DC Circuits II: Kirchhoff's Rules & Loop Equations
DC Circuits II
Physics 2415 Lecture 13
Michael Fowler, UVa
Today’s Topics
•
Kirchhoff’s rules Review
•
RC Circuits
•
Ammeters and Voltmeters
General Circuits: Kirchhoff’s Rules
•
Junction Rule:
when several wires meet at a
point, the total current flowing into the point
must equal the total current flowing out. Charge
cannot disappear, or pile up at a point.
•
Loop Rule:
the total potential (voltage) change
on following wires around a loop to your starting
point must be zero.
•
(
The loop rule is equivalent to saying that if you follow
some random path on a hillside, and get back eventually
to your starting point, your net change in height above
sea level is zero
.)
Using Kirchhoff’s Rules
•
Junction Rule:
as you draw the circuit diagram,
use
this rule as much as you can to get the
smallest number of different unknown currents
(one current for each loop).
•
And, the number can sometimes be reduced by
symmetry.
•
Loop Rule:
the total potential (voltage) change
on following wires
around a loop
to your starting
point must be
zero
: that’s
adding voltages from
batteries to the right
IR
for each resistance
.
Clicker Question
•
A current
I
flows through the
network of resistances (all
different) shown.
•
What is the
minimum
number
of unknown currents it is
necessary to introduce to find
the total effective resistance
using Kirchhoff’s laws?
A.
2
B.
3
C.
4
D.
5
•
.
I
R
1
R
3
R
4
R
2
R
5
I
Clicker Answer
•
A current
I
flows through the
network of resistances (all
different) shown.
•
What is the
miminum
number
of unknown currents it is
necessary to introduce to find
the total effective resistance
using Kirchhoff’s laws?
A.
2
B.
3
C.
4
D.
5
•
.
I
R
1
R
3
R
4
R
2
R
5
I
I-I
1
I
1
I-I
1
-
I
2
I
2
I
1
+
I
2
The Loop Equations
•
Going round both loops
clockwise,
and rearranging:
Set
I
= 1 and solve for
I
1
,
I
2
.
•
.
I
R
1
R
3
R
4
R
2
R
5
I
I-I
1
I
1
I-I
1
-
I
2
I
2
I
1
+
I
2
The Loop Equations
•
Having found
I
1
,
I
2
we can find
the voltage drop: along the path
abc (see figure)
V
drops by
and from
V = IR
we can find the
resistance
R
of this network.
•
If there are
batteries
within a network,
their
emf must obviously be included
in equating the total voltage change
on going around a loop to zero—be
careful about the battery direction!
•
.
I
R
1
R
3
R
4
R
2
R
5
I
I-I
1
I
1
I-I
1
-
I
2
I
2
I
1
+
I
2
a
b
c
Batteries in Series and Parallel
•
For batteries in series, the
voltages add, in parallel, for
identical batteries, they’re
equivalent to a larger battery
of the same voltage.
•
If you put batteries of different
voltages in parallel, the
stronger will charge the
weaker.
RC Circuits
•
Closing switch
S
connects
the plates of a charged
capacitor by a resistance
R
.
How fast does the charge
go down?
Video
!
Charging a Capacitor
•
On closing the switch, charge will
flow from the battery into the
initially empty capacitor. But how
fast?
•
.
Voltmeters and Ammeters
•
Historically, voltmeters and
ammeters passed current through a
small coil between the poles of a
magnet, the coil being free to turn,
but against a small spring. The coil
formed an electromagnet when
current flowed.
•
An ammeter used a low resistance
coil, all the current flowed through
it.
•
A voltmeter used a high resistance
coil, in parallel with the main circuit.
Voltmeters and Ammeters
•
An ammeter used a low resistance coil, all the current
flowed through it: so inserting the ammeter to
measure current slightly reduced the current!
•
A voltmeter used a high resistance coil,
in parallel
with
the main circuit. Inserting the voltmeter in parallel
reduced the resistance
, and therefore the voltage
measured.
•
These meters are in fact obsolete—they’ve been
replaced with DA converters, which affect the
quantities being measured far less.
•
.
RC
Bullet Speed
•
As bullet penetrates gate 1, battery is cut out, capacitor
C
begins to
discharge through
R
: voltage across capacitor .
•
As bullet goes through gate 2, capacitor is isolated, discharge
stops
.
•
The final voltage gives bullet time from gate 1 to gate 2.
RC
Bullet Speed: the Calculation
•
As bullet penetrates gate 1, battery is cut out, capacitor
C
discharges through
R
: voltage .
•
As bullet goes through gate 2, capacitor is isolated,
discharge
stops
, voltage gives time from gate 1 to gate 2.
•
Taking the initial voltage to be 8V and the final reading to
be 4.6V, we have
•
From the diagram,
•
So
•
This gives a bullet speed of about 450m/sec.
Today's lecture covers Kirchhoff's rules for DC circuits to analyze current flow and potential changes. The rules include the Junction Rule and Loop Rule, which help determine currents and voltages in complex circuits. Using these rules, we can simplify circuits and solve for unknown variables efficiently. The lecture also discusses solving circuit problems using Kirchhoff's laws and loop equations, demonstrating how to find total effective resistance and voltage drops in circuits.
