Bồi dưỡng học sinh giỏi vật lý 11 tập 1

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LQI NOIDAU
«B6I DlfONG HOC SINH GI61 VAT L I TRUNG HQC PHO THONG" la
bo sach dung cho hoc sinh kha gioi, hoc sinh cac Idp chuyen Vat l i , cac thay c6 giao
day Vat li d cac trUdng trung hoc pho thong. Bo sach gom 7 cuon:
1. Boi dUdng hQC sinh gioi Vat U10, tap I (Donf^ hoc, Dong lUc hoc, TTnh hoc)
2. Boi dUdng hpc sinh gioi Vqt li 10, tap I I (Cac dinh luat bcio loan, Nhiet hoc)
3. Boi dUdng hQC sinh gioi Vat li 11, t|ip I (D/pn va Dien til) •
4. Boi dUdng hgc sinh gioi Vat li 11, tap I I (Quang hinh)
5. Boi dUdng hgc sinh gioi Vat li 12, tap I {Dao dong va Song ai hoc)
6. Boi dUdng IIQC sinh gioi Vat li 12, tap I I {Dong dien xoay chieu va Dao don^
dien tCe)
7. Boi dUdng hgc sinh gioi Vat li 12, tap I I I {Quang li. Vat li hat nhdn)
Ve cau true, moi cuon sach deu du'dc chia thanh cac phan Idn, trong moi phan
gom nhieu chuyen de, moi chuyen de la mot noi dung kien thiJc tron ven. Moi
chuyen de gom cac phan:
A-T6m tat kien thrfc: Phan nay chiing toi trinh bay mot each c6 he thong
nhi^ng kien thiJc trong tam cua chuyen de tiif cd ban den nang cao trong do chiing toi
chu trong dao sau nhffng kien thiJc nang cao de lam cd sd cho viec giai cac bai tap
cua chuyen de.
B-Nhi?ng chu y khi giai bai tap: Trong phan nay chiing toi neu len nhu'ng
chii y can thiet ve kien thiJc va ki nang giai bai tap. Do la nhu'ng lufu y quan trpng
giiip dinh hu'dng va tranh nhu'ng sai sot khi giai cac bai tap cua chuyen de.
C-Cac bai tap cua chuyen de: He tho'ng bai tap d day kha da dang du'dc s^p
xep ttf de den kho, tCf ddn gian den phiJc tap va di/dc giai kha chi tiet nen rat phii
hdp vdi nhieu do'i tU'dng ban doc.
Trong qua trinh bien soan chiing toi tham khao rat nhieu nguon tai lieu trong
va ngoai niTdc, dac biet la cdc bo sach G/a/ loan Vat U do thay Biii Quang Han lam
chu bien - Nha xuat ban Giao due 1998, bo sach Bai tap va Un gidi Vat li do OS.
Yung Kuo Lim lam chu bien - Nha xuat ban Giao due Viet Nam 2010, bo sach Ca

sd Vat li do David Halliday lam chu bien - Nha xuat ban Giao due 2002... de lam
phong phU them phan kien thiJc cung nhiTphan Idi giai cac bai tap trong bo sich.
Vdi SLf gop siJc cua cac thay c6 giao da va dang cong tac tai cdc tnTdng
chuyen, cac thay c6 giao da tijfng tham gia boi difdng hoc sinh gioi Vat l i cua cac
tinh thanh trong ca ni/dc, hi vong bo sach se la tai lieu tham khao thie't thifc, bo ich
cho nhieu doi tifdng ban doc yeu thich bo mon Vat l i .
Mac dil da dau tiT bien soan kha kl ludng nhiftig nhu'ng han che, sai sdt la dieu
khong the tranh khoi. Rat mong nhan difdc sif dong gop, chia se cua cac thay c6
giao va cac em hoc sinh tren ca nifdc; Moi y kien dong gop xin gufi ve dia chi
hoac khang
Xin tran trong gidi thi^u bo sach den quy thay c6 giao va cic em hoc sinh!
Chu bi6n
ThS. Nguyin Phii Dong

thvrnhat

mdn

TfNH

N»

LlTC TlJfdNG TAC TINH DIEN

Chuxendil:

A-TOMTAxKliNTHtrC

•^^mt^.^ag^

I . Dien tich - S\i tifdng tac giffa cac di^n tich
1. Di^n tich: Cd hai loai dien tich: dien tich du'dng va dien tich am. Cac dien
tich cung loai thi day nhau, cac dien tich khac loai thi hut nhau.
2. Djnh luat Culong: LiTc ti/dng tac giufa hai dien tich diem diJng yen ti le
thuan vdi tich dp Idn ciia hai dien tich va ti le nghjch vdi khoang each giiJa
Chung.

+

k

lq£2|

e'

r^

xjflq •'•

e la h^ng so' dien moi ciia moi triTdng ( 8 = 1 : chan khong hoac khong khi).

+ r la khoang each giffa hai dien tich qi, q2.
/ " ^ j

Chii y : Dinh luat Culong du'dc ap dung cho:
- hai dien tich diem.

^f'^ ^ \
^

- hai qua cau tich dien phan bo deu.
11. Djnh luat bao toan di^n tich

" ^1

i<
t

'•'''^^O

Trong mot he CO lap ve dien, tdng cac dien tich dtfdc bao toan:
q, + q2 + ... = const

j „ ^t^, ^^,(^-,3

B. NHONG C H O t KHI GIAI B A I T A F
-

Khi dp dung djnh luat Culong ve siT tU'dng tac giffa cdc dien tich dffng yen can
chu y:
+ dieu kien ap dung: hai dien tich diem hoac hai qua cau tich dien phan bo deu.
+ cac hien tU'dng thiTc te" thffdng gap:
•

cho hai qua cau nho dan dien nhiT nhau da nhiem dien tiep xiic nhau hoac
no'i vdi nhau

bKng doan

day dan roi tdch rdi ra thi tdng dien tfch se chia

deu cho hai qua cau: q'l = q'2 =
•

' '

khi cham tay vao mot qua cau nho dan dien da tich dien thi qua cau se
mat dien tich va trd thanh trung hoa.
3


Bdi diiOng hgc sinh gi6i Vat ly 11, t$p 1 - Nguyin Phu D6ng

-

Cty TNHH MTV DVVHUhang Vi?t

Khi mot dien tich diem q chju tac dung ctia nhieu life tac dung Fp F j , ... do

c. C A C B A I T A P vi; Lye T U O N G T A C TITOI D I E N

1. Tl/CfNG T A C G I 0 A CAC D I E N T I C H D I E M DlfNG YfeN

cac dien tich diem qi, q2, ... gay ra thi hcJp life tac dung len q la:

1.1. Hai dien tich diem bang nhau dat trong chan khong, each nhau doan R = 4cm.
Li/c day tTnh dien giffa chiing 1^ F = 10'^N.

-f De xac djnh do Idn cua hdp luTc F ta c6 the diTa vao:

**

a) Tim do Idn moi dien tich.

+ djnh l i ham cosin: F^ - Fj^ + F2 + 2FjF2Cosa (a la goc hdp bdi Fj va

).

b) Tim khoang each R, giffa chiing de liTc day tTnh dien la F, = 2,5. lO^^N.
Bai giai

•

F, va F2 cung chieu thi: F = F, + F2

(a = 0, cosa = 1).

•

F, va F2 ngiTdc chieu thi: F = IF, - F2I

(a = n, cosa = -1).

^

a) Do Idn mSi dien tich
_ Vi: + Hai dien tich day nhau nen q, va q2 cung dau.
+ Hai dien tich bang nhau nen: qi = q2.

•

F, va F2 vuong goc thi: F= ,JF^ + F^

•

Fj va F2 cung do Idn (Fi = F2) thi: F = 2Fi cos y .

(a = 90°, cosa = 0).

-

Theo dinh luat Cu-16ng:
-5

F =k

= RJ^
R^

+ phiTdng phap hinh chieu: F = ^jF[+F^
(F, = Fu + F2x +
-

Fy = F,y + F2y +

F2 +

R'

-f Cac life tac dung len dien tich q thi/dng gap la:

8

Vdi khoang each R: F = k - ' ^
R^

(1)

-

Vdi khoang each R,: F' = k - 3 -

(2)

+ liTc cSng day T.

R:

ne'u q, va q2 trai dau; \\ic day ne'u q,

•

va q2 Cling dau).

,

1,3.10"'C

^2

-

+ trong life: P = mg (luon hu'dng xuong).
.SiSl. Q^^Q

1,3.10 ' C

b) Khoang each R, giffa chung de life day tTnh dien la F, = 2,5.10"^'N

...)

... = 6

+ life tTnh dien: F =

=4.10-1 j i ^
9.10^

Vay: Do Idn cua moi dien tich la

Khi mot dien tich q du'ng yen thi help luTc tac dung len q se bang 0:
F = F, +

.. . '1

-

10r5

Suy ra: R, = R. — = 4.

= 8cm.

y 2,5.10"^
,

Vay: De life day tTnh dien giffa hai dien tich la F, = 2,5.10 ''N thi khoang

; u

+ lircdanh6icual6xo:F = k . A / = k ( / - / o ) .
•I

fi

each giffa chiing la R| = 8cm.

FM

1.2. Hat bui trong khong khi d each nhau mot doan R = 3cm, moi hat mang dien
tichq =-9,6.lO-'^C.

_

H :

a) Tinh life tTnh dien giffa hai hat.

'iff''

b) Tinh so electron dirtrong moi hat bui, biet dien tich moi electron la e = 1,6.10-"C.
*"

y-•

Bai giai

a) Lffc tTnh dien giffa hai hat
Vuong goc

Ciang do Idn
Ta cd: F = k

= k^=9.10'.
R'

R^

- ^ . r.
, 1'\

,-13x2
(-9,6.10-'-')

= 9,216.10'^C '

(3.10-^)2

Vay: Life tTnh dien giffa hai hat la F = 9,216.10-'^C
A\

4


Cty TIMHH MTV UWH khang V i ^

B6i diiBng hpc sinh gi6i Vjt ly 11, t?p 1 - IMguygn Phu B6ng

Bai giai

b) So electron diT trong moi hat bui
-9,6.10"
Ta c6:

ne = —

e

=

^,, ,

a) Dp Idn life hU'dng tarn dat len electron
Vi life hU'dng tam trong chuyen dong tron cua electron quanh hat nhan chinh

= 6.10'.

-19

1,6.10

Smi

la life tTnh dien nen:

v'c,fy ::r"!,;..I

Vay: So' electron d\X trong moi hat bui la ne = 6.10^

^-19

1.3. Moi proton c6 khoi liTdng m = l,67.10""kg, dien tich q = 1,6.10""C. Hoi life

FM

=k

^1^2

(-1,6.10~'^).1,6.10
= 9.10".

= 9,2.10-* N

R^

day Culong giiya hai proton Idn hdn li/c hap dan giffa chiing bao nhieu Ian?

(5.10"")2

Vay: Do Idn life hU'dng tam dat len electron la: Fh, = 9,2.10"* N.

Bai giai

*"- ^

b) Van toe va tan so chuyen dong cua electron
-

L\ic day Cu-16ng giffa hai proton la: F = k

= kTa c6: Fh, =

-

Lire hap dan giifa hai proton la: F ' = G

m,m2

= Gm
R^

R^
-

F' G m
Suy ra: — = —.
F
k
Vay:

1,67.10

Vay:

1,6.10

9.10^

'

'

Life day Cu-16ng giffa hai proton Idn hdn life hap dan giCfa chung

« 0,71.10"/s.

Van to'c va tan so' chuyen dong cija electron la Fh, « 2,25.lO' m/s va

nhau bang life F = 1,8N. Dien tich to'ng cong cua hai vat la Q = 3.10"^C. Tinh
dien tich moi vat.

Life tmh dien giiJa hai vat la: F = k

11^2

R^

/.
-

2

Life hap dan giifa hai vat la: F' = G
R^

FR2
^1^2

-G m
R^

=1,6.10-'^

kinhR = 5.10""m.
a) Tinh do Idn life hifdng tarn dat len electron.
b) Tinh van toe va tan so'chuyen dong cua electron.
Coi electron va hat nhan trong nguyen tur hidro tUdng tac theo djnh luat tTnh
dien.

(1)

9.10'^

•;t, .1

-

VT hai dien tich day nhau nen qi va q2 cung dau va ciing du'dng (suy ra tif de
bai). Do do:

1,86.10-kg.

1.5. Electron quay quanh hat nhan nguyen tuT hidro theo quy dao tron vdi ban

= 2.10-10

Matkhac:

6,67.10""

1,86.10"^ kg.

1,8.1^

-

qiq2 =

Vay: De life tTnh dien b^ng life hap dan thi khoi lifdng cua moi vat phai la
m=

^1^2

R^

D e F = F'thi:
.J^

Theo dinh luat Cu-16ng, taed: F = k

= kR^

mjmj

m =

-.1 :v ^ '

Bai giai

Bai giai

k ^ = G i ^
R^
R^

, M ,
2,25.10'm/s

2.3,14.5.10'"

de life tlnh dien bang life hap dan.

-

«

1.6. Hai vat nho mang dien tich dat trong khong khi each nhau doan R = Im, day

1.4. Hai vat nho giong nhau, moi vat thifa mot electron. Tim kho'l lu'dng m6i vat

-

9,1.10

-31

n « 0,71.10"/s.

