: ( II )

: ( II )

1. . :

, ( ).

. () () .

I- : . .

II- ( ):

.

F12 = k*|q1q2|/r122

F12 ;

k = 1/(4pe0e); e ³ 1;

e - ;

e0 = 8,85*10-12 /;

e0 =1/(4p*9*109).

N, ,

F = åF1i, i = 1 ¸ N.

2. :

, , E = F / q.

, F.

. :

= (1/4pe0)*(q/r2), q , .

, , , , , .

/.

: , .

3. :

I- : . .

II- ( ):

.

F12 = k*|q1q2|/r122

F12 ;

k = 1/(4pe0e); e ³1;

e - ;

e0 = 8,85*10-12 /;

e0 =1/(4p*9*109).

8. :

. , .

, - , , . (1)

, .

, (2).

, r, (4pr2). (1), = (1/4pe0)*(q/r2), -  (1/4pe0)*(q/r2)* (4pr2) = q/e0. , , , (2), , , , .

5. :

+q q, l , . , , .

, r+ = r a cos u, ࠠ r- = r + a cos u.

Er Eu:

Er = 1/(4pe0)*(2p.cosu)/r3;

Eu = 1/(4pe0)*(p.sinu)/r3, p = q.l , . .

E2 = Er2 + Eu2 Þ E = 1/(4pe0)*p/r3* *Ö(1+3.cos2u).

, u = p/2, , :

E^ = 1/(4pe0)*p/r3, Er = 0, E^ .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dE1

 
6. :

dE

 


t

 

X

 

L

 

R

 

dr


dq = tdl

 

dE = k*(tdl)/L2

dE1 = dE.cosa = dE(x/4) = =k*t*(x.dl)/(R2+x2)3/2 2pR

E1 = òdE1 = k*t*(x.dl)/(R2+x2)3/2 0òdl = = (2pRtkx)/(R2+x2)3/2 = =k*(Q.x)/ (R2+x2)3/2.

dE1

 
7. , , :

 

 

 

 

 

 

 

 

 

 

 

dr

X

 

L

 

R

 

dq = tdl

 

g

 

dE

 

 

 

 

 

 

 


g -

dQ = gdS = g2prdr

dE1 = k*(dQx)/(r2+x2)3/2 = =kg2p*(xrdr)/(r2+x2)3/2

E1 = kg2px*0òRrdr/(r2+x2)3/2 = =-kg2px(r2+x2)-1/20ôR = =kg2px(1/x1/Ö(R2+x2)) = kg2p(1 x/Ö( R2+x2)).

x<<R, E1 = kg2p .

E = 2pg/(4pe0) = g/(2e0).

9. :

] $ .

= SòdS S ( ).

a - S.

+ , a - , -, a - .

n , .

= SòEdS = /E S / = =SòEndS.

, ,

= ò EdS = ò (q0/(4pr2e0))dS.

. , = (q0/(e04pr2)).òdS = =q0/e0.

, , = q0/e0 , .. , , .

, , , , .. = q0/e0.

10. , .

:

= òndS, En = E1 + E2 + E3 + + = SEni, i = 1 ¸ N.

= oòSEnidS = Sò EnidS = S(qi/e0) = = (Sqi)/e0, i = 1 ¸ N.

( -) : () , , e0.

] r, q = VòrdV. = oòEdS = /E S / = 1/(e0e)*VòrdV, V , , .

e - - , (e = 1 / ).

:

- .

- , , .

= e0eE / /;

= SòdS = / S / = =VòrdV - .

11. :

g.

n

E

E E

E E

, . . oò EndS = (åq)/e0.

, .. . a .

oò EndS = S..ò EndS + S.ò EndS = = /a.. = 900/ = S.ò EndS = E Sò dS = = E 2S = / - / = (1/e0).g.S.

= g/(2e0).

12. :

å=g/e0

 

+g

 

-g

 

å=0

 

å=0

 


-

 

-

 

-

 


+

 

+

 

+

 


, Eå = g/e0.

