2/12/07 184 Lecture 19 1 PHY 184 PHY 184 Spring 2007 Lecture 19 Title: Kirchoff’s Rules for Circuits
Transcript
Slide 1
2/12/07184 Lecture 191 PHY 184 Spring 2007 Lecture 19 Title:
Kirchoffs Rules for Circuits
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2/12/07184 Lecture 192 Results of Midterm 1 (Sec 2) Pretty high
average; will even be higher after the correction set! 3/9 or 33.3%
--> 53% 5/9 or 55.6% --> 69% Correction set: Example
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2/12/07184 Lecture 193AnnouncementsAnnouncements Corrections
Set 1 will open soon Corrections Set 1 is just Midterm 1 with
different numbers You will get 30% credit for those problems you
did incorrectly on Midterm 1 However, you must do all the problems
in Corrections Set 1 to get credit After Corrections Set 1 is due,
we will open Midterm 1 so that you can see your particular
questions and answers This week we will continue with circuits and
start magnetism.
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2/12/07184 Lecture 194 Resistivity of Wires in a Circuit The
circuit on the right shows a resistor of resistance R=6 connected
to an ideal battery providing 12V by means of two copper wires.
Each wire has the length 20 cm and radius 1 mm. We generally
neglect their resistance, the voltage drop across them and the
energy dissipated. Justify by calculating the voltage across R
accounting for the influence of the wire! Idea: For each wire Total
resistance of the circuit is R+2R w
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2/12/07184 Lecture 195 Resistivity of Wires in a Circuit (2)
The circuit on the right shows a resistor of resistance R=6
connected to an ideal battery providing 12V by means of two copper
wires. Each wire has the length 20.cm and radius 1 mm. We generally
neglect their resistance, the voltage drop across them and the
energy dissipated. Justify by calculating the voltage across R
accounting for the influence of the wire! The voltage across R is
V=Ri to be compared to 12 V if we neglect the wires. Neglecting the
resistance of the wire introduces an error of only 0.03%
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2/12/07184 Lecture 196 Clicker Question The circuit on the
right shows a resistor of resistance R=6 connected to an ideal
battery providing 12V by means of two copper wires. Each wire has
the length 20.cm and radius 1 mm. We generally neglect their
resistance, the voltage drop across them and the energy dissipated.
What is the total voltage drop across the two wires (i=1.9993 A,
R=0.0011 each)? A) 5.5 10 -4 V B) 0.01875 V C) 4.4 mV D) 2.2 mV One
wire: V w = iR = 2.2 mV
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2/12/07184 Lecture 197CircuitsCircuits We have been working
with simple circuits containing either capacitors or resistors.
Capacitors wired in parallel Capacitors wired in series
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2/12/07184 Lecture 198 Circuits (2) Resistors wired in parallel
Resistors wired in series We dealt with circuits containing either
capacitors or resistors that could be resolved in systems of
parallel or series components.
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2/12/07184 Lecture 199 Complex Circuits Circuits can be
constructed that cannot be resolved into series or parallel systems
of capacitors or resistors
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2/12/07184 Lecture 1910 Kirchhoffs Rules for Multi-loop
Circuits To handle these types of circuits, we must apply
Kirchhoffs Rules. Kirchhoffs Rules can be stated as Kirchhoffs
Junction Rule The sum of the currents entering a junction must
equal the sum of the currents leaving a junction. Kirchhoffs Loop
Rule The sum of voltage drops around a complete circuit loop must
sum to zero.
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2/12/07184 Lecture 1911 Kirchhoffs Junction Rule Kirchhoffs
Junction Rule is a direct consequence of the conservation of
charge. In a conductor, charge cannot be created or destroyed. At a
junction: all charges streaming into the junction must also leave
the junction i1i1 i2i2 i3i3 i 1 =i 2 +i 3
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2/12/07184 Lecture 1912 Kirchhoffs Loop Rule Kirchhoffs Loop
Rule is a direct consequence of the conservation of electric
potential energy. Suppose that this rule was not valid we could
construct a way around a loop in such a way that each turn would
increase the potential of a charge traveling around the loop we
would always increase the energy of this charge, in obvious
contradiction to energy conservation Kirchhoffs Loop Rule is
equivalent to the law of energy conservation.
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2/12/07184 Lecture 1913 EMF Devices - Directions An emf device
(e.g., a battery) keeps the positive terminal (labeled +) at a
higher electrical potential than the negative terminal (labeled -).
When a battery is connected in a circuit, its internal chemistry
causes a net current inside the battery : positive charge carriers
move from the negative to the positive terminal, in direction of
the emf arrow. Or: Positive charge carriers move from a region of
low electric potential (negative terminal) to a region of high
electric potential (positive terminal) This flow is part of the
current that is set up around the circuit in that same
direction.
