * -
$ ( ) 1 " 0# ) ' ):
					2
		ϕ − ϕ	2	=		!	(	r dr .			(3.4.8)
		1	2				(		)
					1
)-* )+ ) 3 !- $% %&
	)-* ) 5. $ " R ' (1 ' (" ) q. " $
& $ " ( - . " -
(1 * ( & 2 . 3 & , -" (1-
* ) ) *.
	3 !- . "( ' $ '$ ' * ) * ' (-
1 + - " ) *
', 1"+ ' + " -
) + ' (" dq. ' -
1 )) ( ( . 3.8), &
(1 * (, '" ) ( ' (-
" )	dq, 1 # ) $ -
- " ), & *$
" * . (0-
# dE ' ) & 1 0 (.
	!$ & &- ' (" dq,
"( ' (3.4.2), ) 1 + * ':
dϕ =	dq	, -" r = R2 + x2 ,
	4πε0r
-" $ - ' (1-
" " ( ( ":

ϕ =

4πε0

R2 + x2

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