1. The document discusses concepts related to electricity and magnetism including electric fields, electric potential, capacitors, and circuits. 2. Key points include the relationship between electric field strength and charge/distance, methods for calculating electric field and potential, and how electric potential energy depends on charge and electric potential. 3. Formulas are provided for calculating electric force, electric field, electric potential, and capacitance in various circuit configurations. Examples are worked through applying the formulas.

1.    SchoolDD.com
15
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2. 15
15.1 ˂
˂ F F
F F F ˂ F F
F F F F F
F
ˈ ˂ F
1. ˂ 2
- ˂ (+)
- ˂ (-)
2. F ˂ F ˈ 2
- ˈ F F
- ˈ F
.
.
.
F
F F F
F ˆ F www.schoolDD.com 1

3. ˈ
ˈ ˈ ˂
ˈ
F F F
F F
F F F ˈ ˂ F
F F
F F
F ˈ
ˈ
F ( F F F ˈ
F F ˈ )
F + • F
• F
• F F
•
• F
• F
• F ˂
•
•
- •
F
F F
F F
F ˆ F www.schoolDD.com 2

4. F F
F F F F
ˈ F
15.2 F ˂
F FF
F ˂ F F ˈ F F F ˈ F
F
- ˈ ˂ F
- ( > )
- ( > )
- F
- F F F ˂ F F ˈ ˂
15.3 ˂ ˂
˂ (Conductor) F ˂ F F F
F
˂ (Insulator) F ˂ F F F F
- F F F
FF F F F F F
( ) ˈ
F F F F ˂
F ˆ F www.schoolDD.com 3

5. F ( ) F F F F F ˂
F F F F
15.4 ˂
˂ ˈ F ˈ ˂ ˂
F F ˈ F F F F
F F F F F F
( F ) F F
F (- -) (- +) F
> F F F
F F F
F ˆ F www.schoolDD.com 4

6. F ˂
˂ F
ˈ F F F F
˂ ˂ F
F
15.5 F F
F F FF F ˈ F
F
Q1 Q2
F F F ‫ ן‬Q1Q2
r ଵ
F ‫ ן‬୰మ
F F
F F ۹‫ۿ‬૚ ‫ۿ‬૛ F
F=
‫ܚ‬૛
F= F ( , N) ( )
9 2
K= F = 9x10 (N m /C)
Q1 = ˂ 1( F , C)
Q2 = ˂ 2( F , C)
r= F F ( , m)
F 1
˂ Q1 Q2 F +1 F F 2.0 F
F
F ˆ F www.schoolDD.com 5

7. Q1 = +1 ߤC Q2 = +1 ߤC
r = 2.0 m
୏୕భ ୕మ
F F= ୰మ
ଽ୶ଵ଴వ ୶ଵ୶ଵ଴షల ୶ଵ୶ଵ଴షల
= ଶ.଴మ
‫ ׵‬F = 2.25x10 N Ans ( ˈ
-3
+ )
F 2
F AB C q1 = 3 ߤ C , q 2 = 3 ߤ C q3 = -4 ߤC F AB =
3 m. BC. = 3 m. F F B
Q3 = -4 ߤC
C
F2 R
3.0 m
q1 = 3 ߤC q2 = 1 ߤC
A B θ
3.0 m B F1
F B
୏୕భ ୕మ
F F= ୰మ
ଽ୶ଵ଴వ ୶ଷ୶ଵ଴షల ୶ଵ୶ଵ଴షల
F1 = ଷ.଴మ
= 3x10-3 N
ଽ୶ଵ଴వ ୶ଵ୶ଵ଴షల ୶ସ୶ଵ଴షల
F2 = ଷ.଴మ
= 4x10-3 N
F F
B F
R = ඥFଵଶ ൅ Fଶଶ = ඥሺ3x10ିଷሻଶ ൅ ሺ4x10ିଷሻଶ
R = √25x10ି଺
‫ ׵‬R = 5.0x10-3 N Ans
R AB F
୊మ ସ୶ଵ଴షయ ସ
tan θ = ୊భ
= ଷ୶ଵ଴షయ
=
ଷ
‫ ׵‬θ = 53° Ans
F ˆ F www.schoolDD.com 6

