1. The document discusses electric current (I), potential difference (V), resistance (R), and their relationships. Ohm's law states that I=V/R, where R is constant for a given resistor. 2. Resistance depends on the resistivity (ρ) of the material, its length (l), and cross-sectional area (A). R=ρl/A. Materials have different typical resistivity values. 3. In a circuit with both internal and external resistance, the total potential difference (E) across the circuit equals the potential drop across the external resistor (VR) plus the internal resistor (Vr). E=VR+Vr. Using Ohm's law,

1.    SchoolDD.com
16
www.schoolDD.com

2. 16
16.1 ˂
˂ (electric current) ˂
ˈ
. .
F ˂
ˈ ˂ . F F
ˈ F F F F
F F 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 F F ˂ F F
F ˂ F F F F F ˂ ˈ F
16.1.1 ˂
˂ ˂ F ˂
˂
ˈ ˂ (free electron) FF
F F F F F F (Brownian motion)
F ˈ F
F ˆ F www.schoolDD.com 1

3. F F F F ˂ F
˂ F F ˈ F
(drift velocity) F ˂ F ˂
VA VB VA VB
VA = VB VB > VA
F F
F F F F F F ˂
F F F ˂ (VA = VB) (VB > VA) ˂ F
16.1.2 ˂ ˂
˂ ˂ F F
˂
˂ F F ˂ F
F ˈ F F ˈ F F F (A)
˂
‫ۿ‬
I=
‫ܜ‬
I= ˂ (A)
Q= ( ) F (C)
t= F (s)
˂ I ˂ E
F 1
F ˂ 1.25 F F F F F F
F 5 F F
F ˆ F www.schoolDD.com 2

4. I = 1.25 A t=5,Q=?
୕
˂ I= ୲
Q = It
= 1.25x5.0
‫ ׵‬Q = 6.25 C Ans
F 2
F F F F ˂ F F 3.2 F
F F 5.0 ( ˂ 1.6 X10-19 )
F
I = 3.2 A ୕
˂ I= ୲
Q = It
I I
Q = 3.2x5.0 C
F 1 1.6 X10-19 C
‫׵‬ = ଵ.଺௫ଵ଴షభవ
ଷ.ଶ୶ହ.଴
= 1x1020 Ans
˂
F F ˂
I ˂ , F E
E
˂ (I)
F
s = vt
F ˆ F www.schoolDD.com 3

5. ୕
˂ I= ୲ ˂ Q = Ne = (nAs)e = (nAvt)e = nevtA
୬ୣ୴୲୅
I = ୲ .
I = nevA
n= F ( / m3)
e= (1.6 x 10- 19 C)
v= (m/s)
A= (m2)
F 3
F 5.0 x 1028 F F F 2.5 F
F F 0.30 F ˂ F
F
A = 2.5x10-6 m2
I=? I = nevA
= 5x1028x1.6x10-19x0.3x10-3x2.5x10-6
n = 5x1028 ‫ ׵‬I = 6.0 A Ans
v = 0.3x10-3 m/s
16.2 F F ˂ (I) F F (V)
16.2.1 F F
˂ F F F F ˂ F
I‫ן‬V
‫ ׵‬I = KV ( K= F )
୚ ଵ
୍
=୏
୚ ଵ
୍
=R ( ୏
= R)
V=IR F
R= F F ˈ F F F F (Ω)
F ˆ F www.schoolDD.com 4

6. - F F F ˈ F
I I I I
V V V V
F
F 4
F F F F ˂ ˈ F F
V (V)
3
2
1
I (A)
0 0.2 0.4 0.6
F 5
F F F 1.0 F F F F ˂ 1.0 F F
F F
R = 1x106 Ω
F V = IR
I = 1x10-3 A I
= 1x10-3x1x106
‫ ׵‬V = 1x103 V Ans
V=?
F (Resistor)
F ˈ F F F F ˂ F F ˂ F
F 2
1. F (Fixed resistor) F F ˈ F
F F F ˂ F F F
F ˆ F www.schoolDD.com 5