13. :

E=0

 

l

 

t

 

R

 
R t ( ).

r

 


q .

q = l.t q = g.2pR.l

E = t/(2pe0r)

E

Er

~1/r

r

R

R

 

r

 

E=0

 

l

 

r

 
r.

n

E

= E S..òdS = E2prl

q = rV = rpR2l = 1/e0 rpR2l

E = (rR2)/(e02r).


r

l

R


q = rpr2l

= E2prl = (1/e0) rpr2l

E = (rr)/(2e0)

e1 e2, e0*e1(2)

E

1


2

3

r

1 - e1 > e2;

2 - e1 = e2;

3 - e1 < e2.

14. ():

g R:

g

 



|E| - const;

= SoòEndS = E oòdS = E 4pr2 = = (1/e0) g4pR2

q = g 4pR2

E = (gR2)/(e0r2) = q/(4pe0r2)

E = 0

E

Er

~1/r

r

R

g R:

= 4pr2 = (r/e0) 4/3 pR3

q = rV = r 4/3 pR3

E = (gR2)/(e0r2) = q/(4pe0r2)

E = (rr)/(3e0e1)

E


1

Er

2

r

R

r(r):

E = q/(4pe0e2r2)

dq = r(r) 4pr dr

r ;

q = 0òRr(r) 4pr2 dr

E = (4p 0òRr(r) 4pr2 dr)/ /(4pe0e2r2); r

E = (4p 0òr(r) 4pr2 dr)/ /(4pe0e1r2);

:

E = (4p R1òR2r(r) 4pr2 dr)/ /(4pe0e2r2); r

E = (4p R1òr(r) 4pr2 dr)/ /(4pe0e1r2).

 

15. (j):

]$ , q. ]$ q, :

F = 1/(4pe0)*(qq)/r2

, q ,

A12 = 1ò2 F(r)dr = (qq)/(4pe0)r1òr2dr/r2.

, :

A12 = Wp1 Wp2.

, Wp = 1/(4pe0)*(qq)/r.

qВ qВ. , Wp Wp, Wp/q .

j = Wp/q = 1/(4pe0)*q/r , , .

]$ , N . , q, , qN q :

A = i = 1åNAi, Ai = = 1/(4pe0)*(qiq/ri1 - qiq/ri2), ri1 - qi q, ri2 qi q.

Wp q :

Wp = 1/(4pe0)*i = 1åN(qiq)/ri ,

j = 1/(4pe0)*i = 1åN(qi/ri), , , , .

q, j

Wp = qj,

A12 = Wp1 Wp2 = q(j1 - j2).

j ,  A¥ = qj, j , .

16. :

j.

, :

F = qE, Wp = qj.

,

E = - j/x - j/y - j/z, .. :

Ex = -j/x, Ey = -j/y, EZ = -j/z, l: l = = -j/l, .. l.

j = 1/(4pe0)*q/r = / / = 1/(4pe0)*q/Ö(x2+y2+z2).

:

j/x = -q/(4pe0)*x/r3;

j/y = -q/(4pe0)*y/r3;

j/z = -q/(4pe0)*z/r3.

:

E = 1/(4pe0)*q/r2.

, q 1 2, , A12 = 1ò2qEdl A12 = q(j1 - j2), , j1 - j2 = 1ò2Edl. j1 = j2, : oò Edl = 0.

17. :

, , . j(x, y, z) = const.

dl, dj = 0. , , , 0, .. . .. .

. , (Dj = const). .

r, . , .

18. :

. .

]$ :

() 1

- +

- +

- +

- +() 2

- +

- +

-- +

- +

+

- +

0, .. .

j1 - j2 = 0, , .

.


t

j

t.

dj/dt = -Et, ( dj/dt = 0) .

q

= 0

E ~ g

(g - )

, = 0, , .

, = 1/R.

E ~ g ~ C ~ 1/R.

19. , :

, . .

= q/j.

( ):

: q

j = 1/(4pe0)*q/R

C = q/j = 4pe0R

R j

, = Dq/Dj.

-Dq

R

Dq

E+

X E-


+Dq

l

R

Dj - , .

l>>R, .

Dj = j1 - j2

j1 - j2 = Ròl-R Edx

E = E+ + E- = k*Dq/x2 + k*Dq/(l-x)2

:

= 4pe0R

:

q+ q- C = Dq/(j1 - j2) =

= (Dqe0S)/(Dqd) =

= e0S/d

j1 - j2 = E*d =

= gd/e = (Dqd)/(e0S)


j1 j2

:

R1

R2

+q

-q

j1 - j2 = R1òR2E+dr = = Dq/(4pe0) * R1òR2 (1/r2)dr = = Dq/(4pe0)*(1/R1 1/R2).