1,35.10^'' Ian.

-

1-11
9,2.10"'.5.10"

i

m
2,25.10^

27tR
= 1,35.10,36

-19

,2

va n =

-27

6,67.10

mv
R

, ,

^1+^2

- Q = 3.10"^

,-10
2.10

q , + q 2 = 3.10-'

-

-

(2)

d')
^

(2')

/,

-

>

Giai he (1') va (2') ta difdc: r " ® ^ ^ k r - - ^ '
q, = 2.10' C va q2 = l O ' C hoac q, = 10"' C va q2 = 2.10-' C.
Vay: Dien tich moi vat la:
q, = 2.10-' C va q2 = l O ' C hoac q, = 10 C v^ q2 = 2.10

•' '
C.

1.7. Hai qua cau kim loai nho nhu nhau mang cac dien tich qi, q2 dat trong khong
khi each nhau R = 2cm, day nhau bang life F = 2,7.10^N. Cho hai qua cau


Cty TNHH MTV DWHJ
Bdi diJ8ng hpc sinh gi6i Vjt ly 11, tap 1 - Nguyin Phu B6ng

tiep xiic nhau roi lai diTa ve vi tri cu, chiing day nhau bang lye F' = 3,6.10^N.

Tinhqi,q2.

_ Lire tac dung len q2: Fj = Fj2 + F32 =^ F2 =

Bai giai

.F,

- Khi hai qua cau chU'a tiep xuc, ta c6:
F

FR^ ^2,7.10-^(2.10'^)^
= 12.10
k ~ 9.10^
(1) (hai qua cau day nhau)

=k

=> qiq2

= 12.10'^

_ Lire tdc dung len qs: F3 = Fjj + F23
q;q2

.F3

R^

=> F' =

k

(q,+q2)=

R^

-4

±2RJ|^ = ±2.2.10"' 3,6.10
V 9.10^

2. H ; G TONG HglP TAG DyNG LEN MOT D l t N TIGH
1.8. Ba dien tich diem q, = -10"'C, q2 = 5.10"'C, = 4.10"'C Ian lu-dt dat tai A,
B, C trong khong khi, AB = Scm, AC = 4cm, BC = 1cm. Tinh life tac dung len
moi dien tich. , , ,
Bai giai
Ta c6: AB = 5cm, AC = 4cm, BC = 1cm => AB = AC + CB C nkm trong
doan AB.
c
B
e- Life tac dung len qi:
=>F,

=k

1211

AB'

+k

F, = 4 , 0 5 . 1 0 " ' N

•"

"" q3

q2

Fj = F 2 j + F 3 1 =^ Fi = F21 + F31 (F^^pPsi
I3I1

AC^

5.10"V-10"'')

= 9.10'.( ( 5 . 1 0 " ' ) '

(10"')2

'^""g'^hilu)

4.10-'.(-10"^)
( 4 . 1 0 -2x2
"')

=> F3 = F13 + F23

= k I l l s + k ^ ' ' ' ' = 9.10'.(
AC^

(-10"^).4.10"
(4.10 ' ) '

(F,3;F23

+

cung chieu)

5.10 ^4.10"
(10"')'

1.9. Ba dien tich diem q, = 4.10"^C, q2 = -4.10"^C, q3 = 5.10"*C dat trong khong
khi tai ba dinh ABC cua mot tarn giac deu, canh a = 2cm. Xac dinh vectd life
tac dung len q3.
, ; „,,:•<
is^jj.
,
.
Bai giai

=^(q,+q2)= ±8.10-^
(2)
-v-9
- Giai he (1) va (2) ta di^dc: q, = 6.10"' C va qz = 2.10 ' C; q, = -6.10"' C va
q2 = -2.10"' C hoac q, = 2.10"' C va qz = 6.10"' C; q, = -2.10"' C va qz = -6.10"' C.
Vay: Dien tich cua cac qua cau khi chU'a tiep xiic nhau la: qi = 6.10"'C va
q2 = 2.10"'C; q, = -6.10"'C va qs = -2.10"'C hoac q, = 2.10"'C va q2 = 6.10"'C;
qi=-2.10"'Cvaq2 = -6.10"'C.

qi

4.10"^5.10-

.F3 = 2 0 , 2 5 . 1 0 " ' N

q ; = q ^ = ^
^1+^2

qiq2 1 3 ^ 2 = 9.10'. ( - 1 0 " ' ) . 5 . 1 0 "
(5.10-^)'
AB= BC^

.F2= 16,2.1 0 " ' N

- Khi cho hai qua cau tiep xiic nhau roi tach ra xa nhau thi: F' = k
vai:

=k

(Fj2;F32 ngiTdcchieu)

F12-F32

)

Ta c6:
Vi ll

F3 = ^^3 +

=12

F23,

.F,3

vdi

F,3 =

; F23
a"

=k

a

= F23Va a = (F,3,F23)
13'*23 = 120°

4.10"'.5.10
= 45.10-^N
(2.10"')'
Vay: Vectd life tdc dung len q3 c6:
+ diem dat: tai C.
+ phi/dng: song song vdi AB.
+ chieu: tiTAdenB.
+ doldtn:F3 = 45.10"^N.
_i i
1.10. Ba dien tich diem qi = qj = q3 = q =
1,6.10""C dat trong chan khong tai ba
dinh tarn giac deu canh a = 16cm. Xac
djnh lire tac dung len dien tich q3.

Bai giai
Taco: F3 = F,, + F23, vdi
=>F3 = F,3 = F23 = 9.10'

I1I3

Fi3

=k

F23

= k I2I3

a

a


B6i duBng hoc sinh gi6i V$t 1^ 11, tjp 1 - NguySn PhO Dflng

. F,3 =

F23

C t y T N H H MTV DVVti |-,lunu UiAt

va a = (F,3,F23) = 60° => F 3 = 2F,3Cos-^ = 2 k \0

^ F 3 = 2.9.10-/''^-'^7f.^ = 15,6.10-N

(i6.io"2)2

2

I r

64.10"^(-10"'')
= 9.10".
= 36.10^N
F23 = k
(4.10"')^
>/F^TF^ = V(27.10-^)^+(36.10"^)^ =45.10'^N
h2%

Vay: Vectd liTc tac dung len q^ cd:
+ diem dat: tai C.
+ phiTdng: CO (O la trung diem AB).
_
13 _ AC
)
(tan OCB =
F23
BC
+ chieu: tiT C den O.

+ do Idn: F3 = 45.10"'N.

\: -

Fio

= k——;

vdi F20 = F30 ( V I qi

Vay: Vectcf liTc tac dung len q3 c6:
+ diem dat: tai C.
+ phu'dng; vuong gdc vdi AB.
+ chieu: ra xa AB.
+ do l d n : F 3 = 15,6.10-^'N.
1.11. Ba dien tich diem qj = 27.10'^C, qj = 64.10"^C, q3 = -10"'C dat trong khong
khi tai ba dinh tam giac ABC vuong goc tai C. Cho AC = 30cm, BC = 40cm.
Xac dinh vectd liTc tac dung len q3.
Bai giai
Taco: ^ = + , vdi: a = 27.10"".(-10"'')
(F,3,F23) = 90°
111.!
F„ = k
= 9.10^
= 27.10^N
(3.10"')^
AC^

=>F3=

vdi

q i %

1.12. Tai ba dinh tam giac deu canh a = 6cm trong khong khi cd dat ba dien tich

q, = 6.10'"C, q2 = q3 = - 8.10"'C. Xac dinh liTc tac dung len qo = 8.10~'C tai
tam tam giac.
Bai giai
,
T a c d : ^ = F , , + E , o - f 4 = F,o+F23,
1,! ^

=> F23 = 2F20COS

va

F23

= 9.10

F,o

=k

=>F,o = 9.10".

F20

-k

= q3);

b

12%

;F3o

=k

= |h =

=^

va a = {F^^'^o)

= J20°

- = 2k ^ 2 % .cos60° = '2F0

6.10^8.10"^

e.io-^Vi

= 3,6.10^N

92

„.:d3f:! f|.:>f& 'O'ff! f'V
iMs

sill •

Ann
=^ Fo = 3,6.10^ + 4,8.10"^ = 8,4.10"'N

Vay: Vectd life tac dung len qo cd: ,
+ diem dat: tai O.
+ phiTdng: vuong gdc vdi BC.
\
+ chieu: tij" A den BC.
+ doldn:Fo = 8,4.10-^N.
" ^
1.13. Hai dien tich q, = 4.10'^C, q2 = -12,5.10"*C dat tai A, B trong khong khi,
AB = 4cm. Xac dinh lire tac dung len qj = 2.10"'C dat tai C vdi CA 1 AB va
F '
CA = 3cm.
Bai giai
Tacd: F 3 = F , 3 + F 2 3 = ^ F3 = J F ,X' + F,^y
;

Ox n^m ngang, Oy thdng diJng.
Fn

=k

11%

= 9.10'^

4.10^210"^
,-2x2
(3.10-^)

.!)':'
AC

= 8.10^N

5

B


B6i clL0ng hpc sinh gi6i Vjt ly 11, tgp 1 - Nguygn Phd Dfing

F23

=k

1213

Cty TNHH MTV DWH Khang Vi?t

Vay: LiTc tac dung len m 6 i dien tich c6:

(-12,5.10"^).210"^
= 9.10^

= 9.10-^N

(5.10-2)2

BC^

Fx = F , 3 ( x , + F23(x, = 0 + F 2 3 . C O S B = F 2 3 . ^ = 9.10^. ^ = 7,2.10-^N

BC

•

+

chieu: ttr tarn luc giac ra.

+

do Idn: F = k-

+

phU'cfng: hdp vdi A C mot goc P: cosP =

dung len m o i dien tich.

-

j

(8.10^'*)^ +(7,65.10"*)"

tich tai D tren hinh ve.

„2

-(9.10~^)2

T a c o : F = F , + F 2 + F 3 = F , + F 2 3 , v d i : F, = F2 = F 3 = k ^ .

« 0,34 => p « 70°

2

2.8.10"^7,65.10~^
dp l d n : F 3 = 7,65.1 O ^ N .

^'-^ '

'

C

-

vty,,

V''"'

1.14. C6 6 dien tich q b^ng nhau dat trong khong k h i tai 6 dinh luc giac deu canh
a. T i m lire tac dung len m o i dien tich.

D o tinh doi xiJng nen ta chi can khao sat mot dien tich ba't k i , ch^ng han dien

R
'23

\r

„
AD^+HD^-AH^
cosp = •
2.AD.HD

a2 +

Ta c6: F = F, + F3 + F4 + F5 + F,, v d i :

,2
120°

aV3

1V3

2

V

2a.

2 ,

=>F,3 = F, = F3 = kf

=>F^ =

F 3 = k 4 = k - ^ = k - 3 i (c = 2a) D

2^ 2

+

_

w5
2

r

t.

2V

B

a

a

J

4a2

Q2
2
F4 = Ffi = k \ k - 3 - ; b ' = ( 2 a ) ' - a ' = 3 a ^ p = 60°

b^
3a2
F 4 6 = 2F4Cos30° = 2k
3a2

F =F , . F 3 . F . =k

12

= ^ / i k ^

n^m tren diTdng cao H D .

tich tai B tren hinh ve.

(2a)2

r

V i F2 = F 3 ; B D C = 60° =^ F 2 3 = 2F2Cos30° = 2 k ^ . —
2 2

va F^ = F^ + F23 + 2 F 1 F 2 3 C O S P , v d i :

»|

B a i giai

c^

B a i giai

Do tinh doi xu'ng nen ta chi can khao sat mot dien tich ba't k i , ch^ng han dien

2F13F3

F, = F 3 = k 5 - ; a =

12

^ '

d i e m dat: tai C.

tSm luc giac.

(15 + 4 ^ 3 )

q2

c6:

+

+

phU'cfng: AvtUng thang noi dien tich

1.15. Bon dien tich q giong nhau dat d 4 dinh tu" dien deu canh a. T i m life tac

= > F 3 = V(7,2.10~'*)2+(2,6.10-^)2 = 7,65.10-^•N

=> cosP =

d i e m dat: tai cac dien tich.

+
5

AC
Fy = F,3(y, + F23(y, = F , 3 - F23.sinB = F , 3 - F 2 3 . ^
= 8.10"^ - 9.10^. - = 2,6.10^N
BC

Vay: Vectd life tac dung len

+

2

4.k4.k^
4a2'"3a2

V a y : Life tdc dung len m o i dien tich c6:
+ d i e m dat: tai cac dien tich.

(3)

.k4/'^"^^
'" a^ '

12

'

'•u >

+ phiTcfng: hdp v d i mat tur dien mot g6c a: cosa ^ —

2 F E23

—

^3
13


Cty TNHH MTV DVVH Khang ViSt

B6i du8ng hpc sinh gi6i V?t ly 11, tjp 1 - Nguygn Phu B6ng
f

cosa =

I

2>

2

2
2 >

(

—

-

J

a'

(

TiTdng tir doi v d i cac d i e n tich q2, qs va q4.

2\-

I

Vay: D o Idn liTc tac dung len m o i dien tich la F « 0,45.10"'N.

