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Q
+
+
+
+
+
+
+
+
d
-Q
+ -
Capacitance C=Q/V
Unit: farad, F. 1 F=1 coulomb/volt
QUICK QUIZ 16.5
You charge a parallel-plate capacitor, remove it from the
the battery, and prevent the wires connected to the plates
plates from touching each other. When you pull the plates
plates farther apart, do the following quantities increase,
increase, decrease, or stay the same? (a) C; (b) Q; (c) E
E between the plates; (d) ΔV.
QUICK QUIZ 16.5 ANSWER
(a) C decreases
(b) Q stays the same
(c) E stays the same
(d) ΔV increases
Dielectric
• Insulating materials
• Capacitance increased by a factor Κ
Dielectric Constant, K
A measure of how effective it is in reducing an electric field
across the plates
Q
-Q
Vo
Co=Q/Vo
Q
-Q
+
+
+
V=Vo/K
C=Q/V=KQ/Vo=KCo
Electric Field Energy
Work must be done to separate positive and
negative charges against the Coulomb forces
attracting them together. This work is stored as
potential energy or electric field energy in a
capacitor.
e
-
+
Vinitial
V
The average potential
difference during the charge
transfer is:
<V>=(Vfinal+Vinitial)/2
=(Vf+0)/2=Vf/2
+
+Q+
+
+
e
V
The total charge transferred
is Q
- -Q
The work done is
Vfinal
W=U=Q<V>=(1/2) QV
-
=(1/2) CV2=(1/2) Q2/C
(C=Q/V)
Energy Stored in a Capacitor
• Energy stored = ½ Q ΔV
• From the definition of capacitance, this can
be rewritten in different forms
2
1
1
Q
Energy = QΔV = CΔV 2 =
2
2
2C
A Summary of the various formulas for charge Q,
potential difference V, capacitance and energy W
Known Quantities
Unknown
Quantity
Q=
V=
C, V C, Q Q, V
W, C
(2WC)1/2
CV
(2W/C)1/2
Q/C
C=
Q/V
W=
CV2/2 Q2/2C QV/2
W, V W, Q
2W/V
2W/Q
2W/V2 Q2/2W
Questions:
A capacitor is initially charged to 2 V. It is then connected to a
4 V battery. What is the ratio of the final to the initial energy
stored in the capacitor?
(a) 2
(c) 6
X (b) 4
(d) 8
(e) 10
The figure below shows four parallel plate capacitors: A, B, C, and D. Each
capacitor carries the same charge q and has the same plate area A. As
suggested by the figure, the plates of capacitors A and C are separated by a
distance d while those of B and D are separated by a distance 2d.
Capacitors A and B are maintained in vacuum while capacitors C and D
contain dielectrics with Constant κ = 5.
1.Which list below places the capacitors in order of increasing
capacitance?
(a) A, B, C, D (b) B, A, C, D (c) A, B, D, C (d) D, C, B, A (e) B, A, D, C
X
2.Which capacitor has the largest potential difference between its plates?
(a) A
(b) C
(c) A and D are the same (d) B X (e) D and larger than B or C.
Static Electricity
•
•
•
•
•
•
•
•
Coulomb’s Law: F=kQ1Q2/r2
Electric Field: E=F/Q
E=0 inside the a good conductor
Potential Difference or Voltage: V=Ed (uniform E)
Potential due to a point charge: V=kQ/r
Potential Energy:ΔPE=QV
Capacitance: C=Q/V
Electric Field Energy: U=QV/2=CV2/2=Q2/2C
Flowing Electricity
• A conducting path is needed
• A potential difference is also needed
Potential difference is analogous to water pressure
Chapter 17 Electric Currents
Electric Current
Ohm’s Law
Resistivity
Electric Power
Superconductivity
Electric Current
If the net charge Q goes past in the time
interval t, then the average current is
I=Q/t
(Electric Current)
The unit of electric current is the Ampere
(amps), A.
