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Chapter 18
Electrical Energy
and
Capacitance
Chapter 18 Objectives
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Electrical potential
Electric Potential from a Point Charge
Capacitance
Parallel Plate Capacitor
Dielectrics
Electrostatic Force
• Because the Coulomb force is the
same as the gravitational force, it must
also be conservative
• So it fits the rules of conservative
energies
– ΔPE + ΔKE = 0
• Solving you see that change in
potential is opposite to change in
kinetic
– ΔKE = -ΔPE
• Apply Work-Kinetic Energy Theorem
– W = ΔKE
– W = -ΔPE = -qEd
Electric Potential
• The electric potential is the change in potential
energy of a charged object.
– Often referred to as a potential difference.
• Because it is the difference in potential energies of a
charged particle at differing locations.
• This can vary because of the magnitude of charge.
• Larger the charge, the more energy it has!
– Potential energy divided by charge
• SI unit is Volt
– V
• 1 V = 1 J /C
– denoted by V
• So the voltage is the measurement of the ability of the
charge to be moved, not the actual motion itself.
or
ΔV = V2-V1 =
ΔPE
/q = -Ed
Electric Potential Between Two Points
• Recall that in an electric field, electrons are
transferred from positive to negative.
• So particles move from positive locations to
negative locations.
– So a positive charge gains electric potential
energy when it is moved in a direction opposite
the electric field.
• Because it is being pulled away from the “attractive”
point, much like lifting a rock off the ground gives it
more potential energy.
– So a negative charge loses electric potential
energy when it moves in a direction opposite the
electric field.
• Because it is traveling away from the negative center
which is what it wants to do anyways.
Potential Energy Between Points
• The potential energy created from those
two points depends on the work done to
move the charges
– Opposite sign charges attract and work is
negative
– If the work is directly proportional to the separation
between the charges.
» So if the separation gets smaller, the work is
negative.
– Meaning the charges give off energy
• Same sign produces positive potential energy
– Meaning energy added to system
PE
electric
q1q2
= kC
r
This shows the electric potential energy due to point 1 created on point 2.
Electric Potential from Point Charge
• Every point in space has an electric potential, no
matter what charge.
• The potential depends on the size of the charge and
how far the charge is from the reference point.
– Electric potential is a scalar quantity, so direction does
not matter.
• But the sign does.
– So when asked to find the net electric potential, simply
find the algebraic sum of the individual potentials.
q
V = kC
r
As distance increases, potential decreases
Equipotential Surfaces
• A surface on which all points have the
same potential is called an
equipotential surface.
– No work is required to move a charge at
constant speed while on the surface.
– The electric field at every point on the
surface acts perpendicular to that point on
the surface.
• This really tells us that no matter the
surface characteristics, a diagram can
be drawn using each surface as a
single point source.
Capacitance
• We can now set two conducting surfaces, each being
a equipotential surface, close enough to each other to
create an electric field.
• The two surfaces do not have the same potential
difference, therefore work can be done between the two.
– As the two surfaces are charging by an outside voltage
source, electrons are being taken from one surface and
transferred to the other surface through the battery.
• The charging will stop once the plates reach the same
potential difference with each other that the terminals of the
voltage source endure.
– When the voltage source is removed, the capacitor now
becomes the primary voltage source for the circuit.
• So capacitance is defined as the ratio of the charge
between the conducting surfaces and the potential
difference between surfaces.
– Denoted by C
– Measured in Farads, F
• But a Farad is actually a very large number
Q
C=
V
– so we typically measure in the range of F to pF.
Parallel-Plate Capacitor
• The most common design for a capacitor is to place
two conducting plates parallel to each other and
separated by a small distance.
• Distances of millimeters and smaller!
• By connecting opposite leads of a power source to
each plate, the charges begin to line themselves up
according to the potential difference of the battery.
– Remember, the capacitor stops charging once it reaches
the same voltage as the battery.
• Even when the battery is disconnected, the capacitor
will maintain the potential difference of the battery until
the two plates are again connected by a conducting
material.
permittivity
of free
space
C = 0
0 = 8.85 x 10-12 C2/(Nm2)
A
d
surface area of one plate
separation between the plates
Dielectrics
• The material between the
plates of a parallel plate
capacitor can effect the
capacitance of the system.
• A dielectric is an insulating
material that is placed in
between plates of a capacitor
to increase its capacitance.
– Insulators are used because
the plates can realign the
charges on the surface of
the insulator space for the
charge to be stored.
– That gives the opportunity
for more charge to be
transferred to the plates of
the capacitor for more
storage.
+ + + + + + + + + +
- - - - - - - - - -
+ + + + + + + + + +
- - - - - - -
- - -
Dielectric Constant
• Each material is different and has
different abilities to give up electrons to
help increase the capacitance.
– That increase is a multiple factor called
the dielectric constant, .
• So:
C = C0
• This differs from the dielectric
strength, which is the largest electric
field a capacitor can hold.
– There is no relationship between larger
the constant, stronger the field.
Energy Stored in a Capacitor
• Due to the fact that the energy stored in a
capacitor is directly related to the work
required to transfer that charge from plate to
plate, we see the following:
– In order for work to be performed, there must be a
potential difference between plates in order to carry
the charge across.
• W = V• Q
– Thanks to the Work-Kinetic Energy Theorem, and
seeing that Q is the equivalent of mass in the
mechanics world
• W = ½ Q V
– Similar to kinetic energy in the mechanics world
– We combine those to produce a series of equations
that would help to find the energy stored in a
capacitor
• PE = ½ Q V = ½ C(V)2 = Q2 / 2C