7. 1 2 3 4
4 F
F
1
0 0 1 -
2
1 1 101 േ1%
3
2 2 102 േ2%
F 3 3 103 -
4 ˈ F
4 4 104 -
F F
5 5 105 - 3
F 15x10 F
6 6 106 -
േ2 % F 15,000 F േ
F 7 7 - -
300 F F F 14,700 F
8 8 - -
15,300 F
9 9 - -
- - 10-1 േ5%
- - 10-2 േ 10 %
2. F F (Variable resistor) ˈ F F F F F
˂ F F F F F
F F 1 3 F F 2 F F ˂
F F
1 3 F F F 2
A
2 F 1 F 3 F
F F
(Diode)
ˈ F F F F F
˂ ˂ ˂
F ˂ 2
I A F F A F
F ˂ F F ˂
F ˆ F www.schoolDD.com 6

8. - F ˂ F F F ˂ ˈ ˂
F 6
F a F F b F F C
F F F F
F
F a F = 30x102 Ω േ 150 Ω
3 0 102 േ 5%
F
F b F = 18x103 Ω േ 1800 Ω
3
1 8 10 േ 10%
F c F =
16.2.2 F ˂
F ˂ ߩ ˈ F F
˂ F F F F F F ˂
F ˈ F
F F F F ˂ (V) ˂ (I) F
(R) F F
୚ ℓ
‫ן‬
୍ ୅
. F
ℓ
R ‫ן‬
୅
࣋र
R= ߩ F F ˂
‫ۯ‬
F F ߩ = 1.6x10-8 Ω m , = 1.7x10-8 Ω m ߩ
ߩ F = 1010- 1014 Ω m , ߩ = 1014- 1018 Ω m
F 7
F 10 F F F F 1 F
F F F 2 F F F ˈ F F
F ( ߩ = 1.7x10-8 Ω m)
F ˆ F www.schoolDD.com 7

9. d = 1x10-3 m
F ℓ = 10 m గௗమ ଷ.ଵସ୶ሺଵ୶ଵ଴షయ ሻమ
F A= = = 7.85x10-7 m2
ସ ସ
ఘℓ
R= ୅
ଵ.଻୶ଵ଴షఴ ୶ଵ଴
R1 = ଻.଼ହ୶ଵ଴షళ
‫ ׵‬R1 = 0.216 Ω Ans
d = 2x10-3 m
F ℓ = 10 m గௗమ ଷ.ଵସ୶ሺଶ୶ଵ଴షయ ሻమ
A= = = 3.14x10-6 m2
F ସ ସ
ఘℓ
R= ୅
ଵ.଻୶ଵ଴షఴ ୶ଵ଴
R2 = ଷ.ଵସ୶ଵ଴షల
‫ ׵‬R2 = 0.054 Ω Ans
ୖభ ଴.ଶଵ଺
ୖమ
= ଴.଴ହସ
=4
‫׵‬ 1 F F 2 F 4 F Ans
˂ G ˈ F F ˂ F ˈ ( F )-1 F (S)
ଵ
G= ୖ
˂ ߪ ˈ ˈ F F ˂ F ˈ
( F )-1 F F (S/m)
ଵ
ߪ=
ఘ
16.2.3 F F
ˈ F F F
T ---> R
F F F F F
T ---> R
ˈ F F
T ---> R
F ˆ F www.schoolDD.com 8

10. (Superconductivity) F ˂ ˈ F (ߩ = 0) F
F (critical temperature) F (Supper
conductor)
16.3 ˂
16.3.1 ˂ ( E) F F (V)
F F ˂ ( F F ) F F F F
˂ F F F ˂ ˂
F F ˂
F R
I
I I
E I
F ˂
F F F R F ˂
E F r F F ˂ I F F
R
VR
I
V
r E
Vr
˂ F ˂ ( ) F F F ˂ Q
˂ E F F F r R F F ˂ Q F F
F F
F WE = ˂ F
WR = ˂ F F R
Wr = ˂ F F r
F FF WE = W R + W r
QE = QVR + QVr
F E = VR + Vr
F ˆ F www.schoolDD.com 9