2V2

^ J

a = 160°30'

2V^k^.Vik%+ do \dn:F=

Do'i v d i q,: Fi - F^, +

V6k^.

+ F^j + F,,, + F2,i + Vy, + V^., v d i :

F2, = F 4 , = F r i = k \ ; F 3 , =

k

(aVi)'

1.16. ffinh lap phiTdng A B C D , A ' B ' C ' D ' canh a = 6.10"'°m dat trong chan khong,
Xac djnh life tac dung len m o i dien tich, ne'u:

k

a) Co 2 dien tich q , = q2 = 1,6.10-"C tai A, C; 2 dien tich qa = q4 = - 1 , 6 . 1 0 - " C
t a i B ' va D ' .
b) Co 4 d i e n tich q = 1,6.10""C va 4 dien tich - q dat xen ke nhau d 8 dinh cua
hinh lap phiTdng.

F21(x) -

+ F3, + F^,,

Do'ivdiqi: F, =

F31(z)

vdi: F21 = F31 = F41

2a^

F 2 , = k \ ; F 2 , ( y ) = 0;F2,(z) = 0

4

•\2'_.

= 0

F41(x)

„2

: 0; F4i(y) = F41 = k \ ; F 4 i ( z ) = 0
2i

= k

(aV2)^

•= k-

2a^

Fri(x) = 0; Fn(y) = 0; Fn(z) = F n - k -

F21(x) = F21(y) = - F 2 l C 0 S 4 5 °

= -k-

F^r.(x)
j-Uxl =
—.F2.„z)

= -F2-,cos45° = - k - ^ ^ . —

=-—•

'2a2'2

2

4

F2.,(y) = 9

k \

a^

„2

F3UX, = F 3 , ( . , = F3,cos45° =

F4i(y) = F4i(z, = F4iCos45° =

-

= > F i x = F2i(x) + F3i(x) = —

k

4
F ,ly
v -= r2l(y)
F2HV, +
+ F4UV)
r4i(y), = -

2a2

2

2a2

2

^

+•

a^
—

4

a. 2

2

k

4

= Fruy) = Fruz) = F r . c o s A ' A C = k

• 4-l(x) =
' 0; F4-i(y) = F4'i(z) = - F 4 ' , c o s 4 5 ° =
=> F u = F2l(x) + F3i(x) + F4i(x) + Fn(x)

a-

^

a2

F3'i(x)

^k^'

k ^

4~2k ^
4

+

— ^ K —

4

=

k

if

r 3 | ( 2 ) + ^41(2) =

/I

;v'

72

q^

2a^

14

(aV2)'

= -k2a2' 2

a) Ta c6:

riz -

• = k-

F j K x ) = F31(y) = - F 3 l C O S 4 5 °

B a i giai

-

= k3a^

1
Fri = F 3 i = F 4 i =

•

q'4

^2

= q2 = I 3 " I 4

0 Taco:

^

a^

'23^

' 2

a^

a'

4

+ F2-i(x) + FS-KX) + F4-l(x)

4

4

a^

9

a^

= 0

a-

4 '"-2
.9.10\

2

-k

9 •

=0

2
+

Sa^'aVi

9

a^

/iSIO&snU:-

=> F,y = F21(y) + F31(y) + F4|(y) + Fi'Ky) + F2-l(y) + Fs'Ky) + F4'l(y)

2
'—

«

0,45.10"'N

(6.10-'0)2
1

B6i duSng hpc sinh gioi Vat ly 11, tgp 1 - IMguygn Phu D6ng

TCr do:
+ C nam tren du'dng thang A B , ngoai doan A B , ve phia A .
F l z = F2I(z) + F3i(z) + F4i(z) + Fri(z) + F2-l(z) + FS-KZ) + F4-i(z)

' >

*

M

+

BC =

3AC = 3 ( B C -

Vay: Phai dat
" a -

a
2

9

a-

b) Dau va dp Idn cua q3 de qi, q2 cung can bang

C
- De qi va q2 cung can bang thi:

AB^

a-

^.12 = 4cm.

A

B

F|2 + F32

AC

^2

3'

va

F12 = F32.

-7

1,8.10

AB

= 0,45.10"'C

_ V i q , < 0 ; q 2 > 0 ^ q 3 > 0 : qj = 0,45.10"'C.

F,=v^.t(,-:^.:^)k4]=(V^-Vi:?4)ki
a

F21 = F 3 ,

=0

1i'l2
, 1312
, ^3^1
=k
va k
= k
AB'
AC^
BC'

13 = I 2

9

A C =

4 H

^2^1

2

| . 8 = 12cm va

-Q-

a- .

9

BC = | A B =

tai C , vdi A C = 4cm; B C = 12cm thi q3 se nam can bang,

F2, + F31 = 0 va

2

AB)

,^ ,

Vay: De q, va qa cung can bang thi qj = +0,45.lO'^C.
1.18. Tai ba dinh tarn giac deu, ngu'di ta dat 3 dien tich giong nhau qi = q2 = q3 =

F, = ( ^ / 3 - ^ / ^ 5 + l ) • 9 . 1 0 ^ ^ i ^ ^ : l ^ « 0,54.10-^N
3
(6.10-'°)2

-

TiTdng tiT cho cac dien tich khac.

•

q = 6.10~^C. Phai dat dien tich thuf tUqo d dau, la bao nhieu de he can bang?

" ' i.'

Bai giai
>
-

VSy: Do Idn cua life dien tac dung len moi dien tich la F « 0,54.1 O - ' N .
•fW •-'"n
3. SlJ C A N B A N G C U A D I E N T I C H

•

, - . ,1,/. ,|i
^''^

Cac lire dien tac dung vao qo: FJQ , FJQ va FJQ .
De qo can b^ng thi:

•

F,Q +

F2Q

+ FJQ

= 0

Vi qi = q2 = q3 = q = 6. lO^C => qo n^m d tarn
tarn giac ABC.

1.17. Hai dien tich q, = -2.10"^C, q2 = 1,8.10"^C dat trong khong khi tai A va B,
-

AB = / = 8cm. Mot dien tich qa dat tai C. Hoi:

V i tinh doi xiJng cua he nen de he can bang ta

a) C ct dau de q3 n^m can bhng?

Ichi can xet them dieu kien can bang cua mot

b) Da'u va do Idn cua qs de qi, q2 cung can bang.

Itrong ba dien tich kia, chang han q3. De q3 can

*

*

Ib^ng thi: Fo3 + Fj3 + F23 = 6

B^igiai

a) V i tri cua C de qs nhm can b^ng
r ^ F o 3 = F'3 = 2F,3Cos30° = 2k

-

Cac life dien tac dung len qs: F j j . F j j . '

-

De q3 nkm can bing thi: F^J + F23 = 0 => F j j = - F j j => F j j . F j j cung phu'dng,

3k

I0
^2

A
18

9

t-

= V3k

1113

B

jVi

I1I3I 73

(Fo3

2
= k

= Vik

3

q,, q2, q3 >

•Il

=

0 nen

I1I3

(F,3 = F 2 3 = k

Ills

)

I0I3

a

BC^

AC^

[BCJ

I0I3
a'

ngifdc chieu va c&ng do Idn: Fo = F23 <=> k

fAcV

^

3

2

3

.6.10-''= 3,46.10-^C
3
qo <

0.

I Vay: De he can bang tW ph;

j.io-'c.
17

.16


Bfii dugng hpc sinh gi6i V^t ly 11, t^p 1 - NguySn PhCi B6ng

1.19. d moi dinh hinh vuong canh a c6 dat dien tich Q = lO'^C. Xac dinh dau, dp
Idn dien tich q dat d tarn hinh vuong de ca he dien tich can bang?
Bai giai
- Vi dien tich d cac dinh hinh vuong nh\S nhau nen dien tich q dat d tarn hinh
vuong luon can bhng.
- Vi he CO tinh doi xufng nen chi can xet dieu kien can b^ng cua mot trong cac
dien tich con lai, chdng han dien tich dat d D.
- De dien tich dat d D nam can bang thi: Fj4 + F24+F34+Fq= 6
=:>F'4 + F24 = Fq ( F , 4 + F 3 4 - F 4 )

CtyTNHH MW'DWHJ
q2

hay k 1^^ = mg

10"^10
= lO^C.
q = l. mg = 3.10-^'.

VK

\^
V a y : D i e n tich q = 10-*C.

' "

1.21. Hai qua cau kim loai nho giong nhau treo vao mot diem bdi hai day / =
20cm. Truyen cho hai qua cau dien tich tong cpng Q = 8.10"^C, chung day
nhau, cac day treo hdp thanh goc 2a = 90°. Cho g = lOm/sl

a) Tim khoi lu'dng moi qua cau.

b) Truyen them cho mot qua cau dien tich q', hai qua cau van day nhau nhu'ng
g6c giiJa 2 day treo giam con 60°. Tinh q'.
'
Bai giai
;:';•;:! si

a) Khoi liTdng moi qua cau

.

Ta c6: Kho'i lifdng moi qua cau la m; dien tich moi qua cau \k
-7
Q 8.10 = 4.10-'C.

«

- Cac liTc tac dung len mot qua cau: trpng liTc P , liTc c^ng day T v^ life dien F .

- Qua cau nam can bang nen: P + f + F = 6.
- Suy ra: F = P t a n a o k ^ =mg.tan45°(r= l\/2)
kq^
kq^
=>m = (lV2)^g.tan45° 2l2g.tan45°
-T\2

- De Q ct D n^m can bhng thi q < 0

q = -—(2yf2 + \).
4
Vay: De ca he can bang thi q = - ^ ( 2 ^ / 2 +1).
1.20. Hai qua cau kirn loai nho giong nhau moi qua c6 dien tich q khoi li/cfng
m = lOg, treo hd'i hai day ciing chieu dai / = 30cm vao ciing mot diem. GiiJ
qua cau I co djnh theo phifdng th^ng diirng, day treo qua cau II se lech goc
a = 60" so vdi phiTdng thang diifng. Cho g = lOm/sl Tim q.
Bai giai
- Cac life tac dung len qua cau II: trpng life P ,
lUccSngday f va li/cdien F .
H
- Qua cau II nhm can bhng nen: P + f + F = 6.
- Tarn giac liTc "gach gach" 1^ tarn gidc deu n6n: F = P. P
IR,

•.

. .

.

•

9.10 .(4.10'0
m=
2.(2.10')^10.1
= l,8.10-\g=l,8g.
Vay: Kho'i lu'dng cua moi qua cau la m = l,8g.
b) Dien tich truyen them cho mot qua cau
- Khi truyen cho mot qua cau dien tich q' thi
goc giffa hai qua cau giam nen q' < 0. Vi
hai qua cau van day nhau nen (q + q') > 0.
- Dien tich cua qua cau di/Pc truyen
them dien tich la (q + q').
q.(q + q')
= mg.tan30° (r' = 1}
- ri^dngtiTcaua, t a c 6 : F ' = P t a n a ' o k
mg.tan30°.l^
q + q'
kq

1,8.10"^10.^.(2.10"')^
9.10^.4.10 -7

= 1,15.10-'C

'
19


B6i du8ng hoc sinh gioi Vat ly 1 1 , t j p 1 - Nguygn Phu D6ng

Cty TNHH M T V DVVHj^h.nj Viet

V i q > 0 ; q '
+ Cac li/c tac dung vao mot qua cau: trong lire P, lire cSng day f, va li/c

Vay: Dien tich truyen them cho mot qua cau la q' =-2,85.10"^C.

dien Fj.

1.22 Hai qua cau nho bang kim loai giong nhau treo tren hai day dai vao cCing
mot diem, du'cfc tich dien bang nhau va each nhau doan a = 5cm. Cham nhe

+ Dieu kien can bang cua mot qua cau cho: F, - P t a n ^
2
q
a,
= mg.tan —
(1)
a, T
2

tay vao mot qua cau. Tinh khoang each cua chung sau do.
Bai giai
> , ? =T

Goi q, m la dien tich ban dau va khoi liTcJng cua moi qua cau.
-

(2]sin-1)2

Trirdc khi cham tay vao mot qua cau, dieu kien can bang cua mot qua cau cho:

2

2

tana = - « —
P
21
a
~—
a^mg

Trong dien moi E:

(F = k ^ ; P = mg)
a
=> a

+ Cac lire tac dung vao mot qua cau: trong lire P, li/c cSng day

3 ^ 2kq^l

F2 va lire day Ac-si-met F^ .

