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Electrical Potential EnergyIn Chapter 15, we saw that the
gravitational and electrical (Coulomb) forces have similar
forms
This similarity also leads to a similarity between the potential
energies associated with each force
Ue depends on magnitude and sign of a pair of charges Ue is
positive (negative) when q1 and q2 have the same (opposite)
signRemember: potential energy is a scalar
quantityGravityElectricalGravityElectrical(can be obtained directly
through calculus)
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Electrical Potential EnergyComparison of Ug and Ue as a function
of separation distance:
If 2 charges have opposite (same) signs, the potential energy of
the pair increases (decreases) with separation distanceCharges
always move from high to low potential energyPositive (negative)
charges move in the same (opposite) direction as the electric field
UgrUerUerq1q2 < 0q1q2 > 0
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CQ 1: A positively charged particle starts at rest 25 cm from a
second positively charged particle which is held stationary
throughout the experiment. The first particle is released and
accelerates directly away from the second particle. When the first
particle has moved 25 cm, it has reached a velocity of 10 m/s. What
is the maximum velocity that the first particle will reach? 10 m/s
14 m/s20 m/sSince the first particle will never escape the electric
field of the second particle, it will never stop accelerating, and
will reach an infinite velocity.
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Electric PotentialElectric potential is defined as the electric
potential energy per unit chargeScalar quantity with units of volts
(1 V = 1 J/C)Sometimes called simply potential or voltageElectric
potential is characteristic of the field only, independent of a
test charge placed in that fieldPotential energy is a
characteristic of a charge-field system due to an interaction
between the field and a charge placed in the fieldWhen a positive
(negative) charge is placed in an electric field, it moves from a
point of high (low) potential to point of lower (higher) potential
Higher potentialLower potential
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Electric PotentialWhen a point charge q moves between 2 points A
and B, it moves through a potential difference:
The potential difference is the change in electric potential
energy per unit charge:The electric force on any charge (+ or ) is
always directed toward regions of lower electric potential energy
(just like gravity)For a positive (negative) charge, lower
potential energy means lower (higher) potentialHelpful detail: E
points in the direction of decreasing V Electric potential created
by a point charge:Depends only on q and rPotential vs. Potential
Energy
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Example Problem #16.17Solution (details given in class):11.0
kVThe three charges shown in the figure are at the vertices of an
isosceles triangle. Let q = 7.00 nC, and calculate the electric
potential at the midpoint of the base.
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Potential Differences in Biological SystemsAxons (long
extensions) of nerve cells (neurons)In resting state, fluid inside
has a potential that is 85 mV relative to the fluid outside (due to
differences in +/ ion concentrations)A nerve impulse causes the
outer membrane to become permeable to + Na ions for about 0.2
msThis changes polarity of inside fluid to +Potential difference
across cell membrane changes from about 85 mV to +60 mVRestoration
of resting potential involves the diffusion of K and pumping of Na
ions out of cell (active transport)As much as 20% of the resting
energy requirements of the body are used for active transport of Na
ions
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Potential Differences in MedicinePolarity changes across
membranes of muscle cellsMuscle cells have a layer of ions on the
inside of the membrane and + ions on the outsideJust before each
heartbeat, + ions are pumped into the cells, neutralizing the
potential difference (depolarization)Cells become polarized again
when the heart relaxesElectrocardiogram (EKG)Measures potential
difference between points on chest as a function of
timePolarization and depolarization of cells in heart causes
potential differences that are measured by
electrodesElectroencephalogram (EEG) and Electroretinogram
(ERG)Measures potential differences caused by electrical activity
in the brain (EEG) and retina (ERG)
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Potentials and Charged ConductorsWe know that: DU = W (from last
semester) and DU = qDV Combining these two equations:No work is
required to move a charge between two points at the same electric
potentialFor a charged conductor in equilibrium:No work is done by
E if charge is moved between points A and BSince W = 0, VB VA = 0
at surfaceSince E = 0 inside a conductor, no work is required to
move a charge inside conductor (thus DV = 0 inside as
well)Conclusion: Electric potential is constant everywhere inside a
conductor and is equal to its (constant) value at the surface
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CQ 2: Two charged metal plates are placed one meter apart
creating a constant electric field between them. A one Coulomb
charged particle is placed in the space between them. The particle
experiences a force of 100 Newtons due to the electric field. What
is the potential difference between the plates? 1 V 10 V100 V1000
V
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CQ 3: How much work is required to move a positively charged
particle along the 15 cm path shown, if the electric field E is 10
N/C and the charge on the particle is 8 C? (Note: ignore gravity)
0.8 J8 J12 J1200 J
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Equipotential SurfacesAn equipotential surface has the same
potential at every point on the surfaceSimilar to topographic map,
which shows lines of constant elevationSince DV = 0 for each
surface, W = 0 along the surfaceThus electric field lines are
perpendicular to the equipotential surfaces at all points E points
in the direction of the maximum decrease in DV (E points from high
to low potential)Similar to a topographic contour map (slope is
steepest perpendicular to lines of constant elevation)Electric
field is thus sometimes called the potential gradient (meaning
grade or slope)
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Equipotential SurfacesOn a contour map a hill is steepest where
the lines of constant elevation are close togetherIf equipotential
