Lecture 12
Chapter 31
Physics II
07.22.2015
Kirchhoff’s Laws
Course website:
http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII
Lecture Capture:
http://echo360.uml.edu/danylov201415/physics2spring.html
While still only a graduate student, he published a paper that included a pair of rules for the analysis of circuits (Kirchhoff’s laws of circuits).
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
Kirchhoff’s Law
Some circuits are too complicated to analyze
(none of the elements are in series/parallel)
Kirchhoff’s rules are very helpful.
To analyze a circuit means to find:
1. ΔV across each component
2. The current in each component
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
Kirchhoff’s Junction Law
For a junction, the law of conservation of current requires that:
At any junction point, the sum of all currents entering the junction
must equal the sum of all currents leaving the junction.
3
out
2
1
in
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Kirchhoff’s Loop Law
For any path that starts and ends at the same point:
The sum of all the potential differences encountered
while moving around a loop or closed path is zero.
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Department of Physics and Applied Physics
Tactics: Using Kirchhoff’s Loop Law
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ΔV across a battery
Travel direction
Travel direction
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ΔV’s across resistors
Lower V
Higher V
(Because I flows from higher V to lower V)
Current direction
+
_
Δ
Travel direction
Final point
Initial point
according to a travel direction
Current direction
_
+
Δ
Travel direction
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Department of Physics and Applied Physics
Example 31.1. Analyze the circuit
No junction points
Loop rule
=
+
(If our assumption turns out to be wrong, the current will be negative)
2) Choose a travel direction (say, CW) and a start point
Now we can find pot. differences across each resistor
Department of Physics and Applied Physics
‐
Travel direction
1) Assume CW direction of current
95.144, Summer 2015, Lecture 12
=
‐=
+
ConcepTest 1
Loop rule
What is ΔV across the
A) 0V
unspecified circuit element?
B) 1V
C) 2V
D) 3V
+12 V +ΔV - 8 V - 6 V = 0
ΔV= 2 V
Travel direction
Multi-Loop
Circuit
Let’s take a look at how the junction rule and loop rule
help us solve for the unknown values in multi-loop circuits.
In general: if there are N junctions
in a circuit, then there are N-1
independent junction equations
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
Loop rule
I
Travel direction
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Department of Physics and Applied Physics
I
Travel direction
Real Batteries. Internal resistance
To drive a current in a circuit we need a “charge pump”, a device that by
doing work on the charge carriers maintains a potential difference.
Let’s look at a gravitational analog of a battery:
Give me a
break! I do it
as fast as I
can!
∆
Terminal voltage
A person does work to maintain a steady flow of balls through “the circuit”.
However, this guy cannot move balls instantaneously. It takes time.
So there is a natural hindrance to a completely free flow.
To describe this hindrance we can introduce the internal resistance, r.
It is inside a battery and it cannot be separated from the battery.
ε
(EMF, )
Pot. difference of a battery without an internal resistance is called an electromotive force.
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
ConcepTest 2
Wheatstone Bridge
An ammeter A is connected
1) l
between points a and b in the
2) l/2
circuit below, in which the four
3) l/3
resistors are identical. The current
4) l/4
through the ammeter is:
5) zero
Since all resistors are identical,
the voltage drops are the same
across the upper branch and the
lower branch.
I
Thus, the potentials at points a
and b are also the same.
Therefore, no current flows.
V
RC circuit
(discharging)
Let’s apply Kirchhoff’s rule
to discharge RC circuit
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RC circuit (discharging)
The figure shows a
charged capacitor, a switch, and a resistor.
A circuit such as this, with resistors and capacitors,
is called an RC circuit.
At t = 0, the switch closes and the capacitor
begins to discharge through the resistor.
We wish to determine how the current through the resistor will vary as a
function of time after the switch is closed.
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Department of Physics and Applied Physics
RC circuit (discharging)
The figure shows an RC circuit, some time after the
switch was closed.
Kirchhoff’s loop law applied to this circuit
clockwise is:
Q and I in this equation are the instantaneous values of the capacitor charge and the
resistor current.
0
The resistor current
,
The resistor current is the rate at which charge is removed from the capacitor:
where the time constant  is:
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
RC circuit (discharging)
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
ConcepTest 3
Junction Rule
A) Capacitor A.
Which capacitor discharges more
quickly after the switch is closed?
time constant  = RC
  = 12 µs   = 15 µs
   
So the capacitor A discharges faster than B
B) Capacitor B.
C) They discharge at the same rate.
D) Can’t say without
knowing the initial amount of charge.
What you should read
Chapter 31 (Knight)
Sections





31.1
31.2
31.3
31.5
31.9
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Department of Physics and Applied Physics
Thank you
See you tomorrow
95.144, Summer 2015, Lecture 12
Department of Physics and Applied Physics
ConcepTest 5
Junction Rule
A) 2 A
What is the current in branch P?
B) 3 A
C) 5 A
D) 6 A
E) 10 A
The current entering the junction
5A
in red is 8 A, so the current
leaving must also be 8 A. One
P
8A
exiting branch has 2 A, so the
other branch (at P) must have 6 A.
Junction
2A
6A