General Physics (PHY 2140)
Lecture 11
¾ Electricity and Magnetism
9 Direct current circuits
9 Kirchhoff’s rules
9 RC circuits
9 Magnetism
9Magnets
http://www.physics.wayne.edu/~apetrov/PHY2140/
Chapter 18-19
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Department of Physics and Astronomy
announces the Fall 2003 opening of
The Physics Resource Center
on Monday, September 22 in
Room 172 of Physics Research Building.
Hours of operation:
Mondays, Tuesdays, Wednesdays
Thursdays and Fridays
11 AM to 6 PM
11 AM to 3 PM
Undergraduate students taking PHY2130-2140 will be able to get assistance in this
Center with their homework, labwork and other issues related to their physics course.
The Center will be open: Monday, September 22 to Wednesday, December 10, 2003.
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Lightning Review
Last lecture:
1. DC circuits
9 EMF
9 Resistors in series
9 Resistors in parallel
∆V = E − Ir
Req = R1 + R2 + R3 + ...
1
1 1
1
= +
+ + ...
Req R1 R2 R3
Review Problem: The circuit below consists of two
identical light bulbs burning with equal brightness and a
single 12 V battery. When the switch is closed, the
brightness of bulb A
1. increases.
2. remains unchanged.
3. decreases.
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18.4 Kirchhoff’s rules and DC currents
The procedure for analyzing complex circuits is based on
the principles of conservation of charge and energy
They are formulated in terms of two Kirchhoff’s rules:
1. The sum of currents entering any junction must equal the
sum of the currents leaving that junction (current or
junction rule) .
2. The sum of the potential differences across all the
elements around any closed-circuit loop must be zero
(voltage or loop rule).
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a. Junction rule
As a consequence of the Law of the conservation of charge, we have:
•
The sum of the currents entering a node (junction point)
equal to the sum of the currents leaving.
Ia
Id
Ic
Ib
Ia + Ib = Ic + Id
Similar to the water flow in a pipe.
I a, I b, I c , and I d can each be either a positive
or negative number.
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b. Loop rule
As a consequence of the Law of the conservation of energy, we have:
•
The sum of the potential differences across all the
elements around any closed loop must be zero.
1. Assign symbols and directions of currents in the loop
„
If the direction is chosen wrong, the current will come out with a right
magnitude, but a negative sign (it’s ok).
2. Choose a direction (cw or ccw) for going around the loop.
Record drops and rises of voltage according to this:
„
„
„
„
If a resistor is traversed in the direction of the current: +V = +IR
If a resistor is traversed in the direction opposite to the current: -V=-IR
If EMF is traversed “from – to + ”: +E
If EMF is traversed “from + to – ”: -E
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b. Loop rule: illustration
Loops can be chosen arbitrarily. For example, the circuit below contains a
number of closed paths. Three have been selected for discussion.
Suppose that for each element, respective current flows from + to - signs.
-
+ v 2
-
v1
+
-
v3
- v5 +
v4
-
Path 1
+
Path 2
v6
+
+ v7 -
+
Path 3
v12
v10
-
-
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v8
+
+
+
+ v11 -
-
v9 +
7
b. Loop rule: illustration
“b”
•
-
Using sum of the drops = 0
+ v 2
-
v1
+
-
v3
v4
v10
-
+ v11 -
- v7 + v10 – v9 + v8 = 0
• “a”
v8
+
+
v12
Blue path, starting at “a”
+
+ v7 -
+
-
v6
+
+
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- v5 +
-
v9 +
Red path, starting at “b”
+v2 – v5 – v6 – v8 + v9 – v11
– v12 + v1 = 0
Yellow path, starting at “b”
+ v2 – v5 – v6 – v7 + v10 – v11
- v12 + v1 = 0
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Kirchhoff’s Rules: Single-loop circuits
Example: For the circuit below find I, V1, V2, V3, V4 and the power
supplied by the 10 volt source.
30 V
+
+
_
V1
20 Ω
_
10 V
_
"a"
•
+
_
V3
1.