C = (4pe0eR1R2)/(R2-R1).

20. :

. , .

, , . , .

, .

( ).

= +

,

0 = <> = <> + <>

<> = E

:

E = E0 + E

= 0 = <>.

, , .

.

q- l q+


r-                   r+

r- = (i = 1åNriqi-)/( i = 1åNqi-)

r+ = (j = 1åNrjqj+)/( j = 1åNqj+)

. . , , . .

21. :

:

 

E

l +q

Fk

M a

Fk (X)-q

M = Fk*l*sina = q*E*l*sina = = P*E*sina, P .

M = [P x E]

M

dA = Mda = P*E*sina da

dA = dW

W = -P E cosa = -(P E)*

* - c .

:

X

 
+q F+

l

-q DX

F-

DF = (F+) (F-) = q*DE = = q*E/X*l*cosa = P*E/X*cosa = = / , a, , / = = (PEcosa)/X = -W/X.

22. :

, .

P = (i = 1åNPi)/DV


(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

(-+)(-+) (-+)(-+)

g.

P = He0E

H ;

.


E


DS l

n

P

n

d

-g +g

P*DV .

DV = DS*l*cosa

P*DV = P*DS*l*cosa = q*l

q = g*DS

P*DS*cosa*l = g*DS*l

P*cosa = g

g = He0E, .

= 0 + Œ

:

= e0 + = e0E + He0E =

= (1 + H)e0E = ee0E.

23. :

+g0 -g0

0

- +


- +

- +

g0 , 0 ();

, e.

0 = g0/e0

= 0 Π= g0/e0 - g/e0 = = 1/e0(g0 - g);

E = E0 HE E*(1 +H) = E0 E = E0/(1+H) = E0/e;

= e0eE = e0E, .. , .

E = 1/e0*(g0 - g) = E0/e =g0/(e0e);

gC = g0*(e - 1)/e.

25. :

, . .

.

:

1) , .

2) .

3) ():

P

1

Pr 2 3

E

EC

D , P D , . .

(2), , Pr. , .

.

$ , , .

26. :

Et1

e1

n1

En1 a1

dh

Et2

a2

En2 n2

e2

dh 0 S. .

n2*S*cos0o + n1*S*cos180o + = 0, = 0, .. dh 0;

n2*S - n1*S = 0 n2 = n1 e0e2En2 = e0e1En1 En2/En1 = e1/e2.

n , n .

񠠠 dh 0:

E1t

E1

Et2 l

Et1

E2

E2t

E1t l cos0o + E2t l cos180o + + E dh cos90o = 0;

Et1 = Et2; t1/(e0e1) = t2/(e0e2) t1/ t2 = e1/e2 (1 1 , 2 2);

tga1/tga2 = (Et1/ En1)*(En2/Et2) = = En2/En1 = e1/e2.

27. :

, .

w = W/V ;

w = dW/dV - .

[w] = /3;

w :

W = CU2/2 = (e0eSU2)/(2d), 堠 U ;

d ;

V = S*d;

w = W/V =(e0eSU2)/(2d*Sd) = = (e0eU2)/(2d2);

U/d = E;

w = (e0eE2)/2 = E/2 = 2/(2e0e)

w = 1/2 S .

, w , .

W = VòwdV .

:

W = q1*j2 = (q1q2)/(4pe0er) , .

:

Wi = 1/2 qiji;

N :

W = 1/2 i=1åNqiji, i ³ 2.

28. :

, . , l. , , , .

:

<u> = Ö(8kT)/(pm), <u> 105 /.

, <u>, <u>:

j = ne<u>, j ; <u> 10-3 /.

.

<(u + u)2> = <u2 + 2uu + u2> = = <u2> + 2<uu> + <u2> Û

Û <(u + u)2> = <u2> + <u2>, <Dek> = (m<u2>)/2.

29. :

/ / , , , . , .