(1)

mg

21

+ Dieu kien can bang cua mot qua cau cho:
a.
F2 = (P-FA)tan-

Khi cham tay vao mot qua cau, qua cau do se ma't he't dien tich, life dien giiJa
hai qua cau khong con nffa, hai qua cau se cham vao nhau va dien tich lai
du'cfc phan bo deu cho hai qua cau (q' = ~ ) ' hai qua cau lai day nhau va

= (D,-D2)Vg.tan

cau lijc nay ta suy ra:

-

2kq^

3

—

•

.
-1

q

sin

a,
tan—*-

2 «2

2
a,
2«1

n

_

D,-D,

•

D,

tan

'

sm

^

Ot,

2

-J-tan-i

sm^-^tan-^

"4
•w 3,15cm

=> a =
^

TiJ(l) va (2)suyra:

(2)

mg

(2)

a •2 \2
e(2/sin--?)

khoang each giiJa chiing la a', lu'dng tiT, tCf dieu kien can bang ciia mot qua

a

, lire dien

Vay: Gia tri cua £ theo D,, D2, a,, a2 la 8 =


sm^-J-tan—L

D
D, - D ,
'
^

Vay: Khoang each giffa hai qua cau sau khi cham tay la a' « 3,15cm. '
1.23. Hai qua cau nho giong nhau khoi lu'dng rieng Di du'dc treo bang hai day
nhe Cling chieu dai vao cCing mot diem. Cho 2 qua cau nhiem dien giong
nhau, chiing day nhau va cac day treo hdp goc a i . Nhiing he vao chat dien

• 2 "2
0I7
sm-^^tan-^
2
2

b) Dinh D, de a, = a2
Khi a, = a2 thi

E =

D,

eD,

D,-D2

£-1

i> > i i

"I .'Mi(

moi long cd khoi lUdng rieng D2, gdc giila 2 day treo la a2 < a i .
_
Vay: Gia tri cua D, de a, = aj la D, =

a) Tinh e cua dien moi theo Di, D2, a i , a2.
b) Dinh D, de aj = a,.

,

'

'
Bai giai

i>?
,

^ "f
,

20

Trong khong khi:

1-24. Hai dien tich q, = 2.10'^C va q2 = -8.10"'C dat tai A va B trong khong khi,
A B = 8cm. Mot dien tich qa dat d C. Hoi:
^) C 6 dau de q., can bang? Khi qj can bang, qj phai cd dau nhu" the nao de can

a) Tinh e cua dien moi theo Di, D2, tti, a2
-

^^2
E ^

' •

bang nay la can bang ben? khong ben?


Cty TNHH MTV DWH ku,^ ^|^,

B6i dU3ng hoc sinh gi6i V$t ly 11, t$p 1 - Nguygn Phu D6n9

b) Oa'u va do Idn cua qj de he can bang? K h i he can hKng, tW can bang cua he

va k

B a i giai

^AC^^

a) V i t r i cua C de q3 n ^ m can bang va dang can bang
-

<"

V i tri cua C

•

+

Cac \\ic d i e n tac dung l e n qf. F j j . F j , .

+

D e q3 nam can bang thi: F , , +

AB^

AC^

AB^

I ben hay khong ben?

q3 =

•
,

+
! ' '

_

= 0 => F,3 = -F23 => Fi3,F23 cung phiTdng,

q2

BC^

(-8.10"'')

vAB,

• t

= 8.10X
7

V i qi > 0; q2 < 0 => q3 < 0: qj = - 8 . 1 0 ' C .

I

Dang can bling cua he: K h i q3 < 0, can bang cua q3, q,, q2 deu la can b^ng
ben nen can bang cua he la can bang ben.
V a y : D e qi va q2 cQng can bang thi q^ = -8.10'^C va can bhng cua he la can

ngiTdc chieu va cung dp Idn: F o = F23 o

k

C
^2

^BcJ

8

4

bang ben.

BC'

AC^

F'

Chung day nhau, each nhau doan a = 3 V3 cm. T i m q ? Cho g = 1 Om/sl
B a i giai

C nam tren du-dng thang A B , ngoai doan A B , ve phia A .

-

Dang can bang:
+

K h i ba qua cau each nhau mot doan a => he can bang. V i he do'i xi?ng nen chi

• can xet mot qua cau, chang han qua cau tai C.

B C = 2 A C = 2(BC - A B ) => BC = 2 A B = 2.8 = 16cm va A C = - .16 = 8cm.
-

-

,

chieu dai / = 5cm vao cung mot diem O. K h i tich cho moi qua cau dien tich q,

Tur do:

•

,

1.25. Co 3 qua cau cung khoi lu^dng m = lOg treo b^ng 3 sdi day manh cCing

B

A

,

V d i qua cau tai C:
+ Cac lire tac dung len qua cau: cac lire dien F,3,F23; trong lire P3 va liTc

N e u q j < 0: K h i difa q j lech

cang day T 3 .

k h o i vj tri can bang thi hdp
lire (F,3+F23) se
hirdng du^a

+ Qua cau can bang nen: F j j + F23 + P3 + 73 = 6 => F3 + P3 +

CO xu

2/7

trd ve vj tri
=>F'3 = P3tana, vdiP3 = mg;F'3 = 2F,3Cos30° =

can bang cu nen day la
can bang ben.
+

Vik-

N e u q3 > 0: K h i diTa q3 lech

= mg.tana

k h o i vj t r i can b^ng thi hdp
lire ( F j 3 + F 2 3 ) se

-

CO xu

Tarn giac OGC cho: tan a =

(1)
GC
GO

hu^dng diTa q^ ra xa vj tri
v d i : G C = ^ C K = i ^ . ^ ; .
3
3 2
3

can bang cu nen day la
can bhng khong ben.
V a y : Phai dat q3 tai C, v d i A C = 8cm; BC = 16cm thi q3 se nam can bang va

==>tana=

,

^

(2)

can bang do la can bang ben hay khong ben tuy thuoc vao da'u cua q3.
b) Da'u va do \dn cua q3 de q i , q2 cung can bang, dang can bhng cua he
-

Da'u va do Idn cua q3 de he can bang
+

D e q , va q2 cung can bang thi: F21 + F 3 , = 0 va F,2 +F32 - 0
va F,2

= F32.

=> F21 = F3,

-

T i y ( l ) va ( 2 ) s u y r a :

y f s k - ^ =
mga'

= 0
2

2k3-.— =Vsk^
,2

2


B6i duSng hoc sinh gi6i Vgt ly 11, tjp 1 - IMguygn Phu D6ng

mga

= a

L6 xo CO chieu dai / ( 2 L >l> lo) nhuThinh ve. Xac dinh gia trj cua q?

0,01.10.373.10"^

= 3 ^ 3 .10^1
3.9.10%

B a i giai

na, vrs-it'nu'-i' ;

V i he C O tinh doi xifng nen ta ehi can xet mot qua cau, chang han qua cau

(5.10-^)^-^^:'?">!

ben phai (hinh ve):
Cac liTc tac dung l e n qua cau: trong life P ; life dien F ; liTc dan hoi F , ; life
.10'^ = 1,14.10^C
V a y : D i e n tich cua m o i qua cau la q = ± 1,14.10"^C.

quan tinh F^ ; liTc cang day T .

• fjiiVi-.'sri'fiCr

1,26. M o t vong day ban kinh R = 5cm tich dien Q phan bo deu tren vong, vong
dat trong mat phang thang diJng. Qua cau nho m = I g tich dien q = Q diTdc
treo bKng mot day manh each dien vao diem cao nha't cua vong day. K h i can
bang, qua cau nam tren true cua vong day. Chieu dai cua day treo qua cau la

_

Qua cau nam can bang nen: P + F + Fj + F + T = 0

_

Tir(l)suyra:

F - F, = (P + F;,)tana

i

giai

-

Cac lire tac dung len qua cau: trong liTc P ; life dien F ; liTc cang day T .

-

Qua cau nam can bang nen: P + F + T = 0 .

-

Tarn giac lU'c "gach gach" cho: F =

P

'•^isl ,. . - i ^ ^ . g

^2
3
:^ k ^ l = ^mg.

,
, vdi: F = I d F (tong cac liTc dien

.

'

UJ

3mgl

+ 2k'(l-lo)

kq
I'

V a y : D i e n tich cua m o i qua cau la

ZdQ.cosa=

kQ'
mg
—-^.cosa =
tana
kQ^

mg

mgl

sma

R

Q

V

10--\l0.7,2.10"^

= 9.10-^C.

•)OQ:ilmmT

9.10'^.5.10"^

V a y : D i e n tich cua vong day la Q = ± 9. IO"*C.

3mgl
+ 2k'(l-lo)

THirC

I. Di^ntrirt/ng

=7,2.10-1,
" kR

A. T O MT A TKIEN

=1

DIEN TRi/dNG

Chmendil:
(sina = y )

(2)

+k'(/-/o)

tana
cua cac phan tuf nho cua vong day tac dung len q)
P = mg; F = ZdF.eosa =

W i.

k ^ - k ' ( / - / o ) = (mg + m | ) . 1^

/ = 7,2cm, tinh Q.
Bai

(1)

'

1. Di^n triftfng: D i e n triTdng sinh ra bdi dien tich Q la viing khong gian ton
tai xung quanh dien tich Q va tac dung liTc dien l e n dien tich khdc dat
trong no.

M

1.27. Hai qua cau nho cung khoi \\idng m, du'dc tich dien giong nhau q. Chiing

2. CiftJfng dp dipn trvliing: CiTdng dp dien

du'dc no'i vdi nhau bang mot 16 xo nhe each dien, chieu

tn/5ng do dien tich diem Q gay ra tai diem

Q>0

dai tif nhien cua lo xo la /o, dp ciJng k ' . M o t sdi chi,

M each Q mot doan r eo:

0-—^

E

Q<0

each dien, manh, nhe khong dan, c6 chieu dai 2 L , moi

+ D i e m dat: T a i M .

dau day ehi dUdc gan vdi mot qua cau. Cho diem giffa

+ Phu-Png: Dirdng thang noi Q va M .

O cua sdi day chi chuyen dong thang duTng hu'dng len

+ Chieu: HiTdng ra xa Q neu Q > 0; hiTdng ve Q neu Q < 0.

vcti gia toe a , c6 dp Idn bang ^ (g gia toe rPi tif do).

+ Do Idn: E = - .
e r

(k = 9 . 1 0 ' ^ ( ^ ^ ) ; e: h^ng so dien m o i ) .

M


Bfii dUSng hpc sinh gidi Vat ly 11. tjp 1 - Nguygn Phu Dong

Cty TNHH MTV DVVH Khang Vi^t

3 . MO'i quan h$ giffa ctfcfng do di§n trUcfng Itfc d i ^ n trtfcfng: K h i dat dien

+ Neu(Ej,E2)=a

tich thur q trong dien trong dien truf&ng E thi q se chju tac dung ciia life
" i - - ...

dien trUtJng F , v d i : " j '

i--

Tri/cJng hdp dien tich nam can bang trong dien triTdng thi tit d i e u k i e n can

• u

+

C h i e u : q > 0 : F , E cung chieu; q < 0: F , E ngiTOcchieu.

+

Do Idn: F = IqlE.

:

4. N g u y e n l i c h o n g cha't d i $ n t r t f t f n g : N e u trong

*

bingvllirc:

-

ta CO the diTa vao phifdng phap "tam giac life", phiTdng phap hinh chieu nhiT da

E,

M

F = F,+F2+... = 6

diing cf chuyen de 1 de xac dinh cac dai lu'dng can tim theo cac dai lu'dng da cho.

khong gian c6 nhieu dien tich d i e m Q i , Q2, ... thi

Do'i vdi nhffng vat c6 kich thffdc (c6 hinh dang dac biet), de tinh cffdng dp

dien trUdng tdng h d p do cac dien tich nay gay ra

dien trffdng do vat do gay ra ta c6 the dung mot trong hai each sau:

tai d i e m M each Q i , Q2, ... Ian lUdt la r i , Vj, ... la:

+

E = E,+E2+...

1. D i ^ n

thong:

Dien

thong

(thong

liTdng

Cac/i 7: Phi^cJng phap vi phan:
•

E

dien

• TCf tinh doi xilng cua vat ta xac dinh dffdc hufdng va dp Idn cua E .

N = ES.cosa ( a la goc hdp bdi vectd E va phap
tuyen n cua dien tich S )

+

2. D j n h If O s t r o g r a d s k i - Gauss; D i e n thong qua mat k i n c6 gia trj

Cdch 2: Phu'dng phap dung dinh l i 0 - G :
•

bang

tong dai so' cac dien tich c6 mat ben trong mat do chia cho SQ :
, ; - vf^'-"".:

Chia vat thanh nhieu vat raft nho, moi vat nho do dffdc coi nhiT mot dien

do nhieu vat ra't nho (dien tich diem) gay ra: E = l A E j

tru'dng) qua dien tich S la dai lu'dng xac dinh b d i :

N = — E q ; = 47iklq;

•

Tinh dien thong: N = ES.cosa ( a la goc hdp bdi hiTdng cua E va hffdng
phap tuyen n ciia S).

•

^'

Dung djnh l i 0 - G : N = — S q ; = 47tklq;
^0

^0

B. NHtJNQ CHU Y KHI GlAl BAI TAP

C. C A C BAI T A P V £ DIEN

-

1.

Can phan biet giffa yeu cau " t i n h " va "xac d i n h " cffdng dp dien trffdng: tinh
K h i bieu dien vectd cffdng dp dien trffdng do mot dien tich d i e m gay ra can
chu y den dau cua dien tich: Q > 0 ( E

hudng xa Q ) , Q < 0 ( E

hiTdng ve Q ) .

+

N e u E j , E 2 ngi/dc chieu thi E = IE, - E2I.

+

N e u E p § 2 vuong goc thi E = ^E^+E^ .