_
15 Ω
40 Ω
I
+
V2
+
5Ω
+
_
V4
_
For convenience, we start at
point “a” and sum voltage
drops =0 in the direction of
the current I.
+10 – V1 – 30 – V3 + V4 – 20 + V2 = 0 (1)
+
20 V
2. We note that: V1 = - 20I, V2 = 40I, V3 = - 15I, V4 = 5I
(2)
3. We substitute the above into Eq. 1 to obtain Eq. 3 below.
10 + 20I – 30 + 15I + 5I – 20 + 40I = 0
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Solving this equation gives, I = 0.5 A.
(3)
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Kirchhoff’s Rules: Single-loop circuits (cont.)
30 V
+
+
_
V1
20 Ω
_
10 V
_
•
+
_
V3
Using this value of I in Eq. 2 gives:
"a"
_
15 Ω
40 Ω
I
+
V2
+
5Ω
+
_
V4
_
V1 = - 10 V
V3 = - 7.5 V
V2 = 20 V
V4 = 2.5 V
+
20 V
P10(supplied) = -10I = - 5 W
(We use the minus sign in –10I because the current is entering the + terminal)
In this case, power is being absorbed by the 10 volt supply.
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18.5 RC circuits
Charge across capacitor
When switch is closed, current flows because
capacitor is charging
CE
0.63
CE
As capacitor becomes charged, the current
slows because the voltage across the resistor is
ξ - Vc and Vc gradually approaches ξ.
Once capacitor is charged the current is zero
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q = Q (1 − e −t RC )
RC is called the time constant
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Discharging the capacitor in RC circuit
Charge across capacitor
If a capacitor is charged and the switch is closed, then
current flows and the voltage on the capacitor gradually
decreases.
Q
0.37Q
This leads to decreasing charge
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q = Qe
−t RC
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Example : charging the unknown capacitor
A series combination of a 12 kΩ resistor and an unknown capacitor is
connected to a 12 V battery. One second after the circuit is
completed, the voltage across the capacitor is 10 V. Determine the
capacitance of the capacitor.
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A series combination of a 12 kΩ resistor and an unknown capacitor is
connected to a 12 V battery. One second after the circuit is completed, the
voltage across the capacitor is 10 V. Determine the capacitance of the
capacitor.
I
Given:
R =12 kΩ
E = 12 V
V =10 V
C
Recall that the charge is
building up according to
R
q = Q (1 − e −t RC )
Thus the voltage across the capacitor changes as
Find:
V=
C=?
q Q
= (1 − e −t RC ) = E (1 − e −t RC )
C C
This is also true for voltage at t = 1s after the switch is closed,
V
V
− t RC
− t RC
= 1− e
⇒ e
= 1− ⇒
E
E
C=−
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−t
 V
= log 1 − 
RC
 E
t
1s
=−
= 46.5µ F
 V
 10 V 
R log  1 − 
(12, 000 Ω ) log 1 −

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E


V
12


Magnetism
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Magnetism
Magnetic effects from natural magnets have been known for a long
time. Recorded observations from the Greeks more than 2500 years
ago.
The word magnetism comes from the Greek word for a certain type
of stone (lodestone) containing iron oxide found in Magnesia, a
district in northern Greece.
Properties of lodestones: could exert forces on similar stones and
could impart this property (magnetize) to a piece of iron it touched.
Small sliver of lodestone suspended with a string will always align
itself in a north-south direction—it detects the earth’s magnetic field.
Bar Magnet
Bar magnet ... two poles: N and S
Like poles repel; Unlike poles attract.
Magnetic Field lines: (defined in same way as electric field lines,
direction and density)
S
•
N
Does this remind you of a similar case in electrostatics?
Electric Field Lines
of an Electric Dipole
Magnetic Field Lines
of a bar magnet
S
N
Magnetic Monopoles
Perhaps there exist magnetic charges, just like electric charges. Such an
entity would be called a magnetic monopole (having + or - magnetic
charge).
How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
S
N
S
N
S
N
Even an individual
electron has a
magnetic “dipole”!