] v0 w. , w. , E = -mw/e, .. :

j1 - j2 = 1ò2Edl = -1ò2(mw)/edl = = -mwl/e, l . I = (j1 - j2)/R.

q = òdq = -u0ò0ml/(eR)du = = (m/e)*(lu0/R), , .

, , .

30. :

, , . 1853 , s . , , .

:

H = 1/3 nmulCV, V = 3/2 (k/m), H = 1/2 nkul.

H/s = (kmu2)/e2 = 3(k/e)2T = = 2,23*10 ¾ 8*T.

31. , :

, , , . (, ), () ( ).

, , . , , .

, $ . . v , .. , = 0. v u. .. u = v + u, .. <v> = 0, <u> = <u>.

. .

. , , , , .. .

I = dq/dt, dt , dq.

- + . + ¾ ,

I = dq+/dt + |dq¾|/dt.

+ .

. , . j. { dI, dS^} { }:

j = dI/dS^, u.

, , :

I = q/t, q , t.

I = [A].

:

, . . , j, .. . . .

, . , + , .

e = A/q.

F = E**q, * - .

, , + , ():

U12 = j1 - j2 + e12.

, , , :

U = j1 - j2.

, , .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

32. , , :

: , , , U .

I = (1/R)*U, .. , U = j1 - j2; R .

[A] = []/[B].

, . :

R = r(l/S), l , S , r - , .

r = [*].

j ( ) . ,

j = (1/r)*E = sE ( ), s - .

:

, , , :

Q = Uit = / / = RI2t, .

, - t: Q = 0òt RI2dt.

- :

dQ = RI2dt = ((rdl)/dS)(jdS)2dt = = rj2dVdt, dV = dS*dl.

dV dt, - , . V . t:

Q = rj2 .

33. :

, , *, . :

j = s(E + E*) .

, :

1 2:

S


1 2


dL

, j, s, E, E* , ^ 12, ; j, E * .

dl j, E *, :

(*) jL = s(EL + EL*), , + ¾, dL.

- , I = jLS 1 2.

(*) : j = I/S, s = 1/r, :

I(r/S) = EL + EL*, :

I1ò2(r/S)dL = 1ò2ELdL + 1ò2EL*dL Û

Û IR = j1 - j2 + e12 Û Û I = (j1 - j2 + e12)/R .

, .. j1 = j2, : I = e/R, R c .

34. . :

, , 2 . , , .

: , , 0:

åIK = 0, ( ), j 0.

: :

() 2


R1 R2


e1 + + e2

¾

¾

() R3 ()

1 ¾ + 3

:

I1R1 = j1 - j2 + e1,

I2R2 = j2 - j3 + e2, +

I3R3 = j3 - j4 + e3,

I4R4 = j4 - j1 + e4.

åIKRK = åeK II -.

I1 I2

I3

R1 R2 R3


+ +

- -

j1 0 j2

e1 e2


C

I1R1 + I3R3 = -e1

: I2r2 = e2 + j2 -j0&#13;&#10;I2r2 = e2 + j2 -j0&#13;&#10;q = C*Dj&#13;&#10;I1R1 + I2R2 = -e1 +e2

35. :

, . , , (), . (), D .

, , . .

, .

- , , .

:

, , Bi, ; = å Bi.

ôô :

f = k(2I1I2)/l, l .

1 , ôô , 1, , 2*10¾7/.

1 , 1 1.

f = [m0/(4p)]*(2I1I2)/l

2*10¾7 = [m0/(4p)]*2(1*1)/1 m0 = 4p*10¾7 (/).

.

.


I n


, . () .

a = 90 m - ;

a = 0 m = 0;

m ~ I ü

ý m ~ I*S

m ~ S þ

. . : PM=I*S*n

mMAX/PM ~ B.

36. :

, .

I

dB


r

a

dl

:

dH = k(I[dl x dr])/r3 , .

[H] = /; [B] = .

I

(X)

a

da

r

dr

dL


dH = k(I*dL*sina)/r2

dL = dr/sina = rdr/sina = bda/sin2a

r2 = b2/sin2a

dH = I/(4p)*(bda)/sin2a*(sin2a/b2)*sina = = I/(4p)*(sina da)/b;

p

H = I/(4pb) 0ò sina da = I/(2pb);

H = I/(2pb) .