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2/12/0714
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2/12/0715 The flow of electrons is always from anodeto--cathode
outside of the cell (i.e., in the circuit) and from
cathodeto--anode inside the cell. Inside a chemical cell, ions are
carrying the electrons from cathodeto--anode inside the cell. Anode
(negative terminal): Zinc powder Cathode (positive terminal):
Manganese dioxide (MnO 2 ) powder Electrolyte: Potassium hydroxide
(KOH) The half-reactions are: At the cathode 2 MnO 2 + H 2 O + 2 e-
>Mn 2 O 3 + 2 OH- At the anode Zn + 2 OH- > ZnO + H 2 O + 2
e- The overall reaction is: Zn + 2MnO 2 > ZnO + Mn 2 O 3 +
[E=1.5 V]
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2/12/0716 Alkaline battery Al Kaline batter
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2/12/07184 Lecture 1917 Single Loop Circuits We begin our study
of more complicated circuits by analyzing circuits with several
sources of emf and resistors connected in series in a single loop.
We will apply Kirchhoffs Rules to these circuits. To apply these
rules: establish conventions for determining the voltage drop
across each element of the circuit depending on the assumed
direction of current and the direction of the analysis of the
circuit. Because we do not know the direction of the current in the
circuit before we start, we must choose an arbitrary direction for
the current.
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2/12/07184 Lecture 1918 Single Loop Circuits (2) We can
determine if our assumption for the direction of the current is
correct after the analysis is complete. If our assumed current is
negative, then the current is flowing in the direction opposite to
the direction we chose. We can also choose the direction in which
we analyze the circuit arbitrarily. Any direction we choose will
give us the same information.
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2/12/07184 Lecture 1919 Single Loop Circuits (3) If we move
around the circuit in the same direction as the current, the
voltage drop across a resistor will be negative. If we move around
the circuit in the direction opposite to the current, the voltage
drop across the resistors will be positive. If we move around the
circuit and encounter a source of emf pointing in the same
direction, we assume that this source of emf contributes a positive
voltage. If we encounter a source of emf pointing in the opposite
direction, we consider that component to contribute a negative
voltage.
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2/12/07184 Lecture 1920 Circuit Analysis Conventions i is the
magnitude of the assumed current
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2/12/07184 Lecture 1921 Single Loop Circuits We have studied
circuits with various networks of resistors but only one source of
emf. Circuits can contain multiple sources of emf as well as
multiple resistors. We begin our study of more complicated circuits
by analyzing a circuit with two sources of emf (V emf,1 and V emf,2
) and two resistors ( R 1 and R 2 ) connected in series in a single
loop We will assume that the two sources of emf have opposite
polarity.
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2/12/07184 Lecture 1922 Single Loop Circuits (2) We assume that
the current is flowing around the circuit in a clockwise direction.
Starting at point a with V=0, we analyze around the circuit in a
clockwise direction. Because the components of the circuit are in
series, all components have the same current, i
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2/12/07184 Lecture 1923 Start at point a. The first circuit
component is a source of emf V emf,1, which produces a positive
voltage gain of +V emf,1 Next we find resistor R 1, which produces
a voltage drop V 1 given by -iR 1 Continuing around the circuit we
find resistor R 2, which produces a voltage drop V 2 given by -iR 2
Next we meet a second source of emf, V emf,2 This source of emf is
wired into the circuit with a polarity opposite that of V emf,1 We
treat this component as a voltage drop of -V emf,2 rather than a
voltage gain We now have completed the circuit and we are back at
point a. Single Loop Circuits (3)
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2/12/07184 Lecture 1924 Kirchoffs loop rule for the voltage
drops states Generalization: the voltage drops across components in
a single loop circuit must sum to zero. This statement must be
qualified with conventions for assigning the sign of the voltage
drops around the circuit. Single Loop Circuits (4)
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2/12/07184 Lecture 1925 Now lets analyze the same circuit in
the counter- clockwise direction starting at point a The first
circuit element is V emf,2 which is a positive voltage gain The
next element is R 2 Single Loop Circuits (5)
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2/12/07184 Lecture 1926 Single Loop Circuits (6) Because we
have assumed that the current is in the clockwise direction and we
are analyzing the loop in the counter-clockwise direction, this
voltage drop is +iR 2 Proceeding to the next element in the loop, R
1, we use a similar argument to designate the voltage drop as +iR 1
The final element in the circuit is V emf,1, which is aligned in a
direction opposite to our analysis direction, so the voltage drop
across this element is -V emf,1
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2/12/07184 Lecture 1927 Single Loop Circuits (7) Kirchhoffs
Loop Rule then gives us Comparing this result with the result we
obtained by analyzing the circuit in the clockwise direction we see
that they are equivalent. The direction that we choose to analyze
the circuit does not matter.
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2/12/07184 Lecture 1928 Clicker Question The figure shows the
current i in a single-loop circuit with a battery B and a
resistance R. How should the emf arrow be drawn? A) pointing to the
right B) pointing to the left C) doesnt matter
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2/12/07184 Lecture 1929 Clicker Question The figure shows the
current i in a single-loop circuit with a battery B and a
resistance R. How should the emf arrow be drawn? A) pointing to the
right in direction of the current