8. 15.6 ˂
˂ (Electric Field) E F F F F
˂ ˂ ˈ F F +q
F ˈ F F ( N/C)
+Q +q
ሬറ
F ሬറ
E
۴
-Q
E=
+q ‫ܙ‬
ሬറ
F ሬറ
E
r
୏୕భ ୕మ
F= ୰మ
୏୕୯
FF= ୰మ
୊
E=୯
୏୕୯
FE= ୰మ
/q
Q E= ୏୕ ˈ F ˂
୰మ
۹‫ۿ‬
E= Q F F ˈ r
r ‫ܚ‬૛ ଵ
E‫ן‬Q E ‫ ן‬୰మ
+Q -Q
˂ +Q ˂ -Q
F ˆ F www.schoolDD.com 7

9. F 3
˂ Q 1 ߤC F 1 m. F
ˈ
E
E
+Q -Q
r = 1.0 m r = 1.0 m
˂ E = ୏୕
୰మ
ଽ୶ଵ଴వ ୶ଵ୶ଵ଴షల
= ଵ.଴మ
‫ ׵‬E = 9.0x103 N/C
˂ Q F F 9.0x103 N/C Ans
˂ ˂ F +q
F +q F F ˈ r ˈ ˂
Q ˈ ˂ F
Q ˈ ˂ F F
F
4
F X Z Q1 Q2 -4 ߤC F 2 ߤC F
XY F 2.0 YZ F 3.0 ˂ Y
Q2 = 2x10-6 C
Z
E1
Y
3.0 m θ
E
Q1 = -4x10-6 C E2
X Y
2.0 m
˂ F Y
F ˆ F www.schoolDD.com 8

10. ˂ ˈ F ˂ F Y ˈ ˂ F ˂
Q1 Q2
˂ Q1 E1 = ୏୕మభ
୰భ
ଽ୶ଵ଴వ ୶ସ୶ଵ଴షల
= ଶ.଴మ
‫ ׵‬E1 = 9.0x103 N/C ( YX)
˂ Q2 E2 = ୏୕మమ
୰మ
ଽ୶ଵ଴వ ୶ଶ୶ଵ଴షల
= ଷ.଴మ
‫ ׵‬E2 = 2.0x103 N/C ( ZY)
˂ F Y F F
E = ඥEଵଶ ൅ Eଶଶ = ඥሺ9x10ଷ ሻଶ ൅ ሺ2x10ଷ ሻଶ
E = √85x10଺
‫ ׵‬E = 9.2x103 N/C Ans
˂ F YX F
୉మ ଶ୶ଵ଴య
tan θ = ୉భ
= ଽ୶ଵ଴య
= 0.22
‫ ׵‬θ = 12.5° Ans
F ˂
1. +q F ˂ Q
+Q +q ሬറ
E ሬറ
F
-Q F = qE
ሬറ
F ሬറ
E +q
2. q F ˂ Q
+Q -q
ሬറ
F ሬറ
E
F = qE
-Q ሬറ
E -q ሬറ
F
F ˆ F www.schoolDD.com 9

11. - ˂ ˂ F +q F
- F F F F( F , )
15.7 F ˂
F ˂ F F F ˂
F ˂ + F ˂ -
F 1 F ˂ ˈ F ˂ F ˈ
F F +q
E1
E
E2
+Q1 -Q2
F ˂ +Q1 Q2
F ˂
1. F
E + ˂ F
F
E
F ˂
- ˈ F
+ -
F ˆ F www.schoolDD.com 10

12. 2. F F
F ˂ F
F F ˂
3.
F ˂ F F
F F ˂ ˂
F ˈ F
- F ˂ F F F
- F ˂ F ˂ F
- F ˂ F F ˂ F
- ˂ F
F 5
8.4x10-14 F ˂ F
F ˂ ˈ 4x106 F F ( F F
)
+ +
FE F
E E = 4x106
mg
- -
F ‫׵‬ F a=0
F
FE = mg
୊
qE = mg (E = ୯ )
q(4x106) = 8.4x10-14x10
q = 2.1x10-19
‫׵‬ -2.1x10-19 C Ans ( FE F +)
F ˆ F www.schoolDD.com 11

