1. Kirchhoff's
Rules

2. Objectives: To analyze
resistive, direct current
circuits with
Kirchhoff’s rules and
Ohm's law.

3. Junction Rule
‰ The sum of the currents entering any
junction must equal the sum of the currents
leaving that junction
„ -A statement of Conservation of Charge
„ Loop Rule
‰ The sum of the potential differences across
all the elements around any closed circuit loop
must be zero
„ -A statement of Conservation of Energy

4. Junction Rule
The sum of the currents entering any junction must
equal the sum of the currents leaving that junction.
The algebraic sum of the changes in potential across all
of the elements around any closed circuit loop must be
zero.
A junction is any point in a circuit where the current
has a choice about which way to go. The first rule, also
known as the point rule, is a statement of conservation
of charge. If current splits at a junction in a circuit, the
sum of the currents leaving the junction must be the
same as the current entering the junction.

5. Junction Rule:
Σ Iin = Σ Iout

6. • I1 =I2 + I3
• From Conservation of
Charge
• Diagram (b) shows a
mechanical analog

‰ In general, the
number of
times the junction rule
can be used is one fewer
than the number of
junction points in the
circuit

7. The second rule, also known as the
LOOP RULE, is a statement of
conservation of energy. Recall that
although charge is not "used up" as
current flows through resistors in a
circuit, potential is. As current flows
through each resistor of a resistive
circuit the potential drops. The sum
of the potential drops must be the
same as the applied potential.

8. Loop Rule:
∑ ΔV = 0
closed
loop

9. • Traveling around the
loop from
a to b
• In (a), the resistor is
traversed in the
direction of the current,
the potential across the
resistor is – IR
• In (b), the resistor is
traversed in the
direction opposite of the
current, the potential
across the resistor is +
IR

10. LoopRule,
final
• In (c), the source of emf
is traversed in the direction
of the emf (from – to +),
and the change in the
electric potential is +ε
• In (d), the source of emf
is traversed in the direction
opposite of the emf (from
+ to
-), and the change in the
electric potential is -ε

11. Loop Equations from Kirchhoff’s
Rules
• The loop rule can be used as often as
needed so long as a new circuit
element (resistor or battery) or a new
current appears in each new equation
• You need as many independent
equations as you have unknowns

12. Resistor-
Capacitor
Circuits

13. A Resistor-
Capacitor(RC) Circuit
is one where you
have a capacitor and
resistor in the same
circuit.

14. Behavior
of an RC
Circuit

15. Figure 3.Schematic of an RC circuit. The
components in the dotted box are analogous to a
square-wave generator with outputs at points and
. The switch continuously moves between points
and creating a square wave as shown in Figure 4a.

16. Suppose we connect a battery, with voltage, , across a
resistor and capacitor in series as shown by Figure 3.
This is commonly known as an RC circuit and is used
often in electronic timing circuits. When the switch (S)
is moved to position 1, the battery is connected to the
circuit and a time-varying current I (t) begins flowing
through the circuit as the capacitor charges. When the
switch is then moved to position 2, the battery is taken
out of the circuit and the capacitor discharges through
the resistor. If the switch is moved alternately between
positions 1 and 2 , the voltage across points A and B
can be plotted and would resemble Figure 4.

17. Figure 4. A voltage pattern known as a square wave. Moving
the switch in Figure 3 alternatively between positions 1 and
2 can produce this voltage pattern. When the switch is in
position 1, the input voltage is the peak voltage is Vo .
When the switch is moves to position 2 , the input voltage
drops to zero. A function generator more commonly
produces square-wave voltages.

18. This voltage pattern is known as a square wave, for obvious
reasons, and is commonly produced by a function generator. The
function generator is capable of producing voltages that behave
like a sine, square or saw-tooth functions. Additionally, the
frequency of the wave may be varied with the function generator.
The dotted-box in Figure 3 may be thought of as a function
generator with points A and B as outputs.
We will use a two-channel oscilloscope to monitor the important
voltages throughout the experiment. An oscilloscope is an
invaluable tool for testing electronic circuits by measuring
voltages over time, and Figure 5 shows the schematic for
monitoring an RC circuit with an oscilloscope. As shown in the
figure below, the input voltage from the square-wave generator is
monitored by channel one (CH 1) and the voltage across the
capacitor is monitored by channel two (CH 2).

19. Figure 5. The RC circuit diagram. The
oscilloscope's Channel 1 monitors the
function generator while Channel 2
monitors the voltage drop across the
capacitor.

20. The capacitor responds to the square-wave voltage input by
going through a process of charging and discharging. It is
shown below that during the charging cycle, the voltage
across the capacitor . When the switch is in
position , the square-wave generator outputs a zero voltage
and the capacitor discharges. It can also be shown that
during the discharging cycle, the voltage across the capacitor
is
Circuit designers must be careful to ensure that the period of
the square wave gives sufficient time for the capacitor to
fully charge and discharge. It can be shown3 that, as a
general rule of thumb, the time necessary for the capacitor
of an RC circuit to nearly completely charge to Vo, or
discharge to zero, is 4RC .

