Transcript Lecture_7

Chapter 24
Capacitance, Dielectrics,
Electric Energy Storage
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24-4 Electric Energy Storage
A charged capacitor stores electric energy;
the energy stored is equal to the work done
to charge the capacitor:
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24-4 Electric Energy Storage
National Ignition Facility (NIF)
Lawrence Livermore National Laboratory
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NIF
Laser system driven by 4000 300 μF capacitors
which store a total of 422 MJ. They take 60 s to
charge and are discharged in 400 μs.
1) What is the potential difference across each
capacitor?
2) What is the power delivered during the
discharge?
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NIF
Laser system driven by 4000 300 μF capacitors
which store a total of 422 MJ. They take 60 s to
charge and are discharged in 400 μs.
1) What is the potential difference across each
capacitor?
2) What is the power delivered during the
discharge?
Solution:
1) U = CV2/2 →V = (2U/C)1/2
→V = [2(422x106)/4000/300x10-6] ½ = 26.5 kV
2) P = W/t = U/t = 422x106 /400x10-6 ~ 1012 W
= 1000 GW! cf. 1.0-1.5 GW for power plant
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24-4 Electric Energy Storage
Conceptual Example 24-9: Capacitor plate
separation increased.
A parallel-plate capacitor carries charge Q
and is then disconnected from a battery. The
two plates are initially separated by a
distance d. Suppose the plates are pulled
apart until the separation is 2d. How has the
energy stored in this capacitor changed?
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24-4 Electric Energy Storage
The energy density, defined as the energy per
unit volume, is the same no matter the origin of
the electric field:
The sudden discharge of electric energy can be
harmful or fatal. Capacitors can retain their
charge indefinitely even when disconnected
from a voltage source – be careful!
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24-5 Dielectrics
A dielectric is an insulator, and is
characterized by a dielectric constant K.
Capacitance of a parallel-plate capacitor filled
with dielectric:
Using the dielectric constant, we define the
permittivity:
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24-5 Dielectrics
Dielectric strength is the
maximum field a
dielectric can experience
without breaking down.
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24-5 Dielectrics
Here are two experiments where we insert and
remove a dielectric from a capacitor. In the
first, the capacitor is connected to a battery,
so the voltage remains constant. The
capacitance increases, and therefore the
charge on the plates increases as well.
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24-5 Dielectrics
In this second experiment, we charge a
capacitor, disconnect it, and then insert the
dielectric. In this case, the charge remains
constant. Since the dielectric increases the
capacitance, the potential across the
capacitor drops.
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24-6 Molecular Description of
Dielectrics
The molecules in a dielectric, when in an
external electric field, tend to become oriented
in a way that reduces the external field.
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24-6 Molecular Description of
Dielectrics
This means that the electric field within the
dielectric is less than it would be in air, allowing
more charge to be stored for the same potential.
This reorientation of the molecules results in an
induced charge – there is no net charge on the
dielectric, but the charge is asymmetrically
distributed.
The magnitude of the induced charge depends on
the dielectric constant:
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Summary of Chapter 24
• Capacitor: device consisting of nontouching
conductors carrying equal and opposite
charge.
• Capacitance:
• Capacitance of a parallel-plate capacitor:
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Summary of Chapter 24
• Capacitors in parallel:
• Capacitors in series:
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Summary of Chapter 24
• Energy density in electric field:
• A dielectric is an insulator.
• Dielectric constant gives ratio of total field to
external field.
• For a parallel-plate capacitor:
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Chapter 25
Electric Currents and
Resistance
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25-1 The Electric Battery
Volta discovered that
electricity could be
created if dissimilar
metals were
connected by a
conductive solution
called an electrolyte.
This is a simple
electric cell.
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25-1 The Electric Battery
Several cells connected together make a
battery, although now we refer to a single cell
as a battery as well.
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25-2 Electric Current
Electric current is the rate of flow of charge
through a conductor:
The instantaneous current is given by:
Unit of electric current: the ampere, A:
1 A = 1 C/s.
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25-2 Electric Current
A complete circuit is one where current can
flow all the way around. Note that the
schematic drawing doesn’t look much like the
physical circuit!
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25-2 Electric Current
Example 25-1: Current is flow of charge.
A steady current of 2.5 A exists in a wire
for 4.0 min. (a) How much total charge
passed by a given point in the circuit
during those 4.0 min? (b) How many
electrons would this be?
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ConcepTest 25.1
Which is the correct way to
light the lightbulb with the
Connect the Battery
4) all are correct
5) none are correct
battery?