11. E = IR + Ir
୉
F I = ሺୖା୰ሻ
F V F F F F V = VR
F V F F R F F F V؆E F F V=E
F 8
F ˂ 12.0 F F 2.0 F F F 7.0
F F F F F F F F F F F
F F F F F F
R = 20 Ω
VR
I
V=?
r = 2 E = 12 V
Vr
I E = VR + Vr
E = IR + Ir
୉ ଵଶ ଵ
I = ሺୖା୰ሻ = ሺ଻଴ାଶሻ
= ଺
A
V = VR = IR
ଵ
= ଺x70
‫ ׵‬V = 11.67 V Ans
F F F F
V=?
F V ؆ E = 12 V Ans
r = 2 E = 12 V
F ˆ F www.schoolDD.com 10

12. 16.3.2 ˂
˂
R
I V I
E
FQ= ˂ F F t (C)
I= ˂ F F (A)
V= F F F (V)
W= ˂ F F ( J)
W = QV
୕
I= ୲
Q = It V = IR
୚మ ୲
‫ ׵‬W = ItV = I2Rt = ୖ
˂ P ˂ F F F ˈ F F (W)
‫܅‬ ୛ ItV
P= P= ୲
= ୲
= IV
‫ܜ‬
F 9
F 1.5 F2 F F F ˂ 20 F
F 20 F F 15 F ˂ F ˂ 1 F
t = 20 hr
I=20 mA
E = 1.5x2 = 3 V
˂ F W = QV
= ItV
F ˆ F www.schoolDD.com 11

13. = 20x10-3x(20x60x60)x3
‫ ׵‬W = 4320 J
ଵହ୶ଶ
F ˂ F ˂ 1 F = ସଷଶ଴ = 0.007 / Ans
16.4 F F
16.4.1 F F
1. F
ˈ F F F F F
V1 V2 V3
R1 R2 R3
I1 I2 I3
V
I
E
F FF
- ˂ F F F
I = I1 = I2 = I3
- F F ˂ = F F ˂ F
V = V1 + V2 + V3
V = V1 + V2 + V3
V = IR ( F )
IR = I1 R1 + I2 R2 + I3R3
R = R1 + R2 + R3 (I = I1 = I2 = I3)
R = F R1,R2 R3
2. F
ˈ F F F ˈ F
F ˆ F www.schoolDD.com 12

14. V1 = V2 = V3
R1
I1 R2
I2
I3 R3
I I
V
E
F FF
- F F F F F = F F ˂
V = V1 = V2 = V3
- ˂ F = ˂ F F F
I = I1 + I2 + I3
I = I1 + I2 + I3
୚
V = IR I=ୖ
୚ ୚భ ୚ ୚
ୖ
= ୖభ
+ ୖమ + ୖయ
మ య
૚ ૚ ૚ ૚
= +‫܀+ ܀‬ (V = V1 = V2 = V3)
‫܀‬ ‫܀‬૚ ૛ ૜
R = F R1 , R 2 R3
- F F F F F F F
˂ F F F F F ˂ F F
- F F ˈ F F F
F 10
F F F
. FS ʽ . FS ʽ
F ˆ F www.schoolDD.com 13