T A C DVNG L E N D I E N T I C H D I E M

la mat dp dien mat (S: dien tich mat cau)

each be mat qua cau doan 5cm.
B a i giai

dp dien trffdng E , , E 2 t h i ta dung nguyen l i chong cha't dien trffdng de xac

N e u E p E 2 Cling chieu thi E = Ej + E 2 .

TRUCJNG

Cho a = 8,84.10''' C / m l Hay tinh dp Idn cua ciTdng dp dien triTdng tai diem

Trffdng hdp c6 nhieu dien tich d i e m Q i , Q 2 , . . . gay ra tai d i e m M cac ciTdng

+

'

s

ra vdi r la khoang e a c h tCr tam qua cau den d i e m ta xet.

trffdng tong hdp tai M can chii y cac trUdng hdp dac biet sau:

*

DO D I E N TRl/OfNG DO D I ^ N T I C H D I E M G A Y R A . LlJC

T a dat a =

dung de tinh cffdng dp dien trffdng do mot qua cau tich dien phan bo deu gay

dinh cirdng dp dien triTdng tong hdp tai M . D e tinh dp Idn ciTdng dp dien

TRUdNQ

2.1. Q u a cau bang k i m loai, ban kinh R = 5cm dffdc tich dien diTdng q, phan bo deu.

Cong thUiC tinh cffdng dp dien triTdng do dien tich d i e m gay ra cung di/dc

-

CUOING

DIEN

(tinh do Idn), xac dinh (ca d i e m dat, phffdng, chieu va dp Idn).
-

^ '

tich d i e m .
rJy->'y^r
'ii','' • ••' •
• Cffdng dp dien trffdng do vat gay ra la tong hdp cua cffdng dp dien triTdng

11. D i n h h' O s t r o g r a d s k i - Gauss

'{

va E, = E 2 thi E = 2 E , . c o s | .

Chpn mat Gauss la mat cau S' dong tam vdi qua cau, ban kinh r = 10cm.
~

D i e n thong qua m a t s ' l a : N = ES'.cos a = E S ' = E . 4 7 I T ^

~

Theo dinh l i 0 - G ta cd: N = 4 r t k l q i = 47tkoS = 47ika.47tR^ = 167r^R^ka.
=>E.47cr2 = 1 6 7 r ^ R 2 k o = > E = 4 7 i k ( - ) ^ a
=>E = 4.3,14.9.10^(—)^8,85.10-^ = 2 , 5 . l O W / m .
10

27 •

B6i duBng hpc sinh gi6i Vjt ly 11, tjp 1 - Nguy§n Phu D6ng

Vay: Dp Idn cua cUdng dp dien trUdng tai diem each be mat qua ciu doan
5cmla E = 2,6.1 OW/m.
2.2. Proton dirpcdat vao dien tru'dngdeuE= 1,7.1 OV/m.

a) H, trung diem AB.
M each A 1cm, each B 3em.
c) N hdp vdi A, B thanh tarn giac deu.

,•

i''

Bai giiii

a) Tinh gia toe cua proton, biet mp = 1,7.10 " k g .

^ Vectd cirdng dp dien triTdng tai trung diem 11 cua AB

b) Tinh van toe proton sau khi di dU'pe doan du'dng 20cm (van toe dau bang
khong).

Ta c6: E^=E^+E^
Bai giai

.

' '

Vi Ej eung chieu vdi Ej nen EH = E I + E2.

a) Gia toe cua proton: Bo qua trpng li/c tac dung vao proton, gia toe cua proton la:
F
a= —

'm

. . A H = B H - — = - = - = lcm =10-^m
vdi E, = k — i - ; E 2 = k
2
2 2
BH'
AH^

^-19

^ =
mp

1,6.10" m/s'
1,7.10'^'

.

I'

,9 4.10

. E H = 9.10^

Vay: Gia toe eiia proton trong dien tru'dng la a = 1,6.10''* m/s^.
b) Van toe proton sau khi di dU'pe doan du'dng 20cm
Taco: v 2 - V ( ^ = 2 a s =^ v = ^ v ^ + 2as =

+2.1,6.10''*.0,2

=8.10^m/s

^ ! V i electron mang dien tich am nen life dien tru'dng F tac dung len electron
se ngu'de chieu vdi chieu dien tru'dng E nghla la ngu'dc chieu vdi chieu
, chuyen dpng cua electron nen electron se chuyen dpng cham dan deu, cung
chieu vdi chieu du'dng sdc dien tru'dng vdi gia toe:

2a
-

0 - ( 4 10^)^
'
f— = 0,05m = 5cm
2.(-I,6.10'^)

Taco:

E^=E^+E^

-

V i A M = AB + B M = > M nam tren diTdng thang AB, ngoai doan AB, ve phia A.

-

V i E , ngi/dc chieu vdi E2 nen EM = | E I - E J
-10

4.10
= 36.10' V/m.
= 9.10^.
vdi E, = kAM^
(10-')'
,-10

E2= k

BM'

.= 9.10'.

= > E M = 36.10^-4.10-^

+

phufdng: diTdng thang AB.

28

, ,

= 4.10'V/m.

E,

EM

M

E^

A

+ chieu: hiTdng ra xa A (eung chieu vdi E , do E i > E2).
+ dp Idn: EM = 32.10'V/m.
c) Vectd eirdng dp dien trUdng tai diem N
Taed: E N = E , + E 2

AB = a = 2em. Xac dinh vectd eiTdng dp dien tru'dng E tai:

,

Vay: Vectd cu'dng dp d i e n triTdng tai M c6:

(a' = -a = l,6.10'Ws^) va chuyen dpng nhanh dan deu theo chieu ngiTdc lai

2.4. Cho hai dien tich q, = 4.10'°C, qz = ^ . l O ' ^ C dat d A, B trong khong khi,

01}

= 32.10' V/m

d i e m dat: tai M .

D I E N TRUOfNG

,-2x2

rV

(3.10"^)

+

2. S\J CHONG C H A T D I E N TRl/OlNG - D I E N T I C H C A N BANG TRONG

B

b) Vectd cu'dng dp dien triTdng tai diem M

Sau khi diTng lai, dU'di tac dung cua liTc dien tru'dng, electron se thu gia toe a'
(ngu'de chieu vdi dien tru'dng).

EH

+ dp Idn: E H = 72.10'V/m.

„ , £ . : £ ^ . < - ' . ' ^ - ' ° - " ) ; ' " ' > . - 1 . 6 . 1 0 " nvs'
m
m
9,1.10"^'
=

E2

+ chieu: ttj" A den B (eung chieu vdi E, va E^).

deu CLing chieu du'dng siJe. Mo ta chuyen dpng ciia eleetron sau do.
Bai giai

H

^

+ phu'dng: du'dng thang AB.

trU'cJng deu, cUdng dp dien tru'dng E = 910 V/m, V Q eilng chieu diTdng siJc
dien tru'dng. Tinh gia toe va quang du'dng electron chuyen dpng cham dan

©- -

+ diem dat: tai H.

2.3. Electron dang chuyen dpng vdi van toe VQ = 4.10^m/s thi di vao mot dien

v^-v^

E,

Vay: Vectd cu'dng dp dien triTdng tai H eo:

Vay: Van toe proton sau khi di dU'pe doan du'dng 20cm la v = 8.10'' m/s.

va quang du'dng: s =

= 72.10' V/m
i\o~^f

(10-^)^

V 'oi M - ' ' 1
s«:^>5,n,} n

e


Bfii duBng hoc sinh gidi Vat ly 11, t j p 1 - Nguygn Phu D6ng

Cty TNHH MTV DVVH KhangVigt

; N A = N B = a; a = 120° = > E N = E l = E2 = k

Vi

E N = 9.10'.

4.10 -10
(2.10-2)2

= 9.10'V/m

+

d i e m dat: tai M .

+

phiTdng: dirdng thang A B .

+

chieu: hu'dng ra xa A.

+

d p l d n : E M = 40.10'V/m.

K

^2
M

Vectd eirdng dp dien tru'dng tai d i e m N

V a y : Vectcf ciTdng dp dien trirdng tai N c6:
d i e m dat: tai N.

Tacd:

+

phi/Png: dirdng thang A B .

V i qi| = |q2|;NA = N B = a ; a = 60°

+

chieu: tiT A den B.

+

dp Idn: E N = 9 . 1 0 ' V / m .

A

E N = 2Eieos30° = 2 k ^
a

R

2.5. Cho hai dien tich qi = q2 = 4 . 1 0 " ' ° C dat d A, B trong khong k h i , A B = a =
2cm. Xac dinh vectP ciTdng dp dien triTdng E t a i :

,E„ = 2.9.I0'.

,

a) H , trung d i e m A B .

-'°
-2x2
(2.10"')'

cos30°

73

«

N

15,6.10^ V/m

2

Vay: Vectd cu'dng dp dien tru'dng tai N eo:

b) M each A 1cm, each B 3cm.

+ d i e m dat: tai N.

c) N hpp vdti A, B thanh tam giac deu.

Bai giai
a) VectP cu'dng dp dien triTdng tai trung diem H cua A B

= > E H = 9.10^4-'°

phu'dng: vuong goe vdi A B .

+

chieu: hu'dng ra xa A B .

B

A

15,6.10^ V/m.

2.6. Hai dien tich qi = S.IO'^C, qj = -S.IO^^C dat tai A, B trong khong k h i , A B =
nen E H = |E,

^2

El = k;E2=
AH^
-10

+

+ dp Idn: E N *

+E2

V i E , ngirpe chieu v d i
vdi

A

EN=E,+E2

+

Ta c6: Ej^ = E j

B

A

kBH'

-9A0'.

-E^

; AH = BH =
4.10-'°

TT

(10-')'

4cra. T i m vectd cu'dng dp dien tru'dng tai C tren trung triTc A B , each A B 2em,
AB

a

2

2

2

2

^
=0

suy ra life tac dung len q = 2 . 1 0 ' C dat d C.

Bai giai
-

Vectd cu'dng dp dien trUdng tai diem C
Ta c6: E^, = E ,

A

H

E,

+E2

B

V a y : VeetP cu'dng dp dien tru'dng tai H c6 dp Idn bang 0.

Vi

12 ; C A = C B = V C H ' + A H ' ; c o s -2 = cosA =

A H

b) V e c t d eirdng dp dien tru'dng tai diem M
Tacd:

Ej^=E,+E2

Ec = 2 E , c o s -

-

V i A M = A B + B M = > M nlm

-

V i Ej Cling chieu vdi E2

nen E M = E I + E2.

E l = kAM
E2=

= 9.10^.^-^0
(10-')'

= 36.10'V/m.

^2

k.= 9 . 1 0 ^ . ^ : 1 ° — = 4 . 1 0 ^ V/m.
BM'
(3.10-')'

= > E M = 36.10^ + 4 . 1 0 ' = 40.10^ V/m
V a y : V e c t d ciTdng dp dien triTdng tai M eo:
30

2.10,-2

= 9>^.10^(V/m)

[(2.10-')' + ( 2 . 1 0 - ' ) ' ] • 7 ( 2 . 1 0 - ' ) ' + ( 2 . 1 0 - ' ) '

^-10

vdi

A H '

( C H ' + A H ' ) ' V C H ' + A H '

8.10-

Ec = 2.9.10^

VCH' +

AH

=2k

2

tren diTdng thang A B , ngoai doan A B , ve phia A.

A H

V a y : V e c t d cu'dng dp dien triTdng tai C c6:
1+

d i e m dat: tai C.

i+

phi/dng: song song vdi A B .
chieu: tijf A den B.

It dp Idn: E c = 9^2.10^ (V/m).
31


BoiduSng hoc sinh gioi Vat ly 11, tap 1 - Mguyen Phi'i Dong

-

Cty TNHH MTV DVVH Khang Vigt

D o Idn life tac dung len q dat tai C:
Fc =

Ec = 2 . 1 0 ' ' . 9^.\0^

»

k_ii_

25,4.10"'N.

d i e m dat: tai C.

E 3 =

+

phU'dng: song song vdi A B .

'

+

chieu: cting chieu vdi E,;. (do q > 0).

+

do Idn: Fc « 25,4. l O ^ N .

'

Xac dinh vectd E tai M tren trung triTc A B , each A B = 4cm.

A H

= 10 *C; cosa = TT~-~
MA

9.10'*

(32.10"^)^

1024

10^_9T0^
36

\

1024

= 246 V/m.

Vay: D o Idn cu^dng do dien tru'dng tai H la E H = 246 V/m.

tai tarn O hinh vuong trong tru'dng hdp bo'n dien tich Ian liTdt cd dau sau:
=5cm.

a) + + + +.

b) + - + - .

c)+--+.

B a i giai

3
5

V/m.

2.9. Cho bo'n dien tich cdng do Idn q dat tai bon dinh hinh vuong canh a. T i m E

B a i giai •
4^+3'

576

/

V/m.

10,-9

= 9.10".
2

9.10'

2.7. Hai dien tich q, = -IQ-'^C, q2 = lO^C dat tai A , B trong khong k h i , A B = 6cm.

Vi

^3

CH'

=>EH =

Ta c6: M A = M B = V A H ' + H M

k-

36

(18.10"^)^

BH'

Vay: Life tac dung len dien tich q dat tai C c6:
+

1^

10-^

=9.10^

" e n Ei = E2 = k -

MA^

.4.

aV2
V i q i = q2 = q3 = q4 = q ; r, = r j = r j = r4 = - y - nen E i = E2 = E 3 = E 4 .