• Many searches for magnetic monopoles—the existence of which
would explain (within framework of QM) the quantization of electric
charge (argument of Dirac)
• No monopoles have ever been found!
Source of Magnetic Fields?
What is the source of magnetic fields, if not magnetic charge?
Answer: electric charge in motion!
„
e.g., current in wire surrounding cylinder (solenoid)
produces very similar field to that of bar magnet.
Therefore, understanding source of field generated by bar magnet lies
in understanding currents at atomic level within bulk matter.
Orbits of electrons about nuclei
Intrinsic “spin” of
electrons (more
important effect)
Magnetic Fields in analogy with Electric Fields
Electric Field:
„ Distribution of charge creates an electric field E(r)
in the surrounding space.
„ Field exerts a force F=q E(r) on a charge q at r
Magnetic Field:
„ Moving charge or current creates a magnetic field
B(r) in the surrounding space.
„ Field exerts a force F on a charge moving q at r
„ (emphasis this chapter is on force law)
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Magnetic Materials
(a simple look at an advanced topic)
• Materials can be classified by how they respond to an
applied magnetic field, Bapp.
• Paramagnetic (aluminum, tungsten, oxygen,…)
• Atomic magnetic dipoles (~atomic bar magnets) tend to line up
with the field, increasing it. But thermal motion randomizes
their directions, so only a small effect persists: Bind ~ Bapp •10-5
• Diamagnetic (gold, copper, water,…)
• The applied field induces an opposing field; again, this is
usually very weak; Bind ~ -Bapp •10-5 [Exception: Superconductors
exhibit perfect diamagnetism Æ they exclude all magnetic fields]
• Ferromagnetic (iron, cobalt, nickel,…)
• Somewhat like paramagnetic, the dipoles prefer to line up with
the applied field. But there is a complicated collective effect
due to strong interactions between neighboring dipoles Æ they
tend to all line up the same way.
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•
Very strong enhancement. Bind ~ Bapp •10+5
Ferromagnets, cont.
• Even in the absence of an applied B, the dipoles tend to
strongly align over small patches – “domains”.
Applying an external field, the domains align to produce
a large net magnetization.
Magnetic
Domains
• “Soft” ferromagnets
• The domains re-randomize when the field is removed
• “Hard” ferromagnets
• The domains persist even when the field is removed
• “Permanent” magnets
• Domains may be aligned in a different direction by applying
a new field
• Domains may be re-randomized by sudden physical shock
• If the temperature is raised above the “Curie point” (770˚ for
9/29/2003 iron), the domains will also randomize Æ paramagnet
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Mini-quiz
1A
1B
•Which kind of material would you use in a video tape?
(a) diamagnetic
(c) “soft” ferromagnetic
(b) paramagnetic
(d) “hard” ferromagnetic
•How does a magnet attract screws, paper clips,
refrigerators, etc., when they are not “magnetic”?
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Mini-quiz
1A
•Which kind of material would you use in a video tape?
(a) diamagnetic
(c) “soft” ferromagnetic
(b) paramagnetic
(d) “hard” ferromagnetic
Diamagnetism and paramagnetism are far too weak to be
used for a video tape. Since we want the information to
remain on the tape after recording it, we need a “hard”
ferromagnet. These are the key to the information age—
cassette tapes, hard drives, ZIP disks, credit card strips,…
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Mini-quiz
•How does a magnet attract screws, paper clips,
refrigerators, etc., when they are not “magnetic”?
1B
The materials are all “soft” ferromagnets. The external
field temporarily aligns the domains so there is a net
dipole, which is then attracted to the bar magnet.
- The effect vanishes with no applied B field
- It does not matter which pole is used.
S
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N
End of paper clip
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A “bit” of history
IBM introduced the first
hard disk in 1957, when
data usually was stored on
tapes. It consisted of 50
platters, 24 inch diameter,
and was twice the size of
a refrigerator.
It cost $35,000 annually in leasing fees (IBM would not
sell it outright). It’s total storage capacity was 5 MB, a
huge number for its time!
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