13. F 6
F A B F A B F ˂
E A
A B E
B
15.8 F ˂ F F ˂
F ˂
F F F F m F B F F F F A ˈ h
B EPB
m F F F Ep
g mg h
F B F F A F F mgh F
F F A B F
m A EPA EPB - EPA = mgh
EPB - EPA = W୅՜୆
mg
q ˂ E
F ˂ Ep ˂ F F
q
EP
F˂ V F ˂ F 1 F
F ˂ ˈ F F ˈ F F
۳‫ܘ‬ F (V)
V=
‫ܙ‬
F F ˂
F A B ˂ E
E
+q F B B A F
B A
+q F B F ˂ F F A( F
F F F F )
EPB EPA EPB > EPA
F ˆ F www.schoolDD.com 12

14. F F ˂ F F ˂ F F B A ( VB VA ) F F
1 F F A B
E
B A V = ୉୯౦
q ‫ۯ܅‬՜۰
VB VA = VB VA = ୉୯ - ୉୯
౦ా ౦ఽ
‫ܙ‬
VB VA = ୉౦ా ି ୉౦ఽ
୯
୛ಮ՜ా
F A F F(∞) VA = 0 F VB = ୯
F FF F ˂ F 1 F F F
15.8.1 F ˂
F ˂ F F F Q ˈ r F
Q V
۹‫ۿ‬ V= F ˂ F ˈ F
r
V= Q
‫ܚ‬
- F ˂ ˈ F F ˂ F F ˂ ˈ F F ˂ ˈ
F ˂ (+>0>-)
- F ˂ = F ˂ F ( +,-)
୏୕
- F ˂ ˈ F F ( r = ∞ ---> V = ஶ = 0 )
F 7
F ˂ Q F 1 F
-6
. 2.0x10 F . -2.0x10-6 F
Q V=?
r = 1.0 m
. Q = 2x10-6 C
V = ୏୕
୰
ଽ୶ଵ଴వ ୶ଶ୶ଵ଴షల
= ଵ.଴
‫ ׵‬V = 1.8x104 V Ans
F ˆ F www.schoolDD.com 13

15. . Q = -2x10-6 C
V = ୏୕
୰
ଽ୶ଵ଴వ ୶ିଶ୶ଵ଴షల
= ଵ.଴
‫ ׵‬V = -1.8x104 V Ans
F 8
F A, B C 1x10-6 , -2x10-6 3x10-6 F
. F ˂ D
. F 1.2x10-6 F F D
-2x10-6 C
B
0.4 m 3x10-6 C
1x10-6 C C
A
D 0.3 m
0.2 m
. VD = ?
VD ˈ ˂ F A, B C
V = ୏୕
୰
వ షల
F ˂ D A = ଽ୶ଵ଴ ଴.ଶ
୶ଵ୶ଵ଴
= 4.5x104 V
వ షల
F ˂ D B = ଽ୶ଵ଴ ୶ିଶ୶ଵ଴ = -4.5x104 V
଴.ସ
ଽ୶ଵ଴వ ୶ଷ୶ଵ଴షల
F ˂ D C= ଴.ଷ
= 9.0x104 V
VD = (4.5x104) + (-4.5x104) + (9.0x104)
‫ ׵‬VD = 9.0x104 V Ans
.
୛ఽ՜ా
F F ˂ VB VA = ୯
F A F F VA = 0 B F F D F
୛ಮ՜ీ
VD = ୯
Wஶ՜ୈ = qVD
= 1.2x10-6 x9x104
‫ ׵‬W∞՜D = 0.108 J Ans
F ˆ F www.schoolDD.com 14