I

a1

a2

H = [I/(4pb)]*(cosa1-cosa2)

37. :

I

(X)

a

da

r

dr

dL


dH = k(I*dL*sina)/r2

dL = dr/sina = rdr/sina = bda/sin2a

r2 = b2/sin2a

dH = I/(4p)*(bda)/sin2a*(sin2a/b2)*sina = = I/(4p)*(sina da)/b;

p

H = I/(4pb) 0ò sina da = I/(2pb);

H = I/(2pb) .


I

a1

a2

H = [I/(4pb)]*(cosa1-cosa2)

.

I

:

dH

dl R r

X

dH = 1/(4p)*(Idl)/R2

2pR

H = I/(4pR)*0ò dl = I/(2R)

dH ôô = dH sina = dH(R/r)

dH ôô = 1/(4p)*(Idl)/r2*R/r

Hôô = 1/(4p)*(2pR2I)/r3 = = 1/(4p)*(2pm)/r 3, x >> R

Hôô = 1/(4p)*(2pm)/x3

Hôô = 1/2*(2pR3I)/(R2 + x2)3/2, 蠠 (x >> R).

H1

H H2

I I

1 2

, , , ôô .

38. :

.

I

1 2

1 2

4 3

oòH dl = 1ò2Hdl + 2ò3Hdl + 3ò4Hdl + + 4ò1Hdl;

H1ò2dl = H*l = Inl;

H = I*n, n .

.

0.

H1 2 = 0.

39. . :

, , , q, v . F v .

F = q*[v x B];

.

F

B1 q1 v1

(*) ( )

B2

(x) ( ) v2

q2

F

F = q*[v x B];

F = q*[v x B] + q*E

F = 1/(4pe)*(q1q2)/r2

F = qvB = qv*(m0/4p)*(v/r2)*q2 (?)

B2 = m0/(4p)*(I2dl)/r2 = = m0/(4p)*(q2/dt)*(dl/r2) = m0/(4p)*(q2v)/r2

F/F = m0e0v2 = v2/C2.

:

F = e [(u + u), B];

u - ;

u ;

<F> = e [<u>, B];

dV = S*dl;

F = <F>*nS*dl = en [<u>, B] S*dl;

en <u> = j;

F = [j, B] dV;

F. . = F/dV = [j x B];

j*S*dl = I*dl;

dF = I [dl x B] .

40. , :

a

b

FA FA B

(x) (*)

I

FA = IaB

M = IabB = ISB = PMB, . (?)

FA

b

FA


a

(X) n (X) B

FA

FA

F = I [l x B];

M = [PM B];

:


dh

dl1 dl2

B


I

I a

a1

dl1(X)Fˠ dl2(*)

a2 B

I

dF1 = I dl1B sina1 = IB dh

dF2 = I dl2B sina2 = IB dh

dM = dF*a = Iba dh = IB dS

M = ISB = PMB

M = [PM B]

dA = M da = PMB sina da

dA = dWp

A = Wp = 0òaM da = -PMB cosa + const .

a = p/2 Wp = const = 0

Wp = -PMB cosa = -(PM B)

41. :

I

+ I

l

¾ FA

(X) B


I dx

dA = FA dx = IB (l dx) = IB dS = I d;

d , .

() ^ ,

dA = Ibl cosa dx = IBn dS = I d, ..

d = B dS = B cosa dS = Bn dS

, .

I.

1 2

I I

͠ 0

(X) B

2 2

A1 = I ( 0)

2 = I (0 K) (?)

A = A1 + A2 = I ( ) = I D.

I

(X) B

A = -IBS IBS = -2IBS.

42. :

- .

.

B = B0 + B.

, - (i=1åNPMi)/DV:

J = (i=1åNPMi)/DV

[ J ] = A/;

J = c H, c - .

c = c/r = [ 3/], r - .

c = c*n [3/].

44. :

3 :

1) (c < 0, 10¾7¸10¾8 (3/));

2) (c > 0, 10¾6¸10¾7 (3/));

3) (c < 0, 103¸104 (3/)).

. , .

H = B/m0 J = B/m0 - cH

H(1 + c) = B/m0

H = B/(m0m); m = 1 + c.