1)
2)
3)
ConcepTest 25.1
Which is the correct way to
light the lightbulb with the
Connect the Battery
4) all are correct
5) none are correct
battery?
1)
2)
3)
Current can flow only if there is a continuous connection from
the negative terminal through the bulb to the positive terminal.
This is the case for only Fig. (3).
25-2 Electric Current
By convention, current is defined as flowing
from + to -. Electrons actually flow in the
opposite direction, but not all currents consist
of electrons.
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25-3 Ohm’s Law: Resistance and
Resistors
Experimentally, it is found that the
current in a wire is proportional to
the potential difference between its
ends:
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25-3 Ohm’s Law: Resistance and
Resistors
The ratio of voltage to current is called the
resistance:
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25-3 Ohm’s Law: Resistance and
Resistors
In many conductors, the
resistance is independent
of the voltage; this
relationship is called
Ohm’s law. Materials that
do not follow Ohm’s law
are called nonohmic.
Unit of resistance:
the ohm, Ω:
1 Ω = 1 V/A.
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25-3 Ohm’s Law: Resistance and
Resistors
Conceptual Example 25-3: Current and
potential.
Current I enters a resistor R as shown. (a)
Is the potential higher at point A or at point
B? (b) Is the current greater at point A or at
point B?
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25-3 Ohm’s Law: Resistance and
Resistors
Example 25-4: Flashlight bulb
resistance.
A small flashlight bulb draws 300
mA from its 1.5-V battery. (a) What
is the resistance of the bulb? (b) If
the battery becomes weak and the
voltage drops to 1.2 V, how would
the current change?
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25-3 Ohm’s Law: Resistance and
Resistors
Some clarifications:
• Batteries maintain a (nearly) constant
potential difference; the current varies.
• Resistance is a property of a material or
device.
• Current is not a vector but it does have a
direction.
• Current and charge do not get used up.
Whatever charge goes in one end of a circuit
comes out the other end.
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ConcepTest 25.2
You double the voltage
across a certain conductor
and you observe the current
increases three times. What
can you conclude?
Ohm’s Law
1) Ohm’s law is obeyed since the
current still increases when V
increases
2) Ohm’s law is not obeyed
3) this has nothing to do with Ohm’s
law
ConcepTest 25.2
You double the voltage
across a certain conductor
and you observe the current
increases three times. What
can you conclude?
Ohm’s Law
1) Ohm’s law is obeyed since the
current still increases when V
increases
2) Ohm’s law is not obeyed
3) this has nothing to do with Ohm’s
law
Ohm’s law, V = IR, states that the
relationship between voltage and
current is linear. Thus, for a conductor
that obeys Ohm’s law, the current must
double when you double the voltage.
Follow-up: Where could this situation occur?
25-4 Resistivity
The resistance of a wire is directly
proportional to its length and inversely
proportional to its cross-sectional area:
The constant ρ, the resistivity, is
characteristic of the material.
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25-4 Resistivity
This table gives the resistivity and temperature
coefficients of typical conductors, semiconductors,
and insulators.
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25-4 Resistivity
Example 25-5: Speaker wires.
Suppose you want to connect
your stereo to remote
speakers. (a) If each wire must
be 20 m long, what diameter
copper wire should you use to
keep the resistance less than
0.10 Ω per wire? (b) If the
current to each speaker is 4.0
A, what is the potential
difference, or voltage drop,
across each wire?
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25-4 Resistivity
For any given material, the resistivity
increases with temperature:
Semiconductors are complex materials, and
may have resistivities that decrease with
temperature.
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25-4 Resistivity
Example 25-7: Resistance thermometer.
The variation in electrical resistance with
temperature can be used to make precise
temperature measurements. Platinum is
commonly used since it is relatively free from
corrosive effects and has a high melting point.
Suppose at 20.0°C the resistance of a platinum
resistance thermometer is 164.2 Ω. When
placed in a particular solution, the resistance is
187.4 Ω. What is the temperature of this
solution?
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ConcepTest 25.3a
Wires I
Two wires, A and B, are made of the
1) dA = 4dB
same metal and have equal length,
2) dA = 2dB
but the resistance of wire A is four
times the resistance of wire B. How
do their diameters compare?
3) dA = dB
4) dA = 1/2dB
5) dA = 1/4dB
ConcepTest 25.3a
Wires I
Two wires, A and B, are made of the
1) dA = 4dB
same metal and have equal length,
2) dA = 2dB
but the resistance of wire A is four
times the resistance of wire B. How
do their diameters compare?
