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Chapter 28
Direct Current Circuits
1.
2.
3.
4.
5.
R connections in series and in parallel
Define DC (direct current), AC (alternating current)
Model of a battery
Circuits with 2+ batteries – Kirchhoff’s Rules
RC circuit
子曰:"温故而知新"
Confucius says, reviewing helps one learn
new knowledge.
Concepts:
Charge: positive, negative, conserve, induction.
potential V   E  ds E  V  (  i   j   k )V



F
x
x
x
E

F

q
E
Electric:
field
q
   E  dA
flux
Electrostatic equilibrium: no moving charge.
dQ
I

Current: moving charge dt
Capacitance: charge over potential C  QV
Resistance: potential over current R  IV
Resistance and Resistivity (conductivity) and
temperature: ρ  ρ [1 α(T  T )] R  R [1 α(T T )]
B
A
test
E
surface
o
o
o
o
Laws:

qq 
1 q1q2 
F12  k e 1 2 2 r12 
r12
40 r 2
r
Coulomb's: force and charge.
 
q
 E   E  dA 
Gauss’s:
electric flux and charge.
Ohm’s:
electric potential and current.
0
R
V
I
Circuits and components:
Symbols:
Power:
Capacitor:
Resistor:
wire, battery, C, Switch…
current times potential difference: p  I  V
1
1
1
1



C

C

C

C

...
in parallel
in series C C C C
discuss today
eq
1
2
3
eq
1
2
3
 ...
Resistor connections
In series.
Condition:
I  I1  I2
V2
V1
V  V1  V2
V V1  V2 V1 V2
Req 



 R1  R2
I
I
I1
I2
In parallel.
Condition:
I  I1  I2
V  V1  V2
I I
I
I
1
I
1
1

 1 2 1  2 

Req V
V
V1 V2 R1 R2
V1
V2
Resistor connections
In series, I  I1  I2 :
voltage sharing
power sharing
V1 R1

V2 R2
P1 R1

P2 R2
In parallel, V  V1  V2 :
current sharing
I1 R2
I1R1  I2R2 , or 
I2 R1
power sharing
V2
V1
P1 R2
P1R1  P2R2 , or

P2 R1
V1
V2
Resistors connections,
summary

In series
Req  R1  R2  R3  ...

In parallel
1
1
1
1



 ...
Req R1 R2 R3
I  I1  I2  I3  ...
V1:V2 :V3 :...  R1:R2 :R3 :...
P1:P2 :P3 :...  R1:R2:R3 :...
V  V1  V2  V3  ...
I1R1  I2R2  I3R3  ...
PR
1 1  P2R2  P3R3  ...
Resistors in Series – Example


Use the active figure to vary the battery voltage and the
resistor values
Observe the effect on the currents and voltages of the
individual resistors
PLAY
ACTIVE FIGURE
Resistors in Parallel – Example


Use the active figure to vary the battery voltage and the
resistor values
Observe the effect on the currents and voltages of the
individual resistors
PLAY
ACTIVE FIGURE
Combinations of
Resistors



The 8.0-W and 4.0-W
resistors are in series and
can be replaced with their
equivalent, 12.0 W
The 6.0-W and 3.0-W
resistors are in parallel and
can be replaced with their
equivalent, 2.0 W
These equivalent
resistances are in series
and can be replaced with
their equivalent resistance,
14.0 W
More examples
R1
R3
R2
R5
R4
Direct Current and Alternating
Current

When the current direction (not magnitude) in
a circuit does not change with time, the
current is called direct current


Most of the circuits analyzed will be assumed to
be in steady state: with constant magnitude and
direction, like the one powered through a battery.
When the current direction (often also the
magnitude) in a circuit changes with time, the
current is call alternating current.

The current from your car’s alternator is AC.
Model of a battery




Two parameters, electromotive
force (emf), , and the internal
resistance r, are used to model
a battery.
When a battery is connected in
a circuit, the electric potential
measured at its + and –
terminals are called The
terminal voltage V, with V =
 – Ir
If the internal resistance is zero,
the terminal voltage equals the
emf .
The internal resistance, r, does
not change with external load
resistance R, and this provides
the way to measure the internal
resistance.
V
battery
V
load
PLAY
ACTIVE FIGURE
Battery power figure
The power the battery generates
(through chemical reactions):
p  ε  I  (R  r )  I 2
The power the battery delivers to
the load, hence efficiency:
pload  V  I  R  I 2
efficiency =
pload
R

p
Rr
battery
V
load
The maximum power the battery
can deliver to a load
R
2
2
p

ε
We
have
load
From pload  R  I and ε  (R  r )  I
(R  r )2
Where the emf ε is a constant once the battery is given.
From
dpload  1
2R  2