15. 1Ω 2Ω
A B
S
I
3Ω 4Ω
. FS ʽ 1Ω 2Ω
F F R1 R2
A B
R3 R4
I
3Ω 4Ω
F R1 R2 F
F F R 12 = R 1 + R 2 = 1 +2 = 3 Ω
F R3 R4 F
F F R 34 = R 3 + R 4 = 3 +4 = 7 Ω
F F
3Ω
R12
A B
R34
I
7Ω
F R 12 R 34 F
ଵ ଵ ଵ ଵ ଵ ଻ାଷ
F F ୖ
=
ୖభమ
+
ୖయర
=
ଷ
+
଻
=
ଶଵ
ଵ ଵ଴
=
ୖ ଶଵ
‫ ׵‬R = 2.1 Ω Ans
. FS ʽ
F F
1Ω 2Ω
R1 R2
A B
R3 R4
I
3Ω 4Ω
F R1 R3 F
F ˆ F www.schoolDD.com 14

16. ଵ ଵ ଵ ଵ ଵ ଷାଵ
F F ୖభయ
=
ୖభ
+
ୖయ
=
ଵ
+
ଷ
=
ଷ
ଵ ସ
=
ୖభయ ଷ
ଷ
‫ ׵‬R13 = ସ
Ω
F R2 R4 F
ଵ ଵ ଵ ଵ ଵ ଶାଵ
F F ୖమర
=
ୖమ
+
ୖర
=
ଶ
+
ସ
=
ସ
ଵ ଷ
=
ୖమర ସ
ସ
‫ ׵‬R24 = ଷ
Ω
F F
A 3/4 Ω 4/3 Ω B
R13 R24
I
F R 13 R 24 F
F F R = R13 + R24
ଷ ସ ଽ ାଵ଺
= ସ+ଷ = ଵଶ
ଶହ
=
ଵଶ
‫ ׵‬R = 2.08 Ω Ans
16.4.2 F
1. F
ˈ F F F F F
R
I
r1 E1 r2 E2 r3 E3
F FF
- ˂ = ˂ F
E = E1 + E2 + E3
F F F F
F ˆ F www.schoolDD.com 15

17. r = r1 + r2 + r3
2. F
ˈ F F F F
R
I r1 E1
r2 E2
r3 E3
F ˂ F F F E1 = E2 = E3
F FF
- ˂ = ˂ F F F
E = E1 = E2 = E3
- F F F F
૚ ૚ ૚ ૚
‫ܚ‬
= ‫ܚ‬૚
+‫ܚ+ ܚ‬
૛ ૜
- F F ˂ F F ˂
F
F 11
F F ˂ 0.25 F F F 8 F
F F ˂ 0.16 F ˂ F
F ˂ F F F F E r
F ˆ F www.schoolDD.com 16

18. F
R=8Ω
I = 0.25 A V
r1 E1 r2 E2
˂ E = E1 + E2 = E + E = 2E
F r = r1 + r2 = r + r = 2r
F F
R=8Ω
I = 0.25 A
V
r E
V = E Ir
V = IR
F IR = 2E I(2r)
0.25x8 = 2E 0.25(2r)
E 0.25r = 1 ---1
F
R=8Ω
V
I = 0.16 A r1 E1
r2 E2
˂ E = E1 = E2 = E
ଵ ଵ ଵ ଵ ଵ ଶ
F ୰
=
୰భ
+
୰మ
=
୰
+ =
୰ ୰
୰
r = ଶ
F F
R=8Ω
I = 0.16 A
V
r E
F ˆ F www.schoolDD.com 17

19. V = E Ir
V = IR
୰
F IR = E I(ଶ)
୰
0.16x8 = E 0.16(ଶ)
E 0.08r = 1.28 ---2
2-1, -0.08r + 0.25r = 0.28
0.17r = 0.28
‫ ׵‬r = 1.7 Ω Ans
F r 2 F E 0.08(1.7) = 1.28
‫ ׵‬E = 1.41 V Ans
16.5 F ˂ F
˂ F F ˂ F F F
F F ˂
F 12
˂F F a,b c ˂
F E1 = 6 V , E2 = 6 V , r1 = 1 Ω , r2 = 1 Ω , Ra = 7 Ω , Rb = 4 Ω , Rc = 12 Ω
b
a
c
I
r1 E1 r2 E2
F
F b c F F F F a
ଵ ଵ ଵ ଵ ଵ
F b c ୖౘౙ
=
ୖౘ ୖౙ ସ ଵଶ
+ = +
ଷାଵ ସ ଵ
=
ଵଶ
= ଵଶ
= ଷ
‫ ׵‬Rbc = 3 Ω
F a bc R = Ra + Rbc
= 7+3
‫ ׵‬R = 10 Ω
E1 E2 F
˂ E = E1 + E2 = 6 + 6 = 12 V
F ˆ F www.schoolDD.com 18

