Transcript Lecture 6
Lecture 6
Capacitance
Electric Current
Circuits
Resistance and Ohms law
Capacitors in Series
When a battery is connected to the circuit,
electrons are transferred from the left
plate of C1 to the right plate of C2 through
the battery
As this negative charge accumulates on
the right plate of C2, an equivalent
amount of negative charge is removed
from the left plate of C2, leaving it with an
excess positive charge
All of the right plates gain charges of –Q
and all the left plates have charges of +Q
More About Capacitors in
Series
An equivalent capacitor can be found that
performs the same function as the series
combination
The potential differences add up to the battery
voltage
Fig. 16-19, p.551
Fig. 16-20, p.552
Capacitors in Series, cont
V V1 V2
1
1
1
Ceq C1 C2
The equivalent capacitance of a
series combination is always less
than any individual capacitor in the
combination
Demo
Fig. P16-34, p.564
Fig. P16-35, p.564
Problem-Solving Strategy
Be careful with the choice of units
Combine capacitors following the
formulas
When two or more unequal capacitors are
connected in series, they carry the same
charge, but the potential differences across
them are not the same
The capacitances add as reciprocals and the
equivalent capacitance is always less than the
smallest individual capacitor
Problem-Solving Strategy,
cont
Combining capacitors
When two or more capacitors are
connected in parallel, the potential
differences across them are the same
The charge on each capacitor is
proportional to its capacitance
The capacitors add directly to give the
equivalent capacitance
Problem-Solving Strategy,
final
Repeat the process until there is only
one single equivalent capacitor
A complicated circuit can often be reduced
to one equivalent capacitor
Replace capacitors in series or parallel with their
equivalent
Redraw the circuit and continue
To find the charge on, or the potential
difference across, one of the capacitors,
start with your final equivalent
capacitor and work back through the
circuit reductions
Problem-Solving Strategy,
Equation Summary
Use the following equations when working
through the circuit diagrams:
Capacitance equation: C = Q / V
Capacitors in parallel: Ceq = C1 + C2 + …
Capacitors in parallel all have the same voltage
differences as does the equivalent capacitance
Capacitors in series: 1/Ceq = 1/C1 + 1/C2 + …
Capacitors in series all have the same charge, Q,
as does their equivalent capacitance
Fig. 16-21, p.553
Fig. P16-57, p.566
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 QV CV 2
2
2
2C
Fig. 16-22, p.554
Applications
Defibrillators
When fibrillation occurs, the heart produces
a rapid, irregular pattern of beats
A fast discharge of electrical energy through
the heart can return the organ to its normal
beat pattern
In general, capacitors act as energy
reservoirs that can slowly charged and
then discharged quickly to provide large
amounts of energy in a short pulse
Capacitors with Dielectrics
A dielectric is an insulating material
that, when placed between the plates of
a capacitor, increases the capacitance
Dielectrics include rubber, plastic, or waxed
paper
C = κCo = κεo(A/d)
The capacitance is multiplied by the factor κ
when the dielectric completely fills the
region between the plates
Capacitors with Dielectrics
Dielectric Strength
For any given plate separation,
there is a maximum electric field
that can be produced in the
dielectric before it breaks down
and begins to conduct
This maximum electric field is
called the dielectric strength
An Atomic Description of
Dielectrics
Polarization occurs when there is a
separation between the “centers of
gravity” of its negative charge and
its positive charge
In a capacitor, the dielectric
becomes polarized because it is in
an electric field that exists
between the plates
More Atomic Description
The presence of the
positive charge on
the dielectric
effectively reduces
some of the negative
charge on the metal
This allows more
negative charge on
the plates for a given
applied voltage
The capacitance
increases
Fig. 16-30, p.560
Table 16-1, p.557
Fig. 16-1, p.532
Fig. 16-23, p.557
Fig. 16-26, p.558
Fig. 16-28, p.560
Fig. 16-29a, p.560
Fig. 16-29b, p.560
Electric Current
Whenever electric charges of like signs
move, an electric current is said to exist
The current is the rate at which the
charge flows through this surface
Look at the charges flowing perpendicularly
to a surface of area A
Q
I
t
The SI unit of current is Ampere (A)
1 A = 1 C/s
Electric Current, cont
The direction of the current is the
direction positive charge would flow
This is known as conventional current
direction
In a common conductor, such as copper, the
current is due to the motion of the negatively
charged electrons
It is common to refer to a moving
charge as a mobile charge carrier
A charge carrier can be positive or negative
Current and Drift Speed
Charged particles
move through a
conductor of crosssectional area A
n is the number of
charge carriers per
unit volume
n A Δx is the total
number of charge
carriers
Current and Drift Speed,
cont
The total charge is the number of
carriers times the charge per carrier, q
The drift speed, vd, is the speed at
which the carriers move
ΔQ = (n A Δx) q
vd = Δx/ Δt
Rewritten: ΔQ = (n A vd Δt) q
Finally, current, I = ΔQ/Δt = nqvdA
Current and Drift Speed,
final
If the conductor is isolated, the
electrons undergo random motion
When an electric field is set up in
the conductor, it creates an electric
force on the electrons and hence a
current
Charge Carrier Motion in a
Conductor
The zig-zag black
line represents the
motion of charge
carrier in a
conductor
The net drift speed is
small
The sharp changes in
direction are due to
collisions
The net motion of
electrons is opposite
the direction of the
electric field Demo
Electrons in a Circuit
The drift speed is much smaller than
the average speed between collisions
When a circuit is completed, the electric
field travels with a speed close to the
speed of light
Although the drift speed is on the order
of 10-4 m/s the effect of the electric
field is felt on the order of 108 m/s
c = 3 x 108 m/s
Meters in a Circuit –
Ammeter
An ammeter is used to measure current
In line with the bulb, all the charge passing
through the bulb also must pass through
the meter
p.578
Fig. A17-1, p.591
Meters in a Circuit –
Voltmeter
A voltmeter is used to measure voltage
(potential difference)
Connects to the two ends of the bulb
Resistance
In a conductor, the voltage applied
across the ends of the conductor is
proportional to the current through
the conductor
The constant of proportionality is
the resistance of the conductor
V
R
I
Fig. 17-CO, p.568
Resistance, cont
Units of resistance are ohms (Ω)
1Ω=1V/A
Resistance in a circuit arises due to
collisions between the electrons
carrying the current with the fixed
atoms inside the conductor
Georg Simon Ohm
1787 – 1854
Formulated the
concept of
resistance
Discovered the
proportionality
between current
and voltages
Ohm’s Law
Experiments show that for many
materials, including most metals, the
resistance remains constant over a wide
range of applied voltages or currents
This statement has become known as
Ohm’s Law
ΔV = I R
Ohm’s Law is an empirical relationship
that is valid only for certain materials
Materials that obey Ohm’s Law are said to
be ohmic