Transcript Lecture18

Chapter 18: Direct-Current Circuits
Source of EMF
Homework assignment : 9,16,28,41,50
 What
is emf?
• A current is maintained in a closed circuit by a source of emf.
The term emf was originally an abbreviation for electromotive force
but emf is NOT really a force, so the long term is discouraged.
• Among such sources are any devices (batteries, generators etc.) that
increase the potential energy of the circulating charges.
• A source of emf works as “charge pump” that forces electrons to move in
a direction opposite the electrostatic field inside the source.
Source of EMF
 Maintaining
a steady current and electromotive force
• When a charge q goes around a complete circuit and returns to its
starting point, the potential energy must be the same as at the beginning.
• But the charge loses part of its potential energy due to resistance in a conductor.
• There needs to be something in the circuit that increases the potential energy.
• This something to increase the potential energy is called electromotive force
(emf). Units: 1 V = 1 J/C
• Emf (E) makes current flow from lower to higher potential. A device that
produces emf is called a source of emf.
source of emf
-If a positive charge q is moved from b to a inside the

b Fe
-

E
a
E +
Fn
current flow

E
source, the non-electrostatic force Fn does a positive
amount of work Wn=qE on the charge.
-This replacement is opposite to the electrostatic force
Fe, so the potential energy associated with the charge
increases by qVab . For an ideal source of emf Fe=Fn
in magnitude but opposite in direction.
-Wn=qE=qVab, so Vab=E=IR for an ideal source.
Source of EMF
 Internal
resistance
• Real sources in a circuit do not behave ideally; the potential difference
across a real source in a circuit is not equal to the emf.
Vab=E – Ir (terminal voltage, source with internal resistance r)
• So it is only true that Vab=E only when I=0. Furthermore,
E –Ir = IR or I = E / (R + r)
Source of EMF
 Real
battery
cc
I
a
b
r
dd
R
Battery

+
b
a
−
• Real battery has internal resistance, r.
• Terminal voltage, ΔVoutput = (Va −Vb) =  − I r.
•
Vout
 I r
I


R
R

I
Rr
Source of EMF
 Potential
in an ideal resistor circuit
c
d
a
b a
b
c
d
b
Source of EMF
 Potential
in a resistor circuit in realistic situation
c
I
d
R
Battery

r
a
b
-
+
V
ba

r
R

+
-
Ir
IR
0
d
c
ab
ab
Source of EMF
 Example
r  2 ,   12 V, R  4 
A
a
Vcd  Vab
b
V
ammeter
I

Rr
Vab  Vcd .

12 V
 2 A.
42
Vcd  IR  (2 A)(4 )  8 V.
Vab    Ir  12 V - (2 A)(2 )  8 V.
voltmeter
The rate of energy conversion in the battery is I  (12 V)(2 A)  24 W.
The rate of dissipatio n of energy in the battery is Ir 2  (2 A) 2 (2 )  8 W.
The electrical power output is I  I 2 r  16 W.
The power output is also given by Vbc I  (8 V)(2 A)  16 W.
It is also given by IR 2  (2 A) 2 (4 )  16 W.
Resistors in Series

Resistors in series
V
V
IR1  IR2  V  IReq  Req  R1  R2
In general you can extend this formula to : Req  i Ri
The equivalent resistance of a series combination of resistors is
algebraic sum of the individual resistances.
Resistors in Parallel
 Resistors
in parallel
V
V
+
-
+
-
V
V V
1
1
1
I  I1  I 2 
 

 
Req R1 R2
Req R1 R2
I1 R2
V  I1 R1  I 2 R2  
I 2 R1
1
1
In general you can extend this formula to :
 i
Req
Ri
Resistors in Series and Parallel
 Example
1:
Resistors in Series and Parallel
 Example:
(cont’d)
I2
R2
I4
R4
I3
I
R3
V
I  V / Req  12 V/2   6 A
I 3  V / R3  12 V/3   4 A
I 2  I 4  V /( R2  R3 )  12 V/(2  4 )  2 A
Resistors in Series and Parallel
 Example:
(cont’d)
Kirchhoff’s Rules
 Introduction
• Many practical resistor networks cannot be reduced to simple series-parallel
combinations (see an example below).
• Terminology:
-A junction in a circuit is a point where three or more conductors meet.
-A loop is any closed conducting path.
junction
Loop 2
i
i
i2
i1
i
Loop 1
i
i2
junction
Kirchhoff’s Rules
 Kirchhoff’s
junction rule
• The algebraic sum of the currents into any unction is zero:
 I  0 at any junction
Kirchhoff’s Rules
 Kirchhoff’s
loop rule
• The algebraic sum of the potential differences in any loop, including
those associated with emfs and those of resistive elements, must equal
zero.
V  0 for any loop
Kirchhoff’s Rules

Rules for Kirchhoff’s loop rule
 I  0 at any junction
V  0 for any loop
Kirchhoff’s Rules

Rules for Kirchhoff’s loop rule (cont’d)
Kirchhoff’s Rules

Solving problems using Kirchhoff’s rules
Kirchhoff’s Rules

Example 1
Kirchhoff’s Rules

Example 1 (cont’d)
Kirchhoff’s Rules

Example 1 (cont’d)
Kirchhoff’s Rules
Find
all the currents
 Example
2 including directions.
Loop 2
i
i
i2
i1
i
Loop 1
i
Loop 1
i2
Loop 2
0  8V  4V  4V  3i  2i1
 6i 2  4  2i1  0
 6i2  4  2(1A)  0
0  8  3i1  3i 2  2i1
0  8  5i1  3i 2
 6i 2  16  10i1  0
0  12  12i1  0
i 2  1A
multiply by 2
i = i1+ i2
i1  1A
i  2A
R-C Circuits

Charging a capacitor
R-C Circuits

Charging a capacitor (cont’d)
R-C Circuits

Charging a capacitor (cont’d)
R-C Circuits

Charging a capacitor (cont’d)
R-C Circuits

Charging a capacitor (cont’d)
R-C Circuits

Discharging a capacitor
R-C Circuits

Discharging a capacitor (cont’d)
R-C Circuits

Discharging a capacitor (cont’d)
R-C Circuits

Example 18.6 : Charging a capacitor in an RC circuit
An uncharged capacitor and a resistor
are connected in series to a battery.
If E=12.0 V, C=5.00 mF, and R=
8.00x105 , find (a) the time constant
of the circuit, (b) the maximum charge
on the capacitor, (c) the charge on the
capacitor after 6.00 s, (d) the potential
difference across the resistor after
6.00 s, and (e) the current in the resistor
at that time.
5
-6
(a)   RC  (8.00 10 )(5.00 10 F)  4.00 s
(b) From Kirchhoff’s loop rule:
Vbat  VC  VR  0
E
q
 IR  0  Q  CE 60.0 mC
c
when I=0, q=Q at max.
R-C Circuits

Example 18.6 : Charging a capacitor in an RC circuit
An uncharged capacitor and a resistor
are connected in series to a battery.
If E=12.0 V, C=5.00 mF, and R=
8.00x105 , find (a) the time constant
of the circuit, (b) the maximum charge
on the capacitor, (c) the charge on the
capacitor after 6.00 s, (d) the potential
difference across the resistor after
6.00 s, and (e) the current in the resistor
at that time.
t /
6.00 s/4.00s
)  46.6 mC
(c) q  Q(1  e )  (60.0 mC)(1  e
(d) VC  q / C  9.32 V
VR  Vbat  VC  12.0  (9.32 V)  -2.68 V
(e)
I
 VR
 3.4 10 6 A
R