:

B0

(X)(X)

(X)(X)

(X)(X) B


dl

B = B0 + B

B = m0*Il

dPM = Il*S*dl

dPM/dV = J = Il

B = B0 + m0J

H = B/m0 J = B0/m0 = H0 ()

H = H0 H0, 0 ;

H0 = N*J ( )

N = 1 ;

N = 1/3 .

, .

45. :

n

B

m1

b


m2

n

oò BdS = -Bn1S + Bn2 + <Bn>S = 0, (<Bn>S) = 0;

B1n = B2n

.

m0m1H1n = m0m2H2n

H1n/H2n = m2/m1


a1


m1 I a

b


m2 I

a2

m1 > m2

oòH dl = H1t*a - H2t*a + <H>*2b = 0

H1t = H2t

B1t/(m0m1) = B2t/(m0m2) Û B1t/B2t = = m1/m2

tg a1/tg a2 = m1/m2.

46. :

:

u

M r PM

I = en = e (w/2p) = e [u/(2pr)] , .

L = Jw = mr2*u/r = mur . (m - ?)

L M:

PM = IS = I*pr2 = (eur)/2 .

PM/M = -l/(2m) .

åM ¹ 0 .

-åMi ¹ 0 .

, , .



:

~

MS = h/2 .

~

h = h/2p = 1,05*10 34 (*)

() (h), .

:

PMS/MS = - l/m;

~

PMS = - (l h)/(2m);

~

m = (l h)/2m .

, .

:

F = PM (B/x) cos(a), a - . .

48. - :

> 0.

m 1.

(, , .) 1010 , .

:

J


H

B

HC


B

H


m


H

49. , , :

, , . .

I = d/dt ( ).

m, m .

:

I , , .

:

I


(X) n

e + R u

¾

(X) B

I

D

Ie dt = dA .

R , dQ =I2R dt , R, dA = dQ.

R ,

dA = dQ + I d

eI dt = I2R dt + I d

I = (e - d/dt)/R.

().

ei = - d/dt.

:

y = N*1

ei = -dy/dt = -N(d1/dt), y - .

:

(X) B

(e)

U

u

dA = F U dt + F u dt

dA = F U dt - F u dt = e u B U dt - - e U B u dt = 0.

:

, . , . , . . - ,

. .


50. :

, .

y (y - );

y ~ B ~ I y = L*I

L (). , .

, L const.

:

B = m0mnI (n );

= BS, y = N = m0mnISnl = = m0mn2IV;

L = m0mn2V, V .

:

eS = -dy/dt = -(L*dI/dt + I*dL/dt) ;

L const, eS = -L*dI/dt.

51. :

L


R


, .

dA = eSI dt = /- / = -dy/dt Idt = -dyI, dy - dt.

dy = L dI

dA = -LI dI;

A - , .

0

A = òdA = -L ò I dI = LI2/2.

I0

L = m0mn2V

H = nI

A = W = LI2/2 = 1/2*(m0mH2)*V

W . .

W/V = wH = 1/2*(m0mH2) = BH/2 = = B2/(2m0m).

52. :

, . , , .

. :

[DE] = -B/t ( );

D = 0 ( , .. ).

:

[DH] = j + D/t ( );

DD = r (, D ).

, D j c E, a H c B:

D = e0eE;

B = m0mH;

j = sE.

.

:

:

ò E dl = -d/dt SòBdS ( .-. , - . ; , S)

oSòBdS = 0 ( );

:

òHdl = SòjdS + d/dt SòDdS ( );

oSòDdS = Vòr dV ( ).

53. . .

] , , , - . , . , .

:

ei = oòEBdl;

ei = -d/dt,

oòEBdl = -d/dlSòBdS Û

Û oòEBdl = -Sò(B/t)dS Û

Û Sò[ÑEB]dS = -Sò(B/t)dS,

[ÑEB] = -B/t.

q. .. , [ÑEq] = 0.

[ÑEB] ¹ 0 EB, , .

:

= EB + Eq [ÑE] = -B/t.

, . , , . , , , .

:

, $ .

+q -q

i D


S

j = / / = = i/S = (q(t))/S = (q/S)t = g

D = g D = g.

- , .

jCM = (D)

- , - . . , .

j = j + j = j + (D)

oòH dl = Sò j dS + Sò(dD/dt) dS.

