Transcript bYTEBoss potential126web

Electric Potential and Electric
Potential Energy
Learning Objectives
• To introduce the concept of electric
potential energy
• To define the electric potential
• To find and use the electric potential of
point charges, charged spheres, and
parallel-plate capacitors
• To understand charge and energy storage
in a capacitor
Remember Work?
• In high school we learned that
Work = force x displacement*
• Work is a scalar, unlike force
and displacement, which are
vectors.
• If the force and the
displacement are in the same
direction, the work is positive; if
they are in opposite directions,
the work is negative.
• The electric field generates a
force that does work.
*oversimplified, but it
will do for now
Question about Work
Which of the following is true for the proton as it travels from
point A to point B? Blue arrow shows force on proton due
to E field.
A. Work done on the charge is positive.
B. Work done on the charge is negative.
C. No work is done on the charge
Another Question about Work
Which of the following is true for the electron as it
travels from point B to point A this time? Blue
arrow shows force on electron due to E field.
A. Work done on the charge is positive.
B. Work done on the charge is negative.
C. No work is done on the charge.
More about work
• When a conservative force
does work, this work is defined
as the change in potential
energy of an object.
• Conservative forces include
electric force, magnetic force,
gravity, the spring force and
the Tea Party. All other forces
we deal with in this class are
nonconservative.
The change in potential energy
ΔPE = - Wconservative
As the weight goes up, its
potential energy increases, so
ΔPE is positive.
The gravitational force is doing
negative work, since it’s
direction (down) is opposite to
that of the displacement(up)
Knight uses ΔU for the change in
potential energy:
(ΔU = - Wconservative )
The change in potential energy
ΔPE = - Wconservative
As the weight goes down, its
potential energy decreases so
ΔPE is negative.
The gravitational force is doing
positive work, since it’s
direction (down) is same as
that of the displacement
(down).
Question about ΔEPE
Which of the following is true for the electron as it
travels from point B to point A? Blue arrow shows
force on electron due to E field.
A. ΔEPE is positive.
B. ΔEPE is negative.
C. EPE does not change.
Conservation of Energy
W nonconservative = ΔKE + ΔPE
Non-conservative forces that do work include pushes
and pulls, friction, tension and the ACLU.
If no conservative forces do work on a system:
W nonconservative = ΔKE
If no nonconservative forces do work:
ΔKE + ΔPE = 0
Work and EPE
A positive charge is moving
from point A to point B in the
uniform electric field of a
capacitor, as shown in the
drawing. The electric force
does __________ work on
the charge and, as a
consequence, its electric
potential energy
__________ .
A. positive, increases
B. positive, decreases
C. negative, decreases
D. negative, increases
EPE and KE - 1
A small, positively-charged
particle is shot toward a
larger fixed charge, also
positive. There are no nonconservative forces once
the particle is released
What does the particle do?
A.
B.
C.
D.
speed up and gain EPE
slow down and gain EPE
speed up and lose EPE
slow down and lose EPE
fixed
EPE and KE- 2
A small, positively-charged
particle is shot toward a
larger fixed charge, this
time negative.
What does the particle do?
A.
B.
C.
D.
speed up and gain EPE
slow down and gain EPE
speed up and lose EPE
slow down and lose EPE
The Electric Potential Model
• The source charges
have altered the space
around them to create
an electric field (E).
• Any point in the E field
has an electric
potential (V)
associated with it.
• A test charge q (blue)
at that point has an
EPE.
• V = EPE/q
The Electric Potential Model
• The charges of the capacitor
create a constant electric field E.
• At point A, the electric potential
is VA and at point B, it is VB.
• The conservative electric field
does positive work on +q0 as it
moves from A to B.
• EPE decreases, so Δ EPE
(EPEB - EPEA) is negative
• VB is Iower (less) than VA .
The Electric Potential Model
ΔEPE = q0VB - q0VA
ΔEPE/q0 = VB – VA = ΔVAB
ΔVAB is the electrical potential
difference (or potential difference)
WAB /q0 = (- ΔVAB)
19.2 The Electric Potential Difference
DEFINITION OF ELECTRIC POTENTIAL DIFFERENCE
The electric potential difference between 2 points is the change in electric
potential energy experienced by a test charge as it moved between the 2
points, divided by the charge itself.
The change in electric potential energy is the negative of the work done
by the electric field when the charge moves between the 2 points
EPE B EPE A  WAB
VB  VA 