16. 15.8.2 F ˂
V = ۹‫ۿ‬
‫܉‬ a V = ۹‫ۿ‬
‫ܚ‬
r
୏୕
- F ˂ F F F ୟ
a
୏୕
- F ˂ F F F ୰
r F F
F
F 9
10 2 F PQ R F F F
ˈ 5,10 20
. F ˂ PQ R
. F 1x10-9 F F PQ R
. F -1x10-9 F R Q
Q = -2x10-6 C
P Q R
5 cm
10 cm
20 cm
. VP , VQ VR = ?
V = ୏୕
୰
ଽ୶ଵ଴వ ୶ିଶ୶ଵ଴షల
VP = VQ = ଴.ଵ
= -1.8x104 V Ans
వ షల
VR = ଽ୶ଵ଴ ୶ିଶ୶ଵ଴ = -0.9x104 V Ans
଴.ଶ
. W=?
୛ఽ՜ా
F F ˂ VB VA = ୯
F A F F VA = 0 F
F ˆ F www.schoolDD.com 15

17. ୛ಮ՜ా
VB = ୯
Wஶ՜୆ = qVB (B = P)
Wஶ՜୔ = qVP
= 1.0x10-9x-1.8x105
‫ ׵‬W∞՜P = Wஶ՜୕ = -1.8x10-4 J Ans ( ˈ F )
Wஶ՜ୖ = qVR
= 1.0x10-9x-0.9x105
‫ ׵‬W∞՜R = -0.9x10-4 J Ans
. Wୖ՜୕ = ? q = -1x10-9 C
୛
F F ˂ VB VA = ఽ՜ా
୯
Wୖ՜୕ = q (VQ VR) (A = R, B = Q)
= -1.0x10-9(-1.8x105 (-0.9x105))
‫׵‬ Wୖ՜୕ = 0.9x10-4 J Ans ( ˈ F )
15.8.3 F F F F ˂
E
+ -
F +q qE
+ -
B A
+ d
-
+ -
୛ఽ՜ా
F F ˂ VB VA = ୯
+q F A ˂ F qE
˂ ( F ) F F +q A B F (a=0) F
F F qE F ˂
+q A B F
W୅՜୆ = Fd
= qE d
F F F F
qE d
VB VA = ୯
F ˆ F www.schoolDD.com 16

18. VB - VA = Ed
୊
E= ( ୒ x ୫ = େ ୎୫ = ୚
)
E ˈ ˂ F ˈ F F (V/m) ୯ େ ୫ ୫
d ˈ F A B F ˈ (m)
F 10
F F F F 20 F F F F ˈ 400 F
. ˂ F F F
. F 9x10-31 -1.6x10-19 F F
F F F
VB VA = 400 V
B A
+ -
+ -
+ -
+ -
d = 20 cm
.E=?
E VB - VA = Ed
400 = E (0.20)
‫ ׵‬E = 2x103 V/m Ans
. 9x10-31 kg 1.6x10-19 C , a = ?
F F a A --> B ˂ F
F = ma
qE = ma
1.6x10-19(2x103) = 9x10-31(a)
‫ ׵‬a = 3.5x1014 m/s2 Ans
F ˆ F www.schoolDD.com 17

19. F 11
F F F 2 F ˂ F F F
-31 -19
9x10 1.6x10 F F F F F F
F F F ˈ F
+ + + + B
F = qE
F F F ˈ
d = 2 cm + F F ˈ - F
mg F F
A
- - - -
∑ F୷ = ma (a=0)
F = mg
qE = mg
୫୥ ଽ୶ଵ଴షయభ ୶ଵ଴
E= ୯
= ଵ.଺୶ଵ଴షభవ
VB - VA = Ed
ଽ୶ଵ଴షయభ ୶ଵ଴
= ଵ.଺୶ଵ଴షభవ (0.02)
‫ ׵‬VB - VA = 1.12x10-12 V Ans
F ˂
1. F F F A B 1 F A B
E
B A
୛ఽ՜ా
q VB VA = ୯
VB VA
2. F ˂ B 1 F ∞ B
E
B ∞ ୛ಮ՜ా
q VB = ୯
VB Vஶ = 0
3. F ˂ Q r
Q ୏୕
V= ୰
r
F ˆ F www.schoolDD.com 18