54. :

, :

D = e0em0m(2/t2) - .

e0m0 = 1/2, .

D = (em/2)*(2/t2);

:

(2/2) + (2/2) + (2/z2) =

= (em/2)*(2/t2);

(2H/2) + (2/2) + (2/z2) =

= (em/2)*(2/t2);

1/u2 = em/2 .

u = /Öem;

, .

55. :

]$ , .

g = 0; j = 0;

^ x:

1) ^ x y z.

2) , .. ^ .

3) Z = 0, HY = 0.

4) 2E/x2 = (em/C2)*(2EY/t2)

2HZ/x2 = (em/C2)*(2HZ/t2)

DE = (2E/x2) + (2E/y2) + (2E/z2), (2E/y2) = (2E/z2) = 0;

5) E = Em cos(wt kx + a1); (m - ?)

HZ = Hm cos(wt kx - a2);

, a1 = a2. ___

6) Em*Öe0e = Hm*Öm0m;__________

Em/Hm = Öm0e0 = Ö4p*107*9*109*4p = = 120p;

E = Em cos(wt kx);

H = Hm cos(wt kx);

ՠ

Z

56. . :

w = wE + wH = (e0eE2)/2 + (m0mH2)/2 = 1/2(Öe0e*E*Öm0m*H + Öe0e*Öm0m*H*E = = (1/C)*E*H /e = 1, m = 1/ - .

S = w*C .

S = [E H] .

S = Fò S dF - , S .

. .

E* > rj, .

.

57. :

E

j

f.


͠

f. = [j B] = m0m [j H]

__

P = w , w - , ; . __

= (1 + k) w;

k. = S/C2 = [E x H]/C2, k .

k. = m. * C;

m. = S/C3 = w/C2, w - .

= mC2;

, , , .

58. :

, .. - .

-:

, :

E = A cos(wt kr + a);

A = const;

A ~ 1/r, r .

n = C/u; n = Öem = Öe;

:

l = 0,40 ¸ 0,76 ;

f = (0,39 ¸ 0,75)*1015 .

, , , .

I = <ôSô>;

Eme0e = Hmm0m;

S = EH ~ eE2 ~ I ~ eA2.

, , .

:

. .

E

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

59. :

, , , .

I = <ôSô>.

] .

E1 = A1 cos(wt + a1);

E2 = A2 cos(wt + a2);

A2 = A12 + A22 + 2A1A2 cosd, d = = a2 - a1; ___

I = I1 + I2 + 2ÖI1I2 cosd; (I - ?)

d , .

cos[d(t)] = 0, .

A1 = A2 I1 = I2;

d = 0 Iå = 4I;

d = -p Iå = 0;

, Iå = 2I.

.

n1

0

C

n2

wt 0, , , n1,

A1 cos[w*(t S1/u1)];

A2 cos[w*(t S2/u2)];

d = w*(S2/u2 - S1/u1) = = w/C*(S2n2 S1n1) = // u = C/n; w/C = 2pf/C = 2p/l0, l0 // = = (2p/l0)*(S2n2 S1n1) = = (2p/l0)*D, D - , S .

D*(2p/l0) = d;

DMAX = ml0, m = 0, 1, 2, 3,

d ~ m*2p;

cos(d) = (m + 1/2)*2p = (2m +1)*p - .

60. :

:

l >> d


S1


S2

d/2

d

d/2 Dx



l

D = S2 S1;

S12 = l2 + (x d/2)2

S22 = l2 + (x d/2)2

S22 - S12 = (S2 S1)(S2 S1) = 2dx

S1 + S2 2*l,

D = S2 S1 = (2dx)/(2*l)

x = (D*l)/d

xMAX = (ml0*l)/(d*n) = m*(l/d)*l, (?)

m = 0, 1, 2, 3,

l0/n = l - .

xMIN = (m + 1/2)*(l/d)*l

I1 = I2 = I0 (?)