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Presentation Transcript
DC Circuits II Physics 2415 Lecture 13 Michael Fowler, UVa
Todays Topics Kirchhoff s rules Review RC Circuits Ammeters and Voltmeters
General Circuits: Kirchhoffs Rules Junction Rule: when several wires meet at a point, the total current flowing into the point must equal the total current flowing out. Charge cannot disappear, or pile up at a point. Loop Rule: the total potential (voltage) change on following wires around a loop to your starting point must be zero. (The loop rule is equivalent to saying that if you follow some random path on a hillside, and get back eventually to your starting point, your net change in height above sea level is zero.)
Using Kirchhoffs Rules Junction Rule: as you draw the circuit diagram, use this rule as much as you can to get the smallest number of different unknown currents (one current for each loop). And, the number can sometimes be reduced by symmetry. Loop Rule: the total potential (voltage) change on following wires around a loop to your starting point must be zero: that s adding voltages from batteries to the right IR for each resistance.
Clicker Question A current I flows through the network of resistances (all different) shown. What is the minimum number of unknown currents it is necessary to introduce to find the total effective resistance using Kirchhoff s laws? A. 2 B. 3 C. 4 D. 5 . I R1 R2 R3 R4 R5 I
Clicker Answer A current I flows through the network of resistances (all different) shown. What is the miminum number of unknown currents it is necessary to introduce to find the total effective resistance using Kirchhoff s laws? A. 2 B. 3 C. 4 D. 5 . I R1 R2 I-I1-I2 I1+I2 R3 I2 R5 R4 I-I1 I1 I
The Loop Equations Going round both loops clockwise, ) ) ( 2 1 1 I I R I . I ( ( + = 0 I I R I R I R R1 R2 1 4 2 3 + 1 5 I-I1-I2 I1+I2 ) R3 = 0 I R I R I2 2 2 2 3 and rearranging: ( 4 R R I + ) + = R5 R4 R I R I R I 5 R 1 + 3 2 + 4 I-I1 I1 ( ) = R R I R I 2 1 1 2 3 2 1 Set I = 1 and solve for I1, I2. I
The Loop Equations Having found I1, I2 we can find the voltage drop: along the path abc (see figure) V drops by V I R = + . I c R1 R2 ( ) I-I1-I2 + I I R I1+I2 R3 1 5 1 2 2 I2 and from V = IR we can find the resistance R of this network. If there are batteries within a network, their emf must obviously be included in equating the total voltage change on going around a loop to zero be careful about the battery direction! b R5 R4 I-I1 I1 a I
Batteries in Series and Parallel For batteries in series, the voltages add, in parallel, for identical batteries, they re equivalent to a larger battery of the same voltage. If you put batteries of different voltages in parallel, the stronger will charge the weaker.
RC Circuits Video! initial charge Closing switch S connects the plates of a charged capacitor by a resistance R. How fast does the charge go down? ( ) Q t V t RI t C dQ t Q t dt RC RC = Q0 -Q0 C S R ( ) dt dQ t ( ) ( ) = = = R V(t) ( ) ( ) ( ) = = / t RC , Q t Q e 0 is called the decay time. t 0 RC 3RC 2RC
Charging a Capacitor initial charge zero On closing the switch, charge will flow from the battery into the initially empty capacitor. But how fast? . C E R = + = E / , / Q RC e / RI Q C = I + dQ dt R E / / V(t) dQ dt = ( ) / t RC 1 Q C E 0 RC 3RC 2RC
Voltmeters and Ammeters Historically, voltmeters and ammeters passed current through a small coil between the poles of a magnet, the coil being free to turn, but against a small spring. The coil formed an electromagnet when current flowed. An ammeter used a low resistance coil, all the current flowed through it. A voltmeter used a high resistance coil, in parallel with the main circuit. V A
Voltmeters and Ammeters An ammeter used a low resistance coil, all the current flowed through it: so inserting the ammeter to measure current slightly reduced the current! A voltmeter used a high resistance coil, in parallel with the main circuit. Inserting the voltmeter in parallel reduced the resistance, and therefore the voltage measured. V . A These meters are in fact obsolete they ve been replaced with DA converters, which affect the quantities being measured far less.
RC Bullet Speed As bullet penetrates gate 1, battery is cut out, capacitor C begins to discharge through R: voltage across capacitor . As bullet goes through gate 2, capacitor is isolated, discharge stops. The final voltage gives bullet time from gate 1 to gate 2. ( ) V e = / t RC V t 0 gate 2 gate 1 volts C V R 0.022 F 180k 8V
RC Bullet Speed: the Calculation As bullet penetrates gate 1, battery is cut out, capacitor C discharges through R: voltage . As bullet goes through gate 2, capacitor is isolated, discharge stops, voltage gives time from gate 1 to gate 2. Taking the initial voltage to be 8V and the final reading to be 4.6V, we have / t RC e V V = ( ) V e = / t RC V t 0 = = / 4.6/8 0.58. 0 From the diagram, = = 0.022 F, so = 4.0 10 sec. 3 180k , t R C RC So = = = 2.2 10 sec. 3 ln0.58 0.55, t 3 4 10 This gives a bullet speed of about 450m/sec.
