Ch. 25

1
Ch. 25

• Electric Current and DC Circuits

2
Chapter Overview

• Definition of Current
• Ohms Law
• Resistance Conduction in Metals
• Kirchhoffs Laws
• Analysis of DC Circuits
• RC Circuits

3
Current

• Up to this point we have been concerned with
  charges that dont move Static
• When charges do move, then an electric current
  flows
• Current, usually denoted by the letter i is that
  rate at which charge moves. In other words how
  much charges flows past a point per time

4
Current

• i current
• q charge
• t time
• SI Units ampere
• Symbol A
• Fundamental Unit (More on this later)

5
The ampere is a fundamental unit, so the coulomb
is a derived unit. Express the coulomb in terms
of fundamental units

1. A/s
2. As
3. s/A
4. None of he above

1 2 3 4 5
6
Batteries are often rated in amphours. What type
of quantity does an amphour represent?

1. Current
2. charge
3. Electric Potential
4. Capacitance
5. None of the above

1 2 3 4 5
7
Ex. How many coulombs of charge are stored in 60
Ahr battery?
8
Solution

• i?q/?t
• ?qi?t
• ?q60 A x 1 hr
• 60A x 1 hr x 3600 s/1 hr 216000 C

9
Ex. 10000 protons fIow through a detector every
.05 s. What is the current flowing through the
detector?
10
Soln.

• Current i?q/?t
• i10000x1.602x10-19 C/.05 s
• i 3.2 x 10-14 A

Protons
Detector
11
Microscopic View of Current

• At the microscopic level, current is made by
  individual charges moving at a speed vd in the
  material
• A charge will travel the distance x in a time
  given by t x/vd

x
vd
q
12
Microscopic view of current

• The total amount of charge that flows through
  the gray shaded volume in time ?t is ?Q nqV
  where n is the number of charges per volume, V is
  the volume of the gray shaded area, and q is the
  charge of an individual charge
• V xA vd?tA
• So ?Q nqvdA?t

13
Drift Velocity

• I ?Q/?t nqAvd
• vd is the drift velocity. It represents the
  average speed of charges in conductor
• Ex. A copper wire has a radius of .50 mm. It
  carries a current of .25 A. What is the drift
  velocity of the electrons in the wire. Assume 1
  free electron per atom. (?cu 8.93 g /cm3)

14
There will be Avagadros number of charges in 1
mole of copper. the molar mass of copper is 63.5
g. The volume of 1 mole of copper is V 63.5
g/8.93 g/cm3 7.31 cm3 7.31 x 10-6 m3
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n NA / Vmol 6.022 x 1023/7.31 x 10-6 m3
8.24 x 1028 electrons /m3 vd i/nqA .25
A/(8.24 x 1028/m3 x 1.602 x 10-19 As x 3.14 x(5 x
10-4 m)2) 2.4 x 10-5 m/s
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Ohms Law

• What factors determine how much current will flow
  in a circuit (BRST)

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Ohms Law

• Potential Difference and the material properties
  determine current that flows in a circuit

18
Ohms Law

• Current is proportional to potential difference
• Resistance limits the amount of current that
  flows
• Experimental Relationship found by Georg Simon
  Ohm
• Not always true e.g. diodes, transistors,

19
Resistance

• Units ohm denoted by O
• What is the ohm in terms of V and A
• What is the ohm in terms of fundamental units

20
Units

• OV/A
• O kg m2 /(A2 s3 )

21
Resistivity

• What factors determine the resistance of a piece
  of metal? (BRST)

22
Resistivity

• Length L
• Cross sectional Area A
• resistivity ? material property (see table in
  text p. 792)

23
Two different metals with different resistivities
have the same length. Which metal will have the
higher resistance?

1. The metal with the greater ?
2. The metal with the smaller ?
3. The resistances will be the same
4. Cannot be determined

1 2 3 4 5
24
Temperature Dependence of Resistance

• If you measure the resistance of a light bulb
  cold and then measure it when it is glowing, do
  you get the same resistance? (BRST)

25
Temperature Dependence of Resistance

• Resistance increases with increasing temperature
  for metals
• It decreases with increasing temperature for
  semiconductors
• For conductors
• R R20(1 a(T- 20 C))
• a is the Temperature coefficient of resistance
  (see p. 792)

26
Ex. Find the temperature of the filament of a
light bulb (assume W) by measuring the resistance
when cold and glowing.
27
Power

• When we apply a potential difference across a
  resistor, it gets hot
• What determines the power given off by a resistor?