1 A=1coulomb/second
Electrons hop from one atom to the next, but the effects
of electricity travel at the speed of light
300,000 km/s (186,000 mile/s)
Direct Current, DC:
In DC, all the electrons move in the same direction
while the electricity flows. DC is produced by
batteries and similar devices
Alternating Current, AC:
In AC, the direction of electron movement changes
many times each second. AC is generated by
electric generators in power plants
In the US, the frequency of AC is 60 Hz, i.e., the
current changes direction 120 times per second.
This frequency is 50 Hz in much of the rest of the
world
For a current to occur,
• There must be a path between the two points
along which charge can flow (Metals, many
liquids, and plasmas (charged gases) are
conductors)
• There must be a potential difference between the
two points (A superconductor is an exception).
A large potential difference means a large “push”
given to each charge
The Role of Potential Difference
Flow of Water
Electric current
Greater height
Increased flow of water
The higher voltage, the more current
Why is Potential Difference
needed for a current to flow?
Resistance
Ohm’s Law
(German Physicist, Georg Ohm, 1787-1854)
I=V/R
Definition of Resistance
R=V/I
The unit for R is Ohm, Ω
1Ω=1V/1A
Which one of the following circuits has the largest resistance?
X
I
Linear
R=V/I
0
V
The Ohm’s law applies for many normal conductors, but fails
for other materials, and devices.
Non-Ohmic behavior
Factors that govern
R
• Material ρ The ability to carry electric
current varies drastically
• Length L The longer the conductor, the
greater its resistance
• Cross sectional area A The thicker the
conductor, the less its resistance
R=ρL/A
ρ is the resistivity, and depends on the material
used
ρ reflects intrinsic properties of the material
ρ is a function of temperature
The unit of ρ is Ω•m
Question:
The potential difference across the ends of a wire is doubled
in magnitude. If Ohm’s law is obeyed, which one of the
following statements concerning the resistance of the wire is
true?
(a) The resistance is one half of its original value.
(b) The resistance is twice its original value.
(c) The resistance is not changed. X
(d) The resistance increases by a factor of four.
(e) The resistance decreases by a factor of four.
Question:The resistance of a conductor does
not depend on its
(a) mass
(b) length
(c) cross-sectional area
(d) resistivity
Answer: a
Question: A certain wire has a resistance R.
The resistance of another wire, identical
with the first except for having twice its
diameter, is
(a) R/4
(b) R/2
(c) 2R
(d) 4R
Answer: a
Question: A certain piece of copper is to be shaped
into a conductor of minimum resistance. Its
length and cross-sectional area
(a) should be, respectively, L and A
(b) should be, respectively, 2L and A/2
(c) should be, respectively, L/2 and 2A
(d) do not matter, since the volume of copper
remains the same.
Answer: c
Electric Power
Electric energy is readily transformed
into other forms of energy
The work that must be done to take Q
through V
W=QV
Since I=Q/t or Q=It,
W=IVt (Electric work)
Recall P=work done/time interval=W/t,
P=IV (Electric Power)
Using Ohm’s law I=V/R,
P=I2R=V2/R
The unit for Power is Watt, 1W=1J/s
Question: Which draws more current, a 100-W
light bulb or a 75-W bulb? Which has the higher
resistance?
Answer:
Let P1=100 W and P2=75 W.
Since P=IV, V=constant
P1/P2=I1/I2=100/75=4/3
The 100-W bulb draws more current
Also, P=V2/R, where V=constant
P1/P2=R2/R1=100/75=4/3
The 75-W has higher resistance
Question: Electric power is transferred over large
distances at very high voltages. Explain how the
high voltage reduces power losses in the
transmission lines.
Answer: The power transmitted P=IV. So the
higher V, the smaller I.
Since the power lost P=I2R, the smaller I leads to
smaller I2R loss for a given transmission line of
resistance R.
ρ
Conductor
Superconductor
0
Tc
T
The highest Tc is about –150 C or –300 F found in
“high temperature” superconductors