20. F r = r1 + r2 = 1 + 1 = 2 Ω
F F
R
I
V
r E
V = E Ir
V = IR
F IR = E I r
Ix10 = 12 Ix2
‫ ׵‬I = 1.0 A
˂ F F a , Ia = I = 1.0 A Ans
F Ia = Ib + Ic
1.0 = Ib + Ic
Ic = 1.0 - Ib
Vb = Vc
IbRb = IcRc
IbRb = (1.0 - Ib)Rc
Ibx4 = (1.0 - Ib)x12
ଵଶ ଷ
‫ ׵‬Ib = ଵ଺ = ସ = 0.75 A Ans
Ic = 1.0 - Ib = 1.0 - 0.75 = 0.25 A Ans
F 13
F F ˂ 10 F F F F
F F 1.2 F F F 6 F F F F
F F F F F F F
F F F ˂ 6 V.
F F F F F 1.2 V. ˂ F F 10
mA.
F ˆ F www.schoolDD.com 19

21. F F F F F F F F F
F R F
VR VD
R
I F
V
E,r=0
V = VR + VD
0
V = E Ir = E
F E = VR + VD
VR = E VD = 6 1.2
‫ ׵‬VR = 4.8 V
F
VR = I R
୚ ସ.଼
R = ୍౎ = ଵ଴௫ଵ଴షయ
‫ ׵‬R = 480 Ω Ans
F 14
˂ ˂ 2 F F F F F F
. ˂ F
. F F F F
. F
. ˂ F 10
. ˂ F
R = 0.5 Ω
I=2A
r=0 R1
E=6V
.
˂ F F ˂ F F F
‫׵‬I = I = 2 A Ans
F ˆ F www.schoolDD.com 20

22. . VR = ?
F V = IR
VR = 2x0.5
‫ ׵‬VR = 1.0 V Ans
. R1 = ?
E = I r + I R + I R1
6 = 2x0 + 2x0.5 + 2xR1
଺ିଵ
R1 = ଶ
‫ ׵‬R1 = 2.5 Ω Ans
. W=? t = 10 s
W = QV
= QE
୕
= ItE (I = ୲ )
= 2x10x6
‫ ׵‬W = 120 J Ans
. PR = ?
W = QV
୛
P= ୲
QV
P= ୲
QVR ItሺIRሻ
PR = ୲ = ୲
= I2R = 22x0.5
‫ ׵‬PR = 2 W Ans
F 15
F F F a b
I I
R1 = 1 Ω R1 = 1 Ω
a a
E=9V E=9V
R2 = 2 Ω R2 = 2 Ω R3 = 2 Ω
b
b
. .
F ˆ F www.schoolDD.com 21

23. . V2 = ? I
V2 V2 = I R2
F R2 I
E V R
R1 R2 F r=0
R = R1 + R2 = 1 + 2 = 3 Ω
V = IR
0
V = E Ir
F E = IR
9 = Ix3
‫׵‬I = 3 A
V2 = I R2 = 3x2
‫ ׵‬V2 = 6 V Ans
. V2 = ?
R2 R3 F
ଵ ଵ ଵ ଵ ଵ
F ୖ౗ౘ
=
ୖమ
+
ୖయ
=
ଶ
+
ଶ
‫ ׵‬Rab = 1 Ω
R1 Rab F
F R = R1 + Rab = 1 + 1 = 2 Ω
V = IR 0
V = E Ir
F E = IR
9 = Ix2
‫ ׵‬I = 4.5 A
Vab = I Rab = 4.5x1
‫ ׵‬Vab = 4.5 V Ans
16.6 ˂
˂ F F F F F F F F
F (galvanometer) F F F
F
F ˆ F www.schoolDD.com 22