ε 0
2
3
dR
 (R  r ) (R  r ) 
We get R  r to be the condition for
maximum pload , or power delivered to
the load.
Battery power figure
One can also obtain this result
from the plot of
pload 
R
2
ε
(R  r )2
battery
V
Where when R  r
pload reaches the maximum value
The efficiency of the battery at this
point is 50% because
pload
R
efficiency =

p
Rr
load
More complicated circuits, circuits with 2+
batteries: Kirchhoff’s Rules


A typical circuit that goes
beyond simplifications
with the parallel and
series formulas: the
current in the diagram.
Kirchhoff’s rules can be
used to solve problems
like this.
Rule 1:
Kirchhoff’s Junction Rule

Junction Rule, from charge
conservation:


The sum of the currents at any
junction must equal zero
Mathematically:

I 0
junction

The example on the left figure:

I1 - I2 - I3 = 0
Rule 2:
Kirchhoff’s Loop Rule


Choose your loop
Loop Rule, from energy
conservation:
 The sum of the potential
differences across all elements
around any closed circuit loop
must be zero
 Mathematically:
 V  0
V1
V2
Loop direction
Remember two things:
1. A battery supplies power.
 One needs to pay attention the
Potential rises from the “–”
sign (+ or -) of these potential
terminal to “+” terminal.
changes, following the chosen loop 2. Current follows the direction of
direction.
electric field, hence the
decrease of potential.
closed
loop
Kirchhoff’s rules
Strict steps in solving a problem
Step 1: choose and mark
the loop.
Step 2: choose and mark
current directions. Mark the
potential change on resistors.
Step 3: apply junction rule:
–
I2
+
L1
I1
I1  I2  I3  0
Step 4: apply loop rule:
L1: +2.00I3  12.0  4.00I2  0
L2:  8.00  2.00I3  6.00I1  0
Step 5: solve the three equations
for the three variables.
I3
–
+
L2
–
+
One more example
Step 1: choose and mark
the loop.
Step 2: choose and mark
current directions. Mark the
potential change on resistors.
Step 3: apply junction rule:
I1  I2  I3  0
–
L1
+
+
–
Step 4: apply loop rule:
L2
L1: +6.0I1  10.0  4.0I2  14.0  0
L2:  2.0I3  10.0  6.0I1  0
Step 5: solve the three equations
for the three variables.
–
+
RC Circuits,
solve with Kirchhoff’s rules



When a circuit contains a
resistor and a capacitor
connected in series, the circuit
is called a RC circuit.
Current in RC circuit is DC,
but the current magnitude
changes with time.
There are two cases: charging
(b) and discharging (c).
Not a
circuit
charging
Discharging
Charging a Capacitor
When the switch turns to position
a, current starts to flow and the
capacitor starts to charge.
Kirchhoff’s rule says:
+
–
+
Loop
–
ε  Vc  VR  0
Re-write the equation in terms of the
charge q in C and the current I, and then
only the variable q:
q
q
dq
ε   RI  0 and then ε   R
0
C
C
dt
Solve for q:
The current I is
t


RC
q  t   Cε  1 e 


t
dq ε RC
I t  
 e
dt R
Here RC has the unit
of time t, and is called
the time constant.
Charging a Capacitor, graphic
presentation

The charge on the
capacitor varies with time


q(t) = C (1 – e-t/RC)
= Q(1 – e-t/RC)
The current decrease with
time
ε t RC
I( t )  e
R

t is the time constant
 t = RC
Discharging a Capacitor
When the switch turns to position b,
after the capacitor is fully charged
to Q, current starts to flow and the
capacitor starts to discharge.
Kirchhoff’s rule says:
+
–
Loop
+
Vc  VR  0
Re-write the equation in terms of the
charge q in C and the current I, and then
only the variable q:
q
q
dq
 RI  0 and then  R
0
C
C
dt
Solve for q:
The current I is
q  t   Qe
t
RC
t
dq Q RC
I t   

e
dt RC
–
Ii 
Q
RC
RC Circuit, Example


Adjust the values of R
and C
Observe the effect on
the charging and
discharging of the
capacitor
PLAY
ACTIVE FIGURE