Resistors in series and in parallel (review)

R1 R2

V1 V2V

I

21 III 21 VVV

21 RRReq

R1I1

R2I2

I I

10

30

1

15

11

eq

eq

R

R

21 III 21 VVV

R1 R2

?

30

15

2

12

2

1

I

R

R

r

V

r

AV

rRRI 26.0

47

12

11

21

111

RRReq

R1

R2

AV

rRI

eq

0.112

12

3015eqR

Example

r

1

10. Kirchhoff’s rules

1) Kirchhoff's first rule (Junction or current rule)Follows from the conservation of charges

I1I2

I3

outin II

0

0

0

out

in

I

I

I

Example:

I1=I2+I3 0 outin II

?

2

5

3

2

1

I

AI

AI

AI

IAA

3

25

3

3

2

2) Kirchhoff’s second rule (Loop or voltage rule)Follows from the conservation of energy

R1 R2

mmn RI R3

I

V

3) Using Kirchhoff’s rules

outin II

mmn RI

For N junctions write N-1 equations

Write equations only for independent loops

3

1 2 3 04

Example 1: Determine the currents through the elements of this circuit.

R1ε2

R2 R3

I1

I2I3

ε1

• We have 3 different currents We need 3 equations• We have 2 junctions (N=2) We can use N -1=1 equation• We have 2 independent loops We can use 2 equations

4

Junction equation:(Both junction give the same equation)

0321 III

Loop equations:22332

22111 RIRI

RIRI

If we add two equations above, we obtain the equation for the third (big, combined) loop. We do not need this equation – it is dependent from two previous

331121 RIRI

Let us use numbers:

4

3

5

12

24

3

2

1

2

1

R

R

R

V

V

AI

AI

AI

5.0

3.3

8.2

3

2

1

I3 flows opposite to our assumption

Example 1 (continued)

An extra loop equation:

We have three equations for three unknowns, we can solve this system of equations.

5

Example 2:R1

R2

?

15

30

0.6

0.9

2

1

2

1

I

R

R

V

V

ε

I1

I2

I3

222 RIAIII

AV

RI

AVV

RI

ε

9.0

4.015

0.6

5.030

0.60.9

213

2

22

1

211

outin II

mmn RI

1121 RI

321 III

Example 3: In the circuit below, the switch is initially open. When the switch is closed, the current through the bottom resistor:

R

R

9V

9V

9V

A. Increases B. Decreases C. Stays the same 6

1

2

11. Microscopic picture

L

vd+For electrons in copper :drift speed ~ 10-5 - 10-6 m/s random-motion speed ~ 106 m/s

tneAvnALeQ

tvL

d

d

A

n – number of electric carriers per unit volume A – area of cross-sectionAL - volume

t

QI dneAvI

7

Example: A copper wire 2.0 mm in diameter, caries 1.0 A. What is the drift speed of electrons? Assume one electron per atom is free to move.

?

/109.8

/105.63

1002.6

0.12/0.2

0.1

33

3

123

d

mol

A

v

mkg

molkgm

molN

mmmmr

AI

3281024.8/

mm

N

V

Nn

mol

A

mol

A

dneAvI neA

Ivd

smv

mCm

A

neA

Iv

d

d

/104.2

10106.11024.8

0.1

5

2319328

2rA

8