10"^
3
= > E M = 2 E , c o s a = 2.9.10^(5.10"^)^ 5

a) Trufdng hdp dau cua cac dien tich Ian lu'dt la + + + +:
E o = E i + E 2 + E 3 + E 4 = E,3+E24 ^ E o = 0

= 0,432.10^ V/m.

';J '

Vay: Tru'dng hdp dau cua cac dien tich Ian li/dt la + + + + t h i Eo = 0.

Vay: Cifdng do dien tri/dng tai d i e m M c6:

b) Tru'dng hdp dau cua cac dien tich Ian liTdt la + - + - :

+

d i e m dat: tai M .

+

phi/dng: song song vdi A B .

+

chieu: ttr B den A .

+

do Idn: E M = 0,432.10'V/m.

EQ

:\

= Ej + E2 + E 3 + E^ = E,3 + E24 => Eo = 0

Vay: Tru'dng hdp dau cua cdc dien tich Ian lu'dt la + - + - thi Eo = 0.
'

®^

2.8. T a i 3 dinh tarn giac A B C vuong tai A canh a = 50cm, b = 40cm, c = 30cm.
Ta dat cac dien tich q, = q2 = qj = 10"'C. Xac dinh E tai H , H la chan difdng
cao ke tiT A .

c) Trufdng hdp dau cua cac dien tich Ian liTdt la + - - +:
E o = E , + E 2 + E 3 + E 4 = E,3+E24
• Eo = 2E,3Cos45° = 2.2Eicos45° = 4 k

: '

•'

•

B a i giai

' i-u
u
u b
^
40^
= 32cm.
Ta co: C H = b.cosC = b. — = — =
a
a
50
B H = a - H C = 5 0 - 3 2 = 18cm.

a72'

^

A H = VHB.HC =

-

V32.I8 = 24cm.

^3

A ®

— E j ~i~ E2'^ •

V i E, I E 2 3 = > E H =
Vdi:

E,=

+E^3

= 9 . 1 0 ' . _ i ^
(24.10"^)^

32

:9B

D o Idn cua ct/dng do dien tru'dng tai H :
Ej ~i~ ^2

-

Vay: Tru'dng hdp dau cua cac dien tich Ian liTdt la + - - + thi Eo = 4V2 — .

' F

^^Ef+(E2-E3)2
=

^ V / m .

576

Trirdng hdp a

D

C
Trirdng hdp b

D

Trirdng hdp c

C
33


Cty TNHH MTV DWH Khang Vi$t

B6i duSng hgc sinh gi6i V$t ly 11. tjp 1 - IMguy§n Phu D6nq

2.10. Tai ba dinii A, B, C ciia hinh vuong ABCD canh a dat 3 di^n tich q giong

Ta c6: E,^ = E, +

nhau (q > 0 ) . Tinh E tai:
'

a) Tarn O hinh vuong.

, ,

E, =

b)DinhD.

E 2 = k-

. V i q, = qj = q ; A

q

_
= k

M =B M

; cosa = cosA =
a^+h^

A M '

Va^ + h^

Bai giai
^EM

a) CiTcfng do dien tru'dng tai tarn O:
-

VI q, = q2 = q3 = q; r, =

= rj =

nen Ej =

nen

= 2E|COsa = 2 k

kqa

= 2-

E2 = E3.

vay: CiTcJng do dien triTdng E M tai M tren trung triTc cua A B c6:
•

-

Eo = E , + E 2 + E 3 = E , 3 + E 2

+ d i l m dat: tai M .

V i E, \h E 3 ngiTdc chieu nen £ , 3 = 6 nen Eo = Ej.
=> E n =

k

+ phiWng: song song vdi AB. 'i an.i 'nis!
+ chieu: tir A d e n B.

_ 21cq

q

^n&'i/if ;,3 .ev •

,

^

^ ,

+ d61dn:EM= 2
(a2+h2)2

Vay: CiTcJng dp dien triTdng tai tam O la Eo =

b) Gid tri cua h de E M dat ciTc dai

2kq
a

b) CifcJng do dien triTcJng tai dinh D

V i f i = r3 = a; r2 = aV2 nen Ei = E 3 = k — ; E 2 =

A

EM

ciTc dai khi h = 0 va

EM(max) =

Vay: De
k

EM

^

a

B

ciTc dai thi h = 0 va

EM(,„ax) =

.

2a2
-

•J suy ra

(a^+h^)2

Ta c6: Ep = Ej + Ej + E3 = Ej3 + E2
-

kqa

TirEM= 2

2kq

:
q ^

a

,
'Mi

';.r'I.:

c)

2.12. Tai ba dinh ABC cua tu" dien deu SABC canh a trong chan khong c6 ba
dien tich diem q giong nhau (q < 0). Tinh do Idn cu'dng do dien tri/cJng tai dinh

Mat khac, vi E, va E 3 vuong goc nhau nen:

S cua tu" dien. Xac dinh hiTdng cua cu'dng do dien tru'dng nay.
•f

3

-

E,3 =

E.V^ =

V i E , 3 va E2

k4^
CLing

chieu nen:

Bai giai
ED =

E | 3 + E2

2a^

Ta c6: E^ = Ej + E j + E3 = E, + E23.

!

1 , kq

,

-

VI q, = q2 = q3 = q < 0; r i = rz =

-

V I a = ( E j . E j ) = 60° nen t ulri;:;, B . ( v > '

r3

= a nen El = E2

= E3

= k ^ .

2'n2

Vay: CircJng do dien trirdng tai dinh D la E D = ( \ / 2 + - ) ^ .
2.11. Hai dien tich qi = q > 0 va q2 = - q dat tai A , B trong khong khi. Cho A B = 2a.
a) Xac dinh ciTdng do dien tru'dng E M tai M tren trung triTc cua A B , cdch A B

E23 = 2E2Cos30° = 2 k W. : / l = Vik 4
a^ 2-

''

E23 n\m tren duTdng cao SH cua tam giac SBC.

doan h.

b) Xac djnh h de E M dat ciTc dai. Tinh gid tri ciTc dai nay.
Bai giai
a) CiTcJng do dien trUdng E M tai M tren trung trifc cua A B

Suy ra: E^ = E^ + E23 + 2EjE23Cosp,
v<3i cosp =

, -

;

SH^+SA^-AH^
2SA.SH
35


Bi5i du3ng hpc sinh gi6i Vat ly 11, tjp 1 - Nguyin Phu Bfing

2

Cty TNHH MTV DVVH Khang Vi§t

va EBD- = 2EBCOsa2 = 2.

2

cosP =
2a.

V I E ^ c '^""S

2

^/I

+ 2.

k4
V

8^3 k q
9

kq

3
= EAC + EBD'

1673 k q
9

a^

Vay: Do Idn ciTdng do dien triTdng tai tarn O iiinh lap phiTdng la j . ,;

^ y

I6V3 k|q|

va E 5 hiTdng ve tarn tam giac ABC.

.Es= A / 6 -

Vi

^B'D'

8V3 k q
> Eo —
9

N2

= 6

4 kq
3 a2

a

y

2.14. Cho hai dien tich diem qi va q2 dat d A, B trong khong khi, AB = 100cm.
Vay: VecW ciTdng do dien triTdng tai dinh S cua tu" dien c6:

Tim diem C tai do ciTdng do dien triTdng tdng hdp bang khong vdi:
a) qi = 36.10-^C; q2 = 4.10-^C.

+ dpldn:Es=

Bai giai
a) Khiq, = 36.10-'^C;q2 = 4.10-''C

+ hirdng: tiT S den O (ban doc tiT chUng minh!).
2.13. Hinh lap phiTdng A B C D A ' B ' C ' D ' canh a trong chan khong. Hai dien ticii
Qi = q2 = q > 0 dat d A, C; hai dien tich qj = q4 = - q dat 6 B ' , D ' . Tinh do Idn

Ta c6: E^ = Ej + E j . De E^ = 0 => E, = - § 2 , siiy ra:
+ C nam trong doan AB (vi q,, q2 ciing dau).

curdng do dien triTdng tai tam O hinh lap phiTdng.
+ E| = E2 <=> k

Bai giai
Ta c6:

EQ

= E^ + Ec + E 3 , + E^. = E^^ + ^g.^,
=

7a2+(aV2)2

BC

= aVs i
= > A O = C O = B'0

V

^2

=3

vaAC + BC=:AB = 100cm

= D'0=

—
2

=

—^-^

BC

36.10 -6
-6
4.10

AC
A C = A ' C = VAA'^+A'C^

^2

=k
AC^

s

vdi

b) q, = -36.10-'C; q2 = 4.10-*C.

.

C
(1)
(2)

=> AC = 75cm va BC = 25cm

—
2

Vay: Khi q, = 36.10-'C; q2 = 4.10^^C, de E^ = 0 thi AC = 75cm va BC = 25cm.
b) Khi q, = -36.10'C; qj = 4. lO'^C

AO = CO

ndn

EA

= Ec =

CC'_

a

_73

= k
rr\2
.71

3

a2

36

V

y

4 kq

7^

3 a2

3

BC
873 k q

=k

k
AC"

B

^2

^2

1 4.10 -6

= 3

—>

e-

BC

36.10 -6

AC

2

= > EAC = 2 E A C 0 s a i = 2.

+ E, = E, o

4 kq

f

,i .

+ C nam ngoai doan AB, ve phia B (vi q,, q2 trai dau; q, > q 2 ) •

CA'"a73" 3

EB- = ED-

, ,

Taco: E ( , = E | + E 2 . D e E^-^O => E, = - E 2 , suy ra:

= B'0 = D'0;

costti = cosa2 =

B

(3)

1 fllH

I 'i

v a A C - B C = AB = lOOcm
(4)
=> AC = 150cm va BC = 50cm
V§y; Khiq,=-36.10-^C;q2 = 4.10-'^C, de' £ ^ - 0 thi AC = 150cm va BC = 50cm.

37


Cty TNHH MTV D W H Khann Vi$t

B6i du3ng hqc sinh gidi vat ly 11, t?p 1 - NguySn Phu D6n9

B a i giai

2.15. Cho hai dien tich q,, q2 dat tai A va B, A B = 2cm. B i e t q, + q2 = 7.10^^C va
Cdc lyc tdc dung len hon b i :

d i e m C each qi 6cm, each qa 8cm c6 ciTdng do dien triTcJng E = 0. T i m qi, q2:f. ;.;

B a i giai

,

+ Trong li/c P = mg (hi/dng xuo'ng).

.*

Ta c6: + A B + BC = A C => C nam ngoai doan A B nen qi va q2 trai dau.

= Ej +

+ Li/c dien tru-dng: F = qE (hiTdng xuong neu q > 0; hiTdng len neu q < 0).

= 0 => E, = E2 <=> Ic

BC^

8^

H6nbinimcanbang(lc(lijrng)khi:
AC^

BC'

E = mg-DVg

36 ~ 9

16
q2 = - - q ,

B

E
Viq<0

va q, + q2 = 7.1 O^^C =^ q, =-9.10"^C va q2 = 16.1 Q-^C.
V a y : Gia trj cac dien tich q,, q2 la q, = -9.10^^C va

4,1.10'

Hai qud cau nho A va B mang nhiJng dien tich Ian lUdt N

B a i giai

-

a"

2

= V2k

=> k

dai bang nhau. Hai d i e m treo day M v^ N cdch nhau 2cm;
de du'a cde day treo trd ve vj tri thang dufng ngu'di ta phai

*

dung mot dien tru'dng deu c6 hufdng n i o va do Idn bao

B

nhieu?
B a i giai

a'
-

De E D = 0 thi phai dat tai B dien tich q ' sao cho E2 = E ^ .
' => k BD'

2.10^'C va 2 . 1 0 ' C diTdc treo d dau hai sdi day t d each dien
khi can bang, vj t r i cac day treo c6 dang nhi/ hinh ve. H o i

gay ra tai D la: E j j = Ej + E 3 .

V i q, = q3 = q; A D = C D = a nen E.j = 2EiCOs45°.

S

^

Vay: D i e n tich cua bi de no can bhng Id lufng trong dau la q = -2.10"'C.

= 16.10"*C.

dat d B dien tich bao nhieu de curing do dien tru'dng d D bkng khong?

q

•

^

nen q = - 2 . 1 0 ' C .

2.16 Cho hinh vuong A B C D , tai A va C dat cac dien tich q, = qj = q. H o i phai

= ^ E n = 2k

' ,

^ m g - D ^ g = 9.10"^ 10-800.10'*'. 10 ^ 2 10'^C

(1)

Cu'dng do dien tru'dng do qi,

P+ F^+F =0 O F + F = 6

V I P > FA nen P ' = P - FA => F phai hu'dng len => q < 0 va F = P - FA.

64^_I6

AC^

-

1

+ Lire day A c - s i - m e t F ^ - - D V g (hiTdng len)

+ B C > AC
Vi

,

can phai tac dung liTc dien tru'dng ngUdc

E„ ^

= V2k

De diTa cac day treo trd ve vj tri thang dilng
chieu vdi liTc tmh dien va ciing do Idn vdi
lire tinh dien: F ' = F .

= V2k
=>q'=-2^q.