28
Power

• Work q?V
• Power Work/time
• P q?V/t but q/t i
• P i?V

29
Power

• Combine the expression for electric power P
  i?V with Ohms law ?V iR
• P i?V i2R (?V )2/R

30
Ex. A 100 W bulb is designed to emit 100 W when
connected to a 120 V circuit. a) Draw a sketch
and a schematic. b) What is the resistance of
the bulb and the current drawn when connected to
a 120 V outlet? c) Assuming the resistance
doesnt change, what would be the power output if
the bulb was connected to a 240 V circuit?
31
The electric bill

• Power is a rate it tells you how much energy
  per time is being used

32
The electric company bills you in units of kwhr.
What is a kwhr? (TPS)

1. It is a unit of power
2. It is a unit of energy
3. It is a unit of current
4. It is a unit of potential difference
5. Not enough information given

1 2 3 4 5
33
The electric company bills you in units of kwhr.
What is a kwhr? (TPS)

1. It is a unit of power
2. It is a unit of energy
3. It is a unit of current
4. It is a unit of potential difference
5. Not enough information given

1 2 3 4 5
34
EX. How many joules are in a kWhr?
35
For the circuit shown below how does the current
flowing through A compare to that flowing through
B

1. It is the same
2. It is greater at A
3. It is greater at B
4. It cannot be determined

3.0 O
B
A
V
6.0 O
1 2 3 4 5
36
You connect two identical resistors in series
across a 6.0 V battery. How does the current in
the circuit compare to that when a single
resistor is connected across the battery

1. There is no difference
2. The current is twice as large
3. The current is ½ as large
4. Cannot be determined

1 2 3 4 5
37
Resistors in Series

• When we add resistors is series, the current
  decreases since the resistance increases
• We define an equivalent resistance as a single
  resistor which produces the same current when
  attached to the same potential as the combination
  of resistors

38
Series equivalent
39
Equivalent Series Resistance

• We want i to be the same
• V iReq
• V1 iR1, V2 iR2, V3 iR3
• V V1 V2 V3 iR1 iR2 iR3
• iR1 iR2 iR3 i(R1 R2 R3) iReq
• So Req R1 R2 R3

40
Equivalent Series Resistance

• How would this result change if there were four
  resistors in series?
• In general as more resistors are added in series,
  the resistance increases so the current decreases

41
Ex. a) Find the equivalent resistance for the
following circuit. b) Find the current In the
circuit c) Find the potential drop across each
resistor d) Find the power dissipated by each
resistor. e) Find the power Supplied by the power
supply.
42
You connect two identical resistors in parallel
across a 6.0 V battery. How does the current
supplied by the battery compare to that when a
single resistor is connected across the battery?

1. There is no difference
2. The current is twice as large
3. The current is ½ as large
4. Cannot be determined

1 2 3 4 5
43
Resistors in Parallel

• When we add resistors is parallel, the current
  increases
• The effective resistance must then decrease
• How can that be? (BRST)

44
Resistors in Parallel

• There are more branches for current to follow in
  a parallel circuit, so current can be larger

45
Resistors in Parallel

• What is the same for the three resistors shown?
  (GR)

46
Resistors in Parallel

• The potential difference across each resistor is
  the same
• Define i1 V/R1, i2 V/R2, i3 V/R3
• How do the currents combine?

47
Parallel Equivalent Resistance

• We define the equivalent resistance as a single
  resistor that will draw the same current from the
  power supply

48
Parallel Equivalent resistance

• The currents add. Why?
• i i1 i2 i3 V/R1 V/R2 V/R3 V/Req

49
a) Find the equivalent resistance. b) Find the
current flowing through each resistor. c) Find
the current supplied by the power supply
50
Find the equivalent resistance for the following
network
51
Kirchhoffs laws

• The rules for series and parallel resistances are
  examples of Kirchhoffs Laws
• Voltage Law - Sum of the potential differences
  around a closed loop is 0
• Current Law- the sum of currents at a node is 0

52
Voltage Law

• A loop is a closed path in a circuit

53
Current law

• A node is a point in a circuit where several
  wires join

54
For the circuit shown below a) choose currents
for each branch of the circuit. b) For the
choice of currents youve made, label the higher
potential side of each resistor with a and the
lower potent side with a c) Use Kirchoff s
laws to write a closed system of equations for
the currents. d) Solve for the currents. e)
Find the potential difference across each
resistor. f) Find the power dissipated by each
resistor. g) Find the power supplied by each
power supply
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56
RC Circuits

• A resistor in series with a capacitor makes an RC
  circuit
• RC circuits have many applications e.g. camera
  flashes

57
RC Circuits - Charging

• Use Kirchhoffs Voltage Law to analyze the
  circuit shown below

R
V
C
58
RC Circuits

• V VR VC 0
• V iR Q/C 0
• But i dQ/dt
• V - R dQ/dt 1/C Q 0

59
RC Circuits

• Kirchhoffs Voltage Law gives a differential
  equation for the charge
• dQ/dt V/R - Q/RC
• Assuming the capacitor is initially discharged
  the solution (Youll work it out in lab)
• Q CV(1 e-t/(RC))

60
Charging a Capacitor
When t RC 5 s, Vc .63 6V 3.78 V
61
Current in a Charging RC Circuit

• i dq/dt V/R e-t/RC
• The current exponentially decays with the same
  time constant

62
Discharging a Capacitor

• Assume that the capacitor is initially charged
  with Q0 CV0
• What will happen when the switch is closed

R
C
63
Discharging a Capacitor

• -R dq/dt q/C 0
• dq/dt -q/(RC)
• A solution is q CV0 e-t/(RC)
• Solution is exponential decay
• RC is the time constant

64
Ex. a) What fraction of the original charge
remains when t RC? b) At what time is charge
reduced to a fraction f of the initial amount?
65
Current in a Discharging RC Circuit

• i dq/dt -V/R e-t/RC
• The current exponentially decays with the same
  time constant
• It flows in the opposite direction