21. Here it should be noted that the product RC is known
as the time constant,t, and has units of time4. The time
constant is the characteristic time of the charging and
discharging behavior of an RC circuit and represents the
time it takes the current to decrease to of its
initial value, whether the capacitor is charging or
discharging. Over the period of one t, the voltage across
the charging capacitor increases by a factor
Conversely the voltage across the discharging capacitor
decreases by a factor of over the same period,
Put another way, in 1t the voltage across a charging
capacitor grows to 63.2% of its maximum voltage,Vo ,
and in 1t the voltage across a discharging capacitor
shrinks to 36.8% of Vo .

22. Figure 6a. The square wave that drives the RC circuit.
When the switch in Figure 3 is in position , the input
voltage is the peak voltage is . When the switch is moves
to position , the input voltage drops to zero. In this
experiment, this input voltage is read by the
oscilloscope's CH 1.

23. Figure 6b. The voltage drop across the capacitor of Figure 3 as
read by the oscilloscope's CH 2. The capacitor alternately charges
toward and discharges toward zero according to the input voltage
shown in Figure 6a. Here, the frequency (and therefore period) of
the input square wave voltage is exactly such that the capacitor is
allowed to fully charge and discharge. The time constant, , is
equivalent to , and is defined by Equations 11 or 14.

24. The Charging
Process

25. An RC Circuit: Charging
Circuits with resistors and batteries have time-
independent solutions: the current doesn't change as
time goes by. Adding one or more capacitors changes
this. The solution is then time-dependent: the current
is a function of time.
Consider a series RC circuit with a battery, resistor,
and capacitor in series. The capacitor is initially
uncharged, but starts to charge when the switch is
closed. Initially the potential difference across the
resistor is the battery emf, but that steadily drops (as
does the current) as the potential difference across the
capacitor increases.

26. Applying Kirchoff's loop rule:
e - IR - Q/C = 0
As Q increases I decreases, but Q changes
because there is a current I. As the current
decreases Q changes more slowly.
I = dQ/dt, so the equation can be written:
e - R (dQ/dt) - Q/C = 0

27. This is a differential equation that can be solved for Q as a
function of time. The solution (derived in the text) is:
Q(t) = Qo [ 1 - e-t/t ]
where Qo = C e and the time constant t = RC.
Differentiating this expression to get the current as a function of
time gives:
I(t) = (Qo/RC) e-t/t = Io e-t/t
where Io = e/R is the maximum current possible in the circuit.
The time constant t = RC determines how quickly the capacitor
charges. If RC is small the capacitor charges quickly; if RC is large
the capacitor charges more slowly.

28. TIME CURRENT
0 Io
1*t Io/e = 0.368 Io
2*t Io/e2 = 0.135 Io
3*t Io/e3 = 0.050 Io

29. The
DisCharging
Process

30. What happens if the capacitor
is now fully charged and is
then discharged through the
resistor? Now the potential
difference across the resistor is
the capacitor voltage, but that
decreases (as does the current)
as time goes by.

31. Applying Kirchoff's loop rule:
-IR - Q/C = 0
I = dQ/dt, so the equation can be written:
R (dQ/dt) = -Q/C
This is a differential equation that can be solved for Q as a
function of time. The solution is:
Q(t) = Qo e-t/t
where Qo is the initial charge on the capacitor and the time
constant t = RC.

32. Differentiating this expression to get the current as a
function of time gives:
I(t) = -(Qo/RC) e-t/t = -Io e-t/t
where Io = Qo/RC
Note that, except for the minus sign, this is the same
expression for current we had when the capacitor was
charging. The minus sign simply indicates that the charge
flows in the opposite direction.
Here the time constant t = RC determines how quickly the
capacitor discharges. If RC is small the capacitor discharges
quickly; if RC is large the capacitor discharges more slowly.

33. SOURCES:
 physics.wustl.edu/introphys/Phys117_118/Lab_Manual/.../R
C.pdf
 www.physics.isu.edu/~hackmart/spl2kir.pdf
 www.phy-astr.gsu.edu/cymbalyuk/Lecture20.pdf
 www-
physics.ucsd.edu/students/courses/fall2010/.../21B_temp.p
df
 www.cbooth.staff.shef.ac.uk/phy101E&M/Kirchhoff.pdf
 bowlesphysics.com/images/AP_Physics_C_-
_RC_Circuits.pdfphysics.wustl.edu/introphys/Phys117_118/L
ab_Manual/.../RC.pdf
 http://webphysics.davidson.edu/physlet_resources/bu_sem
ester2/c11_RC.html
 http://www.clemson.edu/ces/phoenix/labs/223/rc/index.ht
ml