V a y : Phai dat d B dien tich q ' =

, ,5;
„2

-

E,3

.

-l4lq

V d i qua cau A :

Gid s i q > 0
de cu'dng do dien triTcJng 3 D bang

khong.
2.17. M o t hon bi nho bang kirn loai dU'cIc dat trong dau. Bi c6 the tich V = 10mm\
khoi lU'cfng m = 9.10"'kg. Dau c6 khoi liTcfng rieng D = 800kg/m\t ca difdc dat

E = kAB'

iJi;bl!!i .,0,

2.10-^
= > E = k= 4,5.10" V/m.
= k• = 9.10^.
AB'
MN'
(2.10"^)^
VI q, < 0 nen E ngiTdc chieu v d i F

nghla la cilng chieu v d i F (hirdng tuf

^ a i sang phai).
V d i qua cdu B: TiTdng tir.

^ \

trong mot dien tru'dng deu, E hudng thang duTng tif tren xuong, E = 4,1.10^V/ni- ^%r. De dira cac day treo trd ve vi tri th^ng diJng can phai dung mot dien
trirdng deu c6 hiTdng tCr trai sang phai va cd do Idn E = 4,5.10" V/m.
T i m dien tich cua bi de no can bang Id lijfng trong dau. Cho g = lOm/s^.
39


Cty TNHH MTV DWH Khang Vi$t

Bfli duang hoc sinh gi6i Vjtt 1^ 11, tjp 1 - Nguygn Phu B6ng

Cl/dNG D O D I E N TRl/OfNG D O V ^ T MANG D I E N CO KICK THl/OfC
TAORA

2.19. Mot ban phang rat Idn dat thang dilng, tich dien deu vdi mat dp dien mat a,
a) Xac djnh E do mat phang gay ra tai diem each mat phang doan h. Neu dac
diem cua dien truTdng nay.
^
b) Mot qua cau nho kho'i lUdng m dien tich q cung dau vdi mat phang, diTcfc treo
vao mot diem co djnh gan mat phang bang day nhe khong dan, chieu dai /.
Coi q khong anh hu'dng den sif phan bo' dien tich tren mat phang va khi can
, bang day treo nghieng goc a vdi phu'dng thang di^ng. Tinh q.
Bai giai
a) Cu'dng do dien tru'dng do ban phang gay ra
*
Chon mat Gauss la hinh tru c6 dudng sinh vuong g6c vdti day, hai day hinh
tron CO dien tich S va each deu ban phang doan h.
- Dien thong qua mat Gauss: N = Nj + N2.
+ Phan dien thong qua mat ben: N| = ZEiAScosai = 0 (vl cosai = 0).
+ Phan dien thong qua hai day: N2 = SEiAScosa2 = 2ES.
=> N = 2ES
3.

-I, Theo djnh li Ostrogradski - Gaus: N =
2ES = — laAS =
^0

a.2S

+
<

E=
28„

Vay: Cu'dng do dien tru'dng do mat phang gay
ra tai diem each mat ph^ng doan h:
+ la dien triTctng deu, c6 hu'dng vuong goc

•

+

J
•

E

+ khong phu thuoc vao khoang each ttr diem
ta xet den ban ph^ng.
b) Tinh dien tich q
- Cac life tac dung len q: trong life P, liTc dien
tru'dng F, liTc cang day f .
2mgeo

^'i^^

-1

= E2

<—

+

+

- Vdi hai mat phang: E = Ej+E2
+ Ben trong hai mat phang: E, va E2 cung chieu nen
E = E, + E , = 2- g _ g
; g.,
2e 0 ~ e"0
+ Ben ngoai hai mat phang: E, va E, ngUdc chieu nen
a = 0.
2e.
b) Tru'dng hdp hai mat phang hdp vdi nhau goc a
Vi El = E2 nen:
+ Ben trong hai mat phang:
„-„.a
- CT . a
a . a
E = 2Eism — = 2
. sm — = — .sm — . .
a
a
a
2
28o
2 So
2
.cos— = — cos—.
2 So
2
+ Ben ngoai hai mat phang: E = 2E1COS— = 2 —
E = E, - E, =

•

mg

2mgSQ.tana
Vay: Dp Idn cua dien tich q la
2 20. Tinh ciTdng dp dien tru'dng gay bdi 2 mat phang rpng v6 han:
a) Dat song song, mat dp dien mat a > 0 va -a.
b) Hdp vdi nhau goc a va c6 ciing mat dp dien matCT> 0.
Bai giai
' Si'H'a) Tri^ng hdp hai mat phang dat song song
• Vdi mot mat phang: Chpn mat Gauss la hinh tru c6 dtfdng sinh vuong goc vdi
ddy, hai day hinh tron c6 dien tich S va each deu ban phang doan h.
+ Dien thong qua mat Gauss: N = ZEiAScosa2
< ,•
+ = 2EiS.
+ Theo dinh li Ostrogradski - Gaus:
+
N=-Iqi
h
=> 2E,S = — l a A S = — CT.2S
28,

h

vdi vdi ban phang, c6 do Idn E =

- Tam giac liTc cho: tana = — =

2mg£o.tana

Vdi q > 0

41


Cty TNHH MTV DWH Khang Vi$t

B6\g hgc sinh gi6i Vjt ly 11, tgp 1 - Nguygn Phu P6ng

2.12. Mot ban phang rong v6 han dMc tich dien va dat vao mot didn tri/cfng deu.
Biet cifdng do dien triTdng tong hOp d ben trii vh ben phai cua ban \h Ei, E2
hufdng vuong goc vdi ban, dp Idn Ei vh E2. Hay tinh mat dp dien mat a ciia
ban va life dien tac dung len mot ddn vj dien tich ciia b^n.
)f»n **
a) Mat d6 dien mat cua bin ph^ng
lJ4^4k(»:'
Chpn mat Gauss la hinh tru c6 di/dng sinh vu6ng gdc vdi day, hai day hinh
tron cd dien tich S va each deu ban ph^ng doan h.
- Dien thong qua mat Gauss: N = Ni + N2.
sf + Phan dien thong qua mat ben: Nj = ZEiAScosai = 0 (vi cosai = 0). ' '
+ Phan dien thong qua hai day: N2 = Z E i A S c o s a 2 = E i S + E 2 S = ( E i + E 2 ) S .
=> N = ( E | + E 2 ) S

+

E,

- Theo dinh li Ostrogradski - Gaus: N = — Tq,

4-

^0

<

• => ( E l + £ 2 ) 8 = — S a A S = —
^0

^0

r

+

Vay: Mat dp dien mat a cua ban la a = eo(Ei + E2).
b) LiTc dien tac dung len mot ddn vj dien tich cua ban

-

^0

Lr:;^ M

=>E =

1
1

2ne(,r
Vay: CiTdng dp dien tru^dng gay bdi mot
day thang dai v6 han tich dien deu la
E=

1

•

E2

27re()r
2.23. Hai day dan thing dai v6 han dat song song trong khong khi cich nhau
doan a, tich dien cung da'u vdi mat dp dien dai X.
a) Xae dinh E tai mot diem trong mat phlng doi xiJng giffa hai day, each mat
phang chufa hai day doan h.
b) Tinh h de E ciTc dai va tinh gia trj cure dai nay.
''*'
a) CiTdng dp dien triTdng tai mot diem trong mat phing doi xilng giffa hai day
Chpn hai mat Gauss la hai hinh tru true la eac day dan, hai day ede hinh tru
(1)
2)
la hinh tron cd ban kinh r, chieu cao /.
r, = r2 = r =

= eo(Ei + E2)E2.
Vi F,' ngiTdc chieu vdi F2 nen F' = F, - F j .
F2' = a E j

.2.

Vay: LiTc dien tdc dung len mot ddn vi dien tich cua bdn la F' = — E f - E ^
2 " dien deu
2.22. Tinh cufdng dp dien triTdng gay bdi mot day th^ng dai v6 han tich
(mat dp dien dai X) tai diem each day doan r.
B a i giai

Chpn mat Gauss la hinh tru dong true vdi day, hai ddy hinh tron c6 hin kinh
r, chieu cao/.
re - Dien thong qua mat Gauss: N = Ni + N2.
/
• I: /
+ Phan dien thong qua hai day: N| =i;EiASeosai = 0(vicosai = 0).
+ Phan dien thong qua mat ben: N2 = ZE,AScosa,2= ES = E.27tr/.

E = 2EiCosa = 2.

>

h +—

nen E, = E 2 = E = (bai tren ...)
27xeor

Ef-E^
1

^ N = E.7tr/

=—

^0

YiXi=X2=X;

, vdi F , ' = a E , = eo(E, + E 2 ) E , ,

=>F = S(,(E,+Ej)E,-e„(E,+Ej)E,

E.Tir/ = — Lqi

N =—

B a i giai

i

=> a = 8o(Ei + E2)

Ta cd: F = F, +

Theo dinh H Ostrogradski - Gaus:

X

h

Xh

27ieor r ^ ^ r ^

(h^ + ^ - )
4

1

.1-

I
I

I
/ I
i

TriTdng hdp X >0

Vay: Ci/dng dp dien trUdng tai mot diem trong mat phing doi xtfng giffa hai
day, each mat phing ehiJa hai day doan h la E = ^
4

b) Gia tri cua h de E eifc dai
43


Cty TNHH MTV DWH Khang Vi?t

B6i duSng hgc sinh gi6i Vgt ly 11, tgp 1 - Nguygn Phii D6ng

Vay: Cu:dng dp dien truTdng tai diem each tam qua cau doan r k h i r < R la

X

TCrE =

• => E = E„,ax k h i M
^ 0

"^0

, a
(h^+—)
4
,u2

=

a

.3

h + — min => h = ~
4h

E ^ ^ ;
38

4h

2 25. Ben trong mot qua cau mang dien vdi mat dp dien khoi p c6 mot I 6 hong hinh

1
va E„,ax =
- 0

k h i r > R la E =

( a ^ ^ ^

c l u . Xac dinh dien tru'dng tai mot diem bat k i cua lo hong trong triTdng hdp:

TOoa

a) Lo hong c6 cung tam vdi qua cau.
2

4.^
2

w T a m Or cua qua cau each tam O2 ciia 16 hong mot khoang d.
Bai giai

Vay: Gid t r i cua h de E ciTc dai la h = - va Emax = — ^ •
2
TtEga

a) Tru'dng hdp lo hong c6 ciing tam v d i qua cau

2.24. Qua cau ban kinh R tich dien deu vdi mat do dien k h o i p va dat trong

_

Gpi Ej la ciTdng dp dien triTdng do qua cau

khong k h i . T i n h ciTdng do dien tri/dng tai d i e m each tam qua cau doan r

dac (khong c6 lo hong), mat dp dien khoi p

(trong va ngoai qua cau).

gay ra tai d i e m M ; E j la ciTdng dp dien
Bai glai

tru'dng do qua cau dac (c6 kich thUdc bang

Tac6:

lo hong), mat dp dien k h o i - p gay ra tai

+

Du'dng sijrc dien triTdng la nhiTng du'dng thang hudng doc theo ban kinh qua cau.

d i e m M . Theo nguyen l i chong chat dien

+

D p Idn ciTdng dp dien tru'dng tai cac d i e m nam tren cCing mat cau c6 gia tri

tri/dng, ta c6: E,^ = Ej + E j

nhiT nhau.
-

Chpn mat Gauss la mat cau dong tam vdi qua cau tich dien:
-

Theo ke't qua bai tren, ta c6: Ej = - ^ . f ; E j
3£A

D i e m M nam ben trong qua cau: r i < R:
+

Ef^ = — . r

D i e n thong qua mat cau S , (ban kinh r,) la: N = E S i = E . 4nrj^ .

3EO

+

Theo dinh l i Ostrogradski - Gauss: N =

—Zqi

+ — . r

= 6

3Sn

(r la khoang each tiT hai tam chung O j , O2

36(5

de'n d i e m M )
V a y : K h i O i = 0 2 t h i E^, = 0 .

,,
p. — Ttr;
pV _ ^ 3 1
E.47tr/

=
^0
"0

-

''O

^- -

=>E=-H1

O

b) Tru'dng hdp tam O i cua qua cau each tam O2 cua lo hong mot khoang d

R

3en
-"'O

-

Tirpng tir, ta c6: E , = —

+

D i e n thong qua mat cau S2 (ban kinh T2) la: N = E S 2 = E. 4 ^ 2 .

+

Theo dinh l i Ostrogradski - Gauss: N = —

p.-7tR^

Zq,

-

=

(r, = O ^ ; ^ =

O^M)

38.

0,M

V l E , ~ r , ; E 2 ~ r 2 = > ^ = ii-= - ^
E2
rj O 2 M
hai tam giac O1O2M va M P Q dong dang, tijT do:

R
O

E =

.rj;

38

D i e m M nam ben ngoai qua cau: Vi > R:

r

M

d
=>EM

r,
=

pd

pd

3

r..
45


Cty TNHH MTV DWH Khang Vi$t

B6\g hpc sinh gi6i Vat ly 11. tjp 1 - Mguyen Pliu Dony

Vay: Khi tam O i cua qua cau each tarn O2 cua I 6 hong mot khoang d thi

3, Di?n tM' gSy ra hdi cac di$n tich diem

CO chieu tuf O, den O2 va c6 do Idn EM = — . '•' ' ' •

_ Dien the f^dy ra bdi mot dien tich diem Q:W=—.—

k Q

(

= 0)

e r
(r la khoang each tijf dien tich diem Q den diem ta xet)

2.26. Mot v6 cau ban kinh trong Ri, ban kinh ngoai R2 mang dien tich Q phan bo

_ Dien the f^dy ra bdi h$ di$n tich diem Q,, Q2,

deu theo the tich. Tinh cU'dng do dien trU'cJng tai ncfi each tam qua cau doan r.