24. 16.6.1 F
ˈ ˂ F F F ˂
R R
F A
I I
E,r E,r
˂ F F F F
˂ F I
F ˈ F F F F F (shunt) F
Rs F F G F F I ˈ F F F
F F IG F F F F IS
I IG RG
G
IS
RS
F
F A
F FF F F F F F F
F F F
VS = VG
ISRS = IGRG
୍ ୖ
F RS = ృ୍ ృ
౏
I = IG + IS
F 16
F F F F 1000 F ˂ 50 F
F ˈ F ˂ F 100 F
. F F F F
IG = 50ߤA RG = 1000 Ω
. F F ˈ F I = 100 mA
G
IS
RS = ?
F ˆ F www.schoolDD.com 23

25. . RS = ?
I = IG + IS
IS = I - IG
IS = 100x10-3 50x10-6 A
VS = VG (F )
ISRS = IGRG
୍ ୖ
RS = ృ୍ ృ
౏
ହ଴୶ଵ଴షల ୶ଵ଴଴଴ ଵ଴଴଴
= ଵ଴଴୶ଵ଴షయିହ଴୶ଵ଴షల = ଶ୶ଵ଴య ିଵ
‫ ׵‬RS = 0.5 Ω Ans
. RA = ?
ଵ ଵ ଵ
= + ( F )
ୖఽ ୖృ ୖ౏
ଵ ଵ ଵ ଵାଶ଴଴଴ ଵ
= + = =
ୖఽ ଵ଴଴଴ ଴.ହ ଵ଴଴଴ ଴.ହ
‫ ׵‬RA = 0.5 Ω Ans
- F F F F F F ˂
F
16.6.2 F F
ˈ F F ˂ ˂ F F F F ˂
F F
V
R a R
a b
b
I
I
E,r E,r
˂ F F F F V F R
F F ˂ F
F ˈ F F F F F F
(multiplier) F Rm F F G F F F F
F Vm F ˈ F F F F VG
F ˆ F www.schoolDD.com 24

26. IG Rm RG
G
F
Vm VG
F F V
V = Vm + VG
V = IGRm + IGRG
V = IG(Rm + RG)
୚
Rm = ୍ృ
- RG
F FF F F F F F
F F F F F F F
F 17
F F F 16 F ˈ F F F F F 10 F
. F F F F F
. F F F F F F F F
. F F F F F
IG = 50 ߤA Rm = ? RG = 1000 Ω
G
Vm VG
V
. Rm = ?
V = Vm + VG
V = IGRm + IGRG
10 = 50x10-6xRm + 50x10-6x1000
ଵ଴ି଴.଴ହ
Rm = ହ଴୶ଵ଴షల
‫ ׵‬Rm = 200,000 Ω Ans
F ˆ F www.schoolDD.com 25

27. . F Rm = 0 , VG = ?
VG = IGRG
= 50x10-6x1000
‫ ׵‬VG = 50x10-3 V Ans
. RV = ?
RV = Rm + RG ( F )
= 200.000 + 1000
‫ ׵‬RV = 201,000 Ω Ans
- F F F F F F ˂
F
16.6.3 F F
ˈ F F
F F
R O
x R y
F
x y
F F F
F F F F G F F F R0 E
F F Rx F x y F F
˂ F F F F F F F F ˈ F F
RG
E,r G
R0
I
x y
RX
F F O
ˈ ˂ FF
E = Vr + VRX + VRG + VR0
E = I r + I R X + I RG + I R0
F ˆ F www.schoolDD.com 26