20. 4. F ˂ Q
V = ۹‫ۿ‬
‫܉‬ a V = ۹‫ۿ‬
‫ܚ‬
r
4. F ˂ F +,- 0 F
15.9
(Capacitor) ˈ ˂ F F F
(Capacitance) C F F F (Q)
F F ˂ (V) F ˈ F F F (C /V) (F)
‫ۿ‬
C=
‫܄‬
୏୕
r=a F ˂ V= ୟ
୕
C=୚
୕
= ే్
౗
ୟ
= ୏
‫׵‬ C ‫ן‬ a
- F >
F ˂ F (+) F F ˂ (+) F F (-) F F ˂ (-)
+Q
F F
C -Q
F
V F (+ -) F F Q = CV
F
F
F ˆ F www.schoolDD.com 19

21. F 12
20 F ˂ F
a = 20 cm ୕ ୕
C=୚ = ే్
౗
ୟ
= ୏
଴.ଶ଴
= ଽ୶ଵ଴వ
‫ ׵‬C = 2.2x10-11 F Ans
F 13
˂ F ˂ 10 F ˂ 0.2
V = 10 V ୕
C=୚
C = 0.2 ߤ F
Q = CV
= 0.2x10-6 x10
‫ ׵‬Q = 2.0x10-6 C Ans
F 14
70 F F F 20 F
C = 70 ߤF
+Q -Q ୕
C=୚
V = 20 V
Q = CV
= 70x10-6 x20
‫ ׵‬Q = 1.4x10-3 = 1400 ߤC Ans
15.9.1
U F F F F F (V) ˂
(Q) F ˈ (J)
˂
Q ૚
U = QV
૛
F F
v
F ˆ F www.schoolDD.com 20

22. ୕ +Q C -Q
Q V F F C=୚
Q = CV V
FQ‫ן‬V plot Q V F F
F 15
50 F F F F 10 F
ଵ
U = ଶ QV ---1
୕
FV Q C=୚
Q = CV = 50x10-6 x10 = 500x10-6 C
ଵ
F Q V 1 , U = ଶ 500x10-6x10
‫ ׵‬U = 2.5x10-3 J Ans
15.9.2 F
F F ˂ 2
1. F
ˈ F F F F
C1 C2 C3
Q1 Q2 Q3 Q
V1 V2 V3
V
Q
F F F F ˂ F
FF
1. F F F Q = Q1 = Q2 = Q3
2. F F F F F V = V1 + V2 + V3
V = V1 + V2 + V3
୕ ୕
C=୚ V= େ
୕ ୕భ ୕మ ୕య
େ
= େభ
+େ +େ
మ య
F ˆ F www.schoolDD.com 21

23. ૚ ૚ ૚ ૚
= +۱ +۱
۱ ۱૚ ૛ ૜
2. F
ˈ F F F
V1 = V2 = V3
C1
Q1 C2
Q2
Q3 C3
Q
V
Q
F FF
1. F F F Q = Q1 + Q2 + Q3
2. F F F F F F F V = V1 = V2 = V3
Q = Q1 + Q2 + Q3
୕
C=୚ Q = CV
C V = C1 V1 + C2 V2 + C3 V3
C = C1 + C2 + C3
F 16
10 20 F
C1= 10 ߤF C2= 20 ߤF
F
Q1 Q2 Q
V
Q
F ˆ F www.schoolDD.com 22

24. ଵ ଵ ଵ
େ
= େభ
+େ
మ
ଵ ଵ
= ଵ଴ఓ୊
+ ଶ଴ఓ୊
ଶାଵ ଵ
= ଶ଴ఓ୊
= ଺.଻ఓ୊
‫ ׵‬C = 6.7 ߤF Ans
F
C1= 10 ߤF
Q1
C2= 20 ߤF
Q2
Q
V
Q
= C1 + C2
C
= 10 ߤF + 20 ߤF
‫ ׵‬C = 30 ߤF Ans
F 17
2 4 F F F F 120
F F
. .
. F
C1= 2 ߤF C2= 4 ߤF
Q1 Q2 Q
V = 120 V
Q
୕
Q C=୚
Q =C V ---1
F V = 120 V C
ଵ ଵ ଵ
େ
= େభ
+େ
మ
F ˆ F www.schoolDD.com 23