Goi V i , V2,... la dien the

do cac dien tich Q], Q2,... gay ra tai diem M trong dien trUdng. Dien the
-

Thetichvocaula: V = V 2 - V , = ^7r(R^-R^).

-

Mat do dien tich kh6l cua v6 cau la:

toan phan do he dien tich tren gay ra tai M la:
•

"

-

-

• ^

V = V , + V 2 + . . . . = SVj

•••1 „

UMN

4;r(R5-Rf)

gii?a cifofng dO di^'n trift/ng va hi$u di$n thfi'
= Ed

^

Tai diem ben trong qua cau ( r < R i ) :

E la ci/dng do dien tru'dng cua dien triTdng

E = 0(xem bSi 2.25).
-

Tai diem ben ngoai v6 cau (r > R2):
DR^

E =

,

. _

M , N la hai diem tren cCing mot diTdng siJc;
-

IV''

He thurc tren la npi dung cua nguyen 11 chong cha't dien th6'.

4. LiSn

V

..;.Vi.

•'

• ^'
M

N

deu; d la khoang cdch giffa hai diem doc

^—=^

theo mot di/cfng suTc c6 hieu dien the la U.

^

^

I L T h ^ ' n a n g t l n h di$n

(xem bai 2.24).

The nang cua dien tich q dat tai diem M trong dien tru'dng dac tri/ng cho kha
nang sinh cong cua dien triTdng khi dat dien tich q tai M :
-

Tai diem trong v6 cau (Ri < r < Rj): E =

(xem bai 2.24).

^ osdT

W, = qV
B. NHinSQ C H U Y KHI G l A l B A I T A P

ChuwndeS:

D I E N T H E VA H I E U D I E N T H E

-

k Q
C6ng thiJc tinh dien the gSy ra bc(i mot dien tich diem (V = - . - ^ ) cung diTdc
e r

A . T 6 M T A T KI£N T H C C
I. Di^nth^-Hi$udi$nthg'

O ^

i p dung cho qua cau tich dien phan bo deu vdi r la khoang each tiif tam qua

> -fe f

cau den diem ta xet.

1. Di^n the': Dien the tai diem M trong dien triTdng dSc triTng cho dien

-

tru'dng ve mat diT trCT nSng liTdng va diTcfc do b^ng thiTdng so' giffa cong de
diTa mot dien tich q tuf diem M ra xa v6 ciTc va di$n tich q: V M =

trufdng khong phu thuoc vao dang quy dao di

.

2. Hi^u di^n th6': Hieu dien the giCTa hai di^m M v^ N trong di6n tnrdng dSc

,

..-^-'/f
- 7 ^

2?r

cua diem dau va diem cuo'i cfla quy dao: A = qU.

7^—>•

~ Mo'i quan he giila cong ciia liTc ngoai A ' v^ cong
cua lire dien trircJng A: A ' = - A = - q U .

diTdc do b^ng thi/dng so giffa cong cua life dien lam di chuyen mot dien

•~ Doi vdi vat dan can b^ng di6n can chij ^:

^ M.

-N'

+ Vat dan la vat d^ng the: Cdc diem ben trong vk trdn mat vat dSn c6 c&ng

tich q tit diem M den diem N va do Idn cua dien tich q:
_

^

chuyen cua dien tich ma chi phu thuoc vao vi tri

triTng cho kha nSng thiTc hien cong cua dien trifcfng giffa hai diem d6 va

A^^, .

N

Lire dien triTcJng 1^ life th6' nen cong cfla liTc dien

, dien the.
+ Di?n tich chi phan bo 6 mat ngoai vat dan, tap trung 3 nhffng ch5 loi va nhon.

A
-

The nang tUdng tdc cua h? dien tich diem: Vdi h? g6m cdc dien tich di^m Q i ,

Q 2 , - , the nang ciia h? la:


B6i dugng hpc sinh gi6i vat ly 11, tap 1 - Nguygn Phu D6ng

Cty

W = ^ (Q,V, + Q2V2 +....) = ^'^Qi\ = 1, 2,..., n)
= ^ " M N
kQ
kQ
(Vi = —-+—-+.... la dien the tai diem dat Qj do cac dien tich khdc cfla he
gay ra)

AW,

250.1,6.10"'^

q

-1,6.10""^

,.

TMHH

M T V DVVH

., / .

VSy: Hieu dien the'giu'a hai diem M va N trong dien trirdng la U M N = -250V.
3 4 Electron chuyen dong khong van toe dau tCf A den B trong dien trU'dng deu,
U B A = 45,5V. Tim van toe electron tai B .

l.^.,
......
+ Trirdng hdp he 2 dien tich: W = - ( Q , V , + Q2V2), vdi V, =

, V2 =

Khanu_ViAi

kQ,

•

^U.J

Bai giai

'

IM-..-

:;J

I

,s

fil;^

mv^
i,a

+ Trirdng hdp he 3 dien tich: W = - ( Q i V i + Q2V2 + Q 3 V 3 ) , vdi
2
kQ,
kQ,
kQ, kQ,
kQ, kQ,

Ta cd: Cong cua lire dien trirdng: A = q U ^ = AW^ =

(UAB = - U B A = ^5,5V)

2.(-l,6.10"-).(-45^^^^,^.^^^^^_
m

9,1.10"^'

VSy: VSn toe cua electron tai B la V B = 4.10^ (m/s).
C. C A C BAI T A P V £ DIEN T H ^ - HIEU DIEN T H ^

1. CONG C U A LVC D I E N - D I E N T H E , H I E U D I E N T H E
3.1. Hieu dien the giOfa hai diem M , N trong dien triTdng U M N = lOOV.
a) Tinh cong cua liTc dien trU'dng khi mot electron di chuyen tiT M de'n N.
b) Tinh cong can thiet de di chuyen electron tCr M den N.
Bai giai

3.5. Electron chuyen dong quanh nhan nguyen tuT hidro theo quy dao trdn ban
kinhR = 5.10-''em.
a) Tinh dien the tai mot diem tren quy dao electron.
b) Khi electron chuyen dong, dien trU'dng cua hat nhan cd sinh ra cong khong?
Tai sao?
Bai giai
a) Dien the tai mot diem tren quy dao electron

a) Cong cua liTc dien trU'dng
Ta cd: A = qUwN = -1,6.10~". 100 = -1,6. l O " J.

Tac6:

Vay: Cong cua life dien trirdng khi mot electron di chuyen tir M den N la

v,M.glO''.^'°-".28.8V
E.r
5.10""

Vay: Dien the' tai mot diem tren quy dao cua electron la V = 28,8V.

A = -1,6.10-'^ J.
b) Cong can thiet de di chuyen electron tiT M den N: A = - A = 1,6.10"''' J.
3.2. De di chuyen q = lO^C tif rat xa vao diem M ciia dien trU'dng, can thirc hien
cong A ' = 5.10"'j. Tim dien the d M (goc dien the d 00 ) .

b) Dien trU'dng cua hat nhan cd sinh cong khong?
Khi electron chuyen dong, dien trirdng cua hat nhan khong sinh cong vi
electron chuyen dong theo mot quy dao khep kin.
3.6. Dien tich Q = 5.10'C dat d O trong khong khi.

Bai giai
Ta cd: Cong can thirc hien: A = - A = -q(

V

-

) = -10^(0 - V M ) = 5.10"^ J

a) Can thiTc hien A ' , bao nhieu de diTa q = 4.10^C tiT M each Q doan r i = 40em
den N each Q doan r2 = 25cm.
^) C i n thiTc hien cong A'2 bao nhieu de du'a q tijf M chuyen dong cham ra xa v6
Cling (rj =

00).

Vay: Dien the d diem M la V M = 0,5V.
3.3. Khi bay qua 2 diem M va N trong dien trirdng, electron t5ng toe, dong nSng
tang them 250eV ( l e V = Ue.lO"'"^;). Tinh hieu dien the giffa M va N.
»#
^^'^iai
" ' Ta cd: Cong cua liTc dien trirdng la A = q.U^g = AW^

^) Cong dira q tir M den N

*' '

7

^ Tacd:
g

_g r!

+ Dien the'tai die'm M : V ^ = k-5- = ^"^^ -^-'^
^
e.rj
0,4

'.a

= 112,5 V

»'.'

V.

< .:

49


Cty TNHH MTV DVVH Khang Vigt

B6i duSng hgc sinh gi6i Vat ly 11, tjp 1 - Nguygn Phu D6n9

E)6 bien thien nang liTcfng cua he:
+ Dien the tai diem N:
-

=k—^=
^
8.r2

= 180V

= W2 - W, =-0,108 + 0,27 = 0,162 J > 0 : nang lu-dng ciia he tang.

0,25

Vay: Khi di chuyen hai dien tich ra xa nhau thi nang liTdng cua he tang.

C6ng can thiTc hien de diTa q tuT M den N: A ' , = - A = - q . U M N = - q ( V M - V N )

3 9. Co the tich dien cho vao mot vat dan c6 lap den mot dien the toi da la bao

= ^ A ' , = - 4 . 1 0 " \ 12,5-180) = 2,7.10^J.

nhieu khi chieu vao vat mot chiim tia electron, bay vdi van toe v? Khoi liTcJng

Vay: Cong can thiTc hien de diTa q tuf M den N la A ' , = 2,7.10'*^;.
b) Cong can thiTc hien de diTa q tCr M ra v6 Cling

jn va dien tich e cua electron coi nhUda he't.

, ,

Bai giai

Tac6:V^=0
=> A ' . = - A = - q U ^ ^ = - q ( V ^ - V J = - q V ^

'

"'

Cong can thifc hien de tich dien cho vat dan: A = - A = - q V =

-

mv^
=>-qV=—
2

-hi?

A'2 = -4.10'.112,5 =-4,5.10"^ J.
Vay: Cong can thiTc hien de diTa q tCr M ra v6 cung la A'2 = -4,5.10"^ J.
3.7. Tinh the' nang cua he thong hai dien tich diem qi, q2 each nhau khoang r
trong chan khong.

mvp
2

mv^

=>V,„ax

mv^ ^ , , .

= — - = - — (khi Vo = 0).
-2q

2e

' .

VSy: Co the tich dien cho vat dSn c6 lap den dien the toi da la V =
Bui giai

Taco:

2e
3.10. Electron d each proton doan r = 5,2.10"'cm. Muo'n electron thoat khoi siJc

+ Dien the do qi gay ra: V = k —
r

hut proton no can c6 van toe toi thieu la bao nhieu?

+ The nang cua he dien tich qi, q2: W = q2V =

,

^^^^.

Vay: The nang cua he dien tich q,,q2 la W =
*

r
CM y: Co the dting cong thifc tinh the nang cua he 2 dien tich:
W = 1 (q,V, + q2V2), vdi V, = ^

2

, V2 =

er^,

er^i

eri2

Baigiai
-

Cong cua dien tru-dng tac dung len electron:
,
^,
ke ke^
A =qV=e— =

' '

"

' !
'

- De electron thoat khoi siTc hut proton thi: Wd > A.
mv^

^

=>

er,2

ke^
>

2

nenW=i(q,>.q,H).Hq2
2

.,

r

=> v > J
V mr

o

|2.9.10^(1,6.10~'^)^ , ^ , ^ 5 . , ^
v> J
^=3,2.10^(m/s).
^ 9,1.10"''.5,2.10""

Vay: De electron thoat khoi siJc hut proton thi electron phai c6 van toe toi

r

thieu la v = 3,2.10V/s).

3.8. Hai dien tich q, = 2.10'^C, qj = -3.10"^C each nhau 20cm trong khong khi. Di

3.11. Trong nguyen tO" hidro, electron chuyen dpng quanh hat nhan theo quy dao

chuyen hai dien tich de chiing each nhau 50cm. Nang li/dng cua he hai dien

tron ban kinh R = 5.10~'cm. Tinh nang li/dng can cung cap de ion hoa nguyen

tich tang hay giam. Tinh do bie'n thien n^ng lu'dng cua he.

tur hidro (du'a electron ra xa v6 cifc).

. r . L,
-

Baigiai

The nang ban dau cua he hai dien tich:
^ V ^ ^ kq.q^ _9.10^2.10-^(-3.10-^)_
r,
0,2

-

Bai giai

The nang liic sau cua he hai dien tich:
kq^^9.10^2.10-^(-3.10-^)^
h

0.5

•
^

^

tw

Electron chuyen dong xung quanh hat nhan theo quy dao tron du'di tac dung
cua lire hu-dng tam, life nay chinh la liTe Cu-16ng.
p

1

mv^

ke^

.

,

mr
1
,
1 ke^
Dpng nang ciia electron: W. = —mv^ = —m2
2
mr

ke^
2r

.