3) dA = dB
4) dA = 1/2dB
5) dA = 1/4dB
The resistance of wire A is greater because its area is less than
wire B. Since area is related to radius (or diameter) squared, the
diameter of A must be two times less than the diameter of B.
,
Rρ
A
25-5 Electric Power
Power, as in kinematics, is the energy
transformed by a device per unit time:
or
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25-5 Electric Power
The unit of power is the watt, W.
For ohmic devices, we can make the
substitutions:
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25-5 Electric Power
Example 25-8: Headlights.
Calculate the resistance of a 40-W
automobile headlight designed for 12 V.
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25-5 Electric Power
What you pay for on your electric bill is
not power, but energy – the power
consumption multiplied by the time.
We have been measuring energy in
joules, but the electric company
measures it in kilowatt-hours, kWh:
1 kWh = (1000 W)(3600 s) = 3.60 x 106 J.
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25-5 Electric Power
Example 25-9: Electric heater.
An electric heater draws a steady 15.0
A on a 120-V line. How much power
does it require and how much does it
cost per month (30 days) if it operates
3.0 h per day and the electric company
charges 9.2 cents per kWh?
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25-6 Power in Household Circuits
Conceptual Example 25-12: A dangerous
extension cord.
Your 1800-W portable electric heater is too
far from your desk to warm your feet. Its
cord is too short, so you plug it into an
extension cord rated at 11 A. Why is this
dangerous?
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25-7 Alternating Current
Current from a battery
flows steadily in one
direction (direct current,
DC). Current from a
power plant varies
sinusoidally (alternating
current, AC).
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25-7 Alternating Current
The voltage varies sinusoidally with time:
,,
as does the current:
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25-7 Alternating Current
Multiplying the current and the voltage gives
the power:
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25-7 Alternating Current
Usually we are interested in the average power:
.
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25-7 Alternating Current
The current and voltage both have average
values of zero, so we square them, take the
average, then take the square root, yielding the
root-mean-square (rms) value:
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25-7 Alternating Current
Example 25-13: Hair dryer.
(a) Calculate the resistance and the peak current
in a 1000-W hair dryer connected to a 120-V line.
(b) What happens if it is connected to a 240-V line
in Britain?
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
Electrons in a conductor have large, random
speeds just due to their temperature. When a
potential difference is applied, the electrons
also acquire an average drift velocity, which is
generally considerably smaller than the
thermal velocity.
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
We define the current density (current per
unit area) – this is a convenient concept
for relating the microscopic motions of
electrons to the macroscopic current:
If the current is not uniform:
.
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
This drift speed is related to the current in the
wire, and also to the number of electrons per unit
volume:
and
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
Example 25-14: Electron speeds in a wire.
A copper wire 3.2 mm in diameter carries a 5.0A current. Determine (a) the current density in
the wire, and (b) the drift velocity of the free
electrons. (c) Estimate the rms speed of
electrons assuming they behave like an ideal
gas at 20°C. Assume that one electron per Cu
atom is free to move (the others remain bound
to the atom).
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
The electric field inside a current-carrying
wire can be found from the relationship
between the current, voltage, and resistance.
Writing R = ρ l/A, I = jA, and V = El , and
substituting in Ohm’s law gives:
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25-8 Microscopic View of Electric
Current: Current Density and Drift
Velocity
Example 25-15: Electric field inside a wire.
What is the electric field inside the wire of
Example 25–14? (The current density was
found to be 6.2 x 105 A/m2.)
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25-9 Superconductivity
In general, resistivity
decreases as
temperature decreases.
Some materials,
however, have
resistivity that falls
abruptly to zero at a
very low temperature,
called the critical
temperature, TC.
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25-9 Superconductivity
Experiments have shown that currents, once
started, can flow through these materials for
years without decreasing even without a
potential difference.
Critical temperatures are low; for many years no
material was found to be superconducting above
23 K.
Since 1987, new materials have been found that
are superconducting below 90 K, and work on
higher temperature superconductors is
continuing.
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Summary of Chapter 25
• A battery is a source of constant potential
difference.
• Electric current is the rate of flow of electric
charge.
• Conventional current is in the direction that
positive charge would flow.
• Resistance is the ratio of voltage to current:
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Summary of Chapter 25
• Ohmic materials have constant resistance,
independent of voltage.
• Resistance is determined by shape and
material:
• ρ is the resistivity.
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Summary of Chapter 25
• Power in an electric circuit:
• Direct current is constant.
• Alternating current varies sinusoidally:
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Summary of Chapter 25
• The average (rms) current and voltage:
• Relation between drift speed and current:
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