25. ଵ ଵ
= ଶఓ୊
+ ସఓ୊
ଶାଵ ଵ
= ସఓ୊
= ଵ.ଷଷఓ୊
‫ ׵‬C = 1.33 ߤF
F 1, Q = 1.33x10-6 x120
‫ ׵‬Q = 160x10-6 C Ans
ଵ
U U = ଶ QV
ଵ
U = ଶQ V
ଵ
= ଶ (160x10-6)(120)
‫׵‬U = 0.0096 J Ans
. F
C1= 2 ߤF
Q1
C1= 4 ߤF
Q2
Q
V
Q
C C = C1 + C2
= 2 ߤF + 4 ߤF
‫ ׵‬C = 6 ߤF
୕
Q C= ୚
Q =C V
Q = 6.0x10-6 x120
‫ ׵‬Q = 720x10-6 C Ans
ଵ
U U = ଶ QV
ଵ
U = ଶQ V
ଵ
= ଶ (720x10-6)(120)
‫׵‬U = 0.043 J Ans
F ˆ F www.schoolDD.com 24

26. F 18
140 F F1 F F
F F 1,000 F
C1
Q1 C2
Q2
Qn Cn
Q =1C
V = 1000 V
Q
F n
F C = C1 + C2 + C3 + Cn
F C1 = C2 = C3 = C n = 140 ߤF
C = 140 (n) ߤF
୕
C= ୚
୕
C = ୚
-6 ଵ.଴
140 (n) x10 = ଵ଴଴଴
ଵ଴଴଴
n = ଵସ଴ = 7.1
‫ ׵‬n=8 Ans
F 19
C1 6 C2 12 C3 8
F F F 100 F F F F
C1
100 V C3
C2
F ˆ F www.schoolDD.com 25

27. C1 C2 F F F C3 C1 C2
ଵ ଵ ଵ
େ
= େభ
+େ
మ
ଵ ଵ ଵ
େభమ
= ଺ఓ୊
+ ଵଶఓ୊
ଶାଵ ଵ
= ଵଶఓ୊
= ସఓ୊
‫ ׵‬C12 = 4 ߤF
୕
Q12 C= ୚
Q12 = C12V
= 4x10-6 x100
‫ ׵‬Q12 = 400 ߤC
C1 C2 F ‫ ׵‬Q1 = Q2 = Q12 = 400 ߤC Ans
୕
Q3 C= ୚
Q3 = C3V
= 8x10-6 x100
‫ ׵‬Q12 = 800 ߤC Ans
୕ ୕
F F F C= ୚ େ
V=
୕భ ସ଴଴ఓେ
V1 = େభ
= ଺ఓ୊
= 66.67 V Ans
୕మ ସ଴଴ఓେ
V2 = େమ
= ଵଶఓ୊
= 33.33 V Ans
(Check V1 + V2 = V , 66.67 + 33.33 = 100 V --> OK! )
୕య ଼଴଴ఓେ
V3 = େయ
= ଼ఓ୊
= 100 V Ans
(Check V3 = V , 100 V = 100 V --> OK! )
ଵ
F U = ଶ QV
ଵ
U1 = ଶ Q1 V1
ଵ
= ଶ (400x10-6)(66.67)
‫ ׵‬U1 = 0.013 J Ans
ଵ
U2 = ଶ Q2 V2
ଵ
= ଶ (400x10-6)(33.33)
‫ ׵‬U2 = 0.0067 J Ans
F ˆ F www.schoolDD.com 26

28. ଵ
U3 = ଶ Q3 V3
ଵ
= ଶ (800x10-6)(100)
‫ ׵‬U3 = 0.04 J Ans
F ˆ F www.schoolDD.com 27