I = 2I0(1 + cos d) = 4I0 cos2(d/2);

d ~ D ~ x, I ~ cos2x;

:

Dx = (l/d)*l

61. :

.


q1 S1


q1 q1


n q2 q2 nS2 q2

d


S1 = 2d tgq2*sinq1

S2 = (2dn)/cosq2

D = nS2 S1 = 2d*[(n2 sinq1*sinq2*n)/(cosq2*n)] = = /sinq2*n = sinq1/ = = 2d[(n2 sin2q1)/(n*cosq2)] = = /n*cosq2 = Ön2 n2*sin2q2/ = = 2d*Ön2 sin2q1;

: ________

D = 2d*Ön2 sin2q1 -l/2;

ml = D; _______

max: 2d*Ön2 sin2q1 = (m + 1/2)l, m = 0, 1, 2, 3,

m min .

62. :

D = 2b + l/2

R


r

b

R2 = (R + b)2 + r2 /R >>b/ R2 2Rb + r2;

B = r2/(2R);

D = r2/R + l/2;

DMAX = ml = /m = 0, 1, 2, 3, / = = 2m*(l/2);

DMIN = (m + 1/2)l = (2m + 1)*(l/2);

r = ÖmlR, m , ;

m , .

63. :

.

.

.

A cos(wt kx + a)

A(t), w(t), a(t) , .

- , .

A1 cos[w(t)t + a1(t)];

A2 cos[w(t)t + a2(t)];

w(t) = w0 + Dw(t)

A2 = A12 + A22 + 2A1A2 cos[d(t)];

d = a2(t) - a1(t) + Dw(t);

Dw(t) = Dw2(t) - Dw1(t).

64. :

t () .

d(t) = -p ¸ p;

cos[d(t)] = 0 ;

cos[d(t)] ¹ 0 .

t - , p.

t << t ;

t >> t ;

t t .

, (-). .

4. :

 


a

Ӡ


ՠ

t da

r1 r R r2

a1 a2


dl


r*dr

a a


dl

dl = (r*dr)/sina

r = R/sina

dl = (R*da)/sin2a

dE = t*dl/(4pe0r2)

dEx = dE cosa = [t*dl/(4pe0r2)]*cosa= = [(tRda*sin2a)/(sin2a*4pe0R2)]*cosa = = [t/(4pe0R)]*cosa*da;

dEy = dE sina = [t/(4pe0R)]*sina*da;

Ex = [t/(4pe0R)]*a1òa2cosada = = [t/(4pe0R)]*(sina2 - sina1);

Ey = [t/(4pe0R)]*a1òa2sinada = = [t/(4pe0R)]*(cosa1 - cosa2);

E = ÖE2x + E2y;

:

a1 = 0; a2 = 180;

Ex = 0; Ey = t/(2pe0R).

10. :


Ӡ


Ex dx Ex+(Ex/x)dx

dz

dy

ՠ

Z

= [Ex + (Ex/x)dx]dydz cos 0 + + EX dydz cos180;

X = (Ex/x)dxdydz = (Ex/x)dV;

= (E/y)dxdydz;

Z = (EZ/z)dxdydz;

oSòEdS = + + Z = (Ex/x + + E/y + EZ/z)dV;

lim [(oSòEdS)/V] = div E

V 0

div E = (Ex/x + E/y + EZ/z)

:

oSòEdS = q/e0e = (VòrdV)/(e0e);

divE = r/(e0e);

divD = r.

24. :

+ + + + + + + + + + + + + + + +

- -

+ +

+ - - -

+ +


¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾

E = E0 E

E0/e = E0 - E

E = E0 (1 1/e) = E0 [(e - 1)/e]

e - 1 = c - ;

= 0*(c/e);

= s/e0;

0 = s/e0;

s = s*(c/e), s ;

s - .

47. :

PM B

a

u N


a

dM=Ndt

M

Msina

r

I

PM

N = [PM B]

dM = N*dt

dM/(M*sina) = dj

N = PM*B*sina

|dM| = PM*B*sina*dt

(PMBsina dt)/(Msina) = dj

dj/dt = wL ;

w = (PM/M)*B = (l/2m)*B, ;

w .

.

:

: PM = Ipr2 = e*(wL/2p)*pr2 = = -(e2/4m)*Br2;

<PM> = -(e2/4m)*B<r2> = = -(e2/6m)*Br;

<r2> = 2/3*r2;

i=1åN<PM> = -(e2/6m)Bi=1åNri2;

X = åPM/(VH);

X = [(-e2*m0*NA)/(6m)]*I=1åNri2.

, . < 0.


 
2012 , , .