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Lecture 11-1
Electric Current
Current = charges in motion
q dq
Magnitude
I  lim

x 0 t
dt
rate at which net positive charges
move across a cross sectional surface
I   J d A
A
Units:
[I] = C/s = A (ampere)
Current is a scalar, signed quantity, whose
sign corresponds to the direction of motion of
net positive charges by convention
J = current density
(vector) in A/m²
Lecture 11-2
Ohm’s Law
Resistance
(definition)
V
R
I
R
I
V
constant R
Ohm’s Law
L
R
A
Power dissipation :
dU
P
 IV  I 2 R  V 2 / R
dt
Lecture 11-3
EMF – Electromotive Force
• An EMF device is a charge pump that can maintain a potential
difference across two terminals by doing work on the charges
when necessary.
Examples: battery, fuel
cell, electric generator,
solar cell, fuel cell,
thermopile, …
• Converts energy (chemical, mechanical, solar, thermal, …)
into electrical energy.
 Within the EMF device, positive charges
are lifted from lower to higher potential.
 If work dW is required to lift charge dq,

dW

dq
   Volt
EMF
Lecture 11-4
Resistors in Series
 The current through devices
in series is always the same.
i
R1
R2
i
i
Req

ε
ε
  iR1  iR2  0
  iReq  0
Req  ( R1  R2 )
For multiple
resistors in series:
Req  R1  R2  R3  ...
Same equation for parallel connected capacitors
Lecture 11-5
Real Battery = Resistors in Series
 The current through devices in series is always the same.
i
Req

terminal
voltage
ε
  iReq  0
  ir  iR  0
Req  r  R
internal
resistance
 i

Rr
,
Vb  Va    ir
R

Rr
Lecture 11-6
Resistors in Parallel

  i1  i2  i3  Req
  i1R1  i2 R2  i3R3
Devices in parallel has the
same potential drop

Generally,




1
1
1
1



or
 

R1 R2 R3 Req
Req R1 R2 R3
1
1

Req
i Ri
Same equation for
capacitors connected in
serial
Lecture 11-7
Kirchhoff’s Rules
Kirchhoff’s Rule 1: Loop Rule
 When any closed loop is traversed completely in a circuit,
the algebraic sum of the changes in potential is equal to zero.
 V
i
0
 Coulomb force is conservative
loop
Kirchhoff’s Rule 2: Junction Rule
 The sum of currents entering any junction in a circuit is equal to
the sum of currents leaving that junction.
I  I
i
in
out
j
 Conservation of charge
 In and Out branches
 Assign Ii to each branch
Lecture 11-8
Circuit Analysis Tips
• Simplify using equivalent resistors
• Label currents with arbitary directions
•If the calculated current is negative, the real direction is opposite to the one
defined by you.
• Apply Junction Rule to all the labeled currents.
•Useful when having multiple loops in a circuit.
• Choose independent loops and define loop direction
•Imagine your following the loop and it’s direction to walk around the circuit.
• Use Loop Rule for each single loop
•If current I direction across a resistor R is the same as the loop direction,
potential drop across R is ∆V = −I×R, otherwise, ∆V = I×R
•For a device, e.g. battery or capacitor, rely on the direction of the electric
field in the device and the loop direction to determine the Potential drop
across the device
• Solve simultaneous linear equations
Lecture 11-9
Loop Example with Two EMF Devices
 V
i
0
loop
 IR1  IR2   2  Ir2  IR3  1  Ir1  0
 I
1   2
R1  R2  R3  r1  r2
If 1 <2, we have I<0 !?
This just means the actual current flows reverse to the assumed
direction. No problem!
Lecture 11-10
Finding Potential and Power in a Circuit
Va  0  12  I  1 V 
But what is I? Must
solve for I first!
I
12  4
 0.5 ( A)  0
1 5  5 1 4
Va  12  0.5  1  11.5(V )
Vb  Va  I  5  9(V )
P12V
The rest?
Just means
0 V here
supplied by
 12  0.5  6(W ) 12V battery
PR  0.52  16  4 (W ) dissipated by
into 4V battery
(charging)
resistors
P4V  4  0.5  2(W )
Lecture 11-11
Charging a Battery
• Positive terminal to positive terminal
• Charging EMF > EMF of charged device
good
battery
(12V)
Say, R+r1+r2=0.05 (R is for jumper cables).
Then,
12  11(V )
I
 20( A)
0.05 ()
battery being
charged (11V)
P 2  11  20  220 (W )
power into battery 2
• If connected backward,
12  11
I
 460 ( A)
0.05
 Large amount of gas produced
 Huge power dissipation in wires
Lecture 11-12
Using Kirchhoff’s Laws in Multiple Loop Circuits
• Identify nodes and use Junction Rule:
i3  i1  i2
• Identify independent loops and use Loop Rule:
1  i1R1  i2 R2   2  i1R1  0
 2  i1  i2  R1  i2 R2   2  i1  i2  R1  0
 2  i1  i2  R1  i1R1  1  i1R1  i1  i2  R1  0
Only two are
independent.
Lecture 11-13
Warm-up quiz
I1+I2
I2
• What’s the current I1 ?
I1
(a). 2.0A
(b). 1.0A
(c). -2.0A
(d). -1.0A
(e). Need more information to
calculate the value.
Lecture 11-14
Answer for the Warm-up quiz
I1+I2
I2
• Sketch the diagram
• Simplify using equivalent resistors
I1
• Label currents with directions
• Use Junction Rule in labeling
• Choose independent loops
• Use Loop Rule
Replace by equivalent
R=2 first.
• Solve simultaneous linear equations
18  12( I1  I 2 )  6 I1  0
 3I1  2 I 2  3
3I 2  21  2 I 2  6 I1  0
6 I1  5I 2  21
 I 2  3( A), I1  1( A)
Lecture 11-15
Ammeter and Voltmeter
Ammeter: an instrument used to
measure currents
• It must be connected in series.
• The internal resistance of an
ammeter must be kept as small as
possible.
Voltmeter: an instrument used to
measure potential differences
• It must be connected in parallel.
• The internal resistance of a
voltmeter must be made as large
as possible.
Lecture 11-16
Galvanometer Inside Ammeter and Voltmeter
Galvanometer: a device that detects small currents
and indicates its magnitude. Its own resistance Rg
is small for not disturbing what is being measured.
galvanometer
Ammeter: an instrument
used to measure currents
shunt resistor
Voltmeter: an instrument
used to measure potential
differences
galvanometer
Lecture 11-17
PHYS241 – Quiz 11A
What is the current through R1 ?
30
30
a. 0.575A
b. 0.5A
c. 0.75A
d. 0.33A
e. 1.5A
45V
R1
R3
R2
30
45V
Lecture 11-18
PHYS241 – Quiz11B
What is the current through R2 ?
10
10
a. 0.33A
b. 2.5A
c. 0.75A
d. 1.5A
e. 0.5A
15V
R1
R3
R2
10
15V
Lecture 11-19
PHYS241 – Quiz 11C
What is the current through R3 ?
20
20
a. 0.375A
b. 0.5A
c. 0.75A
d. 1A
e. 1.5A
30V
R1
R3
R2
20
30V