surfaces are drawn such that the potential difference between
adjacent surfaces is constant, then the surfaces are closer
together where the field is stronger
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Examples of Equipotential Surfaces
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CQ 4: Interactive Example Problem:Drawing Equipotential
Lines(PHYSLET Physics Exploration 25.1, copyright Pearson Prentice
Hall, 2004)Which equipotential plot best represents the electric
field pattern shown?Plot 1Plot 2Plot 3Plot 4
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CapacitanceA capacitor is a device that stores electrical
potential energy by storing separated + and charges2 conductors
separated by vacuum, air, or insulation+ charge put on one
conductor, equal amount of charge put on the other conductorA
battery or power supply typically supplies the work necessary to
separate the chargeSimplest form of capacitor is the parallel plate
capacitor2 parallel plates, each with same area A, separated by
distance dCharge +Q on one plate, Q on the otherIf plates are close
together, electric field will be uniform (constant) between the
platesCharging A Capacitor
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CapacitanceFor a uniform electric field, the potential
difference between the plates is (see Example Problem #16.6) DV =
Ed E is proportional to the charge, and DV is proportional to E
therefore the charge is proportional to DVThe constant of
proportionality between charge and DV is called capacitance
Capacity to hold charge for a given DV 1 F is very large unit:
typical values of C are mF, nF, or pFCapacitance depends on the
geometry of the plates and the material between the platesUnits: C
/ V = Farad (F)(for plates separated by air)
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Capacitors in Circuits and ApplicationsCapacitors are used in a
variety of electronic circuitsExample of circuit diagram consisting
of capacitors and a battery shown at rightMany practical uses of
capacitorsSome computer keyboard keys have capacitors with a
variable plate spacing below themMicrophones using capacitors with
one moving plate to create an electrical signalConstant potential
difference kept between plates by a batteryAs plate spacing
changes, charge flows onto and off of platesThe moving charge
(current) is amplified to generate signalTweeters (speakers for
high-frequency sounds) are microphones in reverseMillions of
microscopic capacitors used in each RAM computer memory chipCharged
and discharged capacitors correspond to 1 and 0 states
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CQ 5: Interactive Example Problem:Fun With Capacitors(PHYSLET
Physics Exploration 26.2, copyright Pearson Prentice Hall, 2004)If
a constant electric potential is maintained between the plates of
the capacitor, what happens to the charge on the capacitor?The
charge gets smaller.The charge gets larger.The charge stays the
same.The capacitor discharges.
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Combinations of CapacitorsCapacitors can be combined in circuits
to give a particular net capacitance for the entire circuitParallel
CombinationPotential difference across each capacitor is the same
and equal to DV of the battery Qtot = Q1 + Q2 + Q3 + Total
(equivalent) capacitance:
Series CombinationMagnitude of charge is the same on all plates
DV (battery) = DV1 + DV2 + DV3 + Total (equivalent)
capacitance:
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Example ProblemSolution (details given in class):1.8 102 mC (4.0
mF capacitor)89 mC (2.0 mF capacitor)Capacitors C1 = 4.0 mF and C2
= 2.0 mF are charged as a series combination across a 100V battery.
The two capacitors are disconnected from the battery and from each
other. They are then connected positive plate to positive plate and
negative plate to negative plate. Calculate the resulting charge on
each capacitor.
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Example Problem #16.35Solution (details given in class):2.67
mF24.0 mC (each 8.00-mF capacitor), 18.0 mC (6.00-mF capacitor),
6.00 mC (2.00-mF capacitor)3.00 V (each capacitor)Find (a) the
equivalent capacitance of the capacitors in the circuit shown, (b)
the charge on each capacitor, and (c) the potential difference
across each capacitor.
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Energy Stored in a Charged CapacitorIts easy to tell that a
capacitor stores (releases) energy when it charges (discharges)The
energy stored by the capacitor = work required to charge the
capacitor (typically performed by a battery or power supply)As more
and more charge is transferred between the plates, the charge,
voltage, and work done by battery increases (DW = DVDQ)Total work
done = total energy stored:
Defibrillators typically release about 1.2 kJ of stored energy
from capacitor with DV 5 kV
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Capacitors with DielectricsA dielectric is an insulating
material Rubber, plastic, glass, nylonWhen a dielectric is inserted
between the conductors of a capacitor, the capacitance
increasesCapacitance increases for a parallel-plate capacitor in
which a dielectric fills the entire space between the plates k =
dielectric constant (ratio of capacitance with dielectric to
capacitance without dielectric)For any given plate separation d,
there is a maximum electric field (dielectric strength) that can be
produced in the dielectric before it breaks down and conductsSee
Table 16.1 for values of k and dielectric strength for various
materials
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Capacitors with DielectricsThe molecules of the dielectric, when
placed in the electric field of a capacitor, become
polarizedCenters of positive and negative charges become
preferentially oriented in the field (see figure below at
left)Creates a net positive (negative) charge on the left (right)
side of the dielectric (see figure below at right)This helps
attract more charge to the conducting plates for a given DVSince
plates can store more charge for a given voltage, the capacitance
must increase (remember C = Q / DV )
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Capacitors with DielectricsTo increase capacitance while keeping
the physical size reasonable, plates are often made of a thin
conducting foil that is rolled into a cylinderDielectric material
is sandwiched in betweenHigh-voltage capacitor commonly consists of
interwoven metal plates immersed in silicone oilVery large
capacitances can be achieved with an electrolytic capacitor at
relatively low voltagesInsulating metal oxide layer forms on the
conducting foil and serves as a (very thin) dielectric