qo
qo
qo
EPE   WAB
V 

qo
qo
SI Unit of Electric Potential Difference: joule/coulomb = volt (V)
19.2 The Electric Potential Difference
DEFINITION OF ELECTRIC POTENTIAL
The electric potential at a point is the electric potential energy
of a small test charge divided by the charge itself:
EPE
V
qo
SI Unit of Electric Potential: joule/coulomb = volt (V)
Recall that any value for a potential energy is based on an arbitrary zero .
The same is true for the electric potential. Only the differences ΔEPE
and ΔV can be measured absolutely in terms of the work done between 2
points.
Potential and Potential Energy
As the +1 nC charge moves toward the
source charges, it gains 1 μJ of potential
energy. The change in potential is:
A. 1 KV B. 1 μV C. 1 mV D. 1 nV
Sources of Electric Potential Difference
A potential
difference is
created by
separating
positive charge
from negative
charge.
Measuring Electric Potential Difference
• The voltmeter
measures the potential
difference (ΔV)
between 2 points.
• Since it is the difference
that is important, not
the actual value, the
black lead is 0 volts and
the value read by the
voltmeter is equal to
(ΔV).
Work and potential difference
In each of the cases below, the same +q0
moves from points A to B. In which case
is the most work done by the electric field?
A. Case 1
C. Case 3
B. Case 2
D. All equal work
19.4 Equipotential Surfaces and Their Relation to the Electric Field
An equipotential surface is a surface on
which the electric potential is the same everywhere.
For a point charge, this would be a spherical surface;
all the places in space at a distance r from the charge.
kq
V
r
The potential difference is related to the work done
to move a test charge in an electric field
The net electric force does no work on a test charge
if that charge were moved along an equipotential
surface.
When looking at a 2 dimensional depiction of
equipotential surfaces, they appear as lines
(red lines in this picture).
The green lines are electric field lines.
• The equipotential
lines tell us how much
EPE a test charge of 1
C would have at that
distance from the blue
source charge. lines.
• The field lines show
the direction and the
relative magnitude of
the force that a positive
test charge would
experience at that
distance from the blue
source charge
An electric dipole
19.4 Equipotential Surfaces and Their Relation to the Electric Field
• The electric field lines created by any charge or group of charges is
everywhere perpendicular to the associated equipotential surfaces or lines
• The electric field lines points in the direction of decreasing potential.
• One way to investigate the relationship between the electric field and
the electric potential is with the parallel plate capacitor.
Parallel Plate Capacitor
The parallel plate capacitor
consists of 2 plates of area
A,with equal and opposite
charges (+q,-q) separated
by a small distance, d.
The electric field, E inside this
a capacitor is constant. It
points from the positive
plate to the negative plate,
parallel to the force a
positive test charge would
experience.
The two plates act as
equipotential surfaces.
q
E
o A
Electric Potential of the Capacitor
The electric potential is the
same anywhere on each
(imaginary) equipotential
surface, and decreases in
the direction of the electric
field.
A positive test charge would
have the greatest EPE at the
positive plate and the least
EPE at the negative plate.
The test charge would have the
same EPE anywhere on an
equipotential surface.
Relationship between Electric Field and
Electric Potential of the Capacitor
If the charge travels in the direction of the
electric field, ∆V is negative,
If the charge travels opposite to the electric
field, ∆V is positive.
This is the same for both negative and positive
charge.
E = - ∆V/ ∆s
For the capacitor, we can simplify and write this
as
E = V/d
where E is the magnitude of the electric field
strength, V is the potential of the positive
plate (with negative plate = 0V) and d the
plate spacing.
The electric field E can be expressed in units of
volts/meter (V/m).
Graphical representations of electric
potential inside a capacitor
• the green dashed contour
lines represent
equipotential surfaces.
• Any point on one of
these surfaces is at the
same potential.
• These surfaces are
perpendicular to the
direction of E.
• E points in the direction of
decreasing potential
Stop to think
Rank, in order, from
largest to smallest,
the potentials Va to Ve
EOC# 35
The inner and outer surfaces
of a cell membrane carry
a negative and a positive
charge, respectively.
Because of these
charges, a potential
difference of about 0.070
V exists across the
membrane. The thickness
of the cell membrane is
8.00 x10-9 m. What is the
magnitude of the electric
field in the membrane?
outer
+
+
+
+
+
+
+
+
+
+
+
+
inner
_
_
_
_
_
_
_
_
_
_
_
_
E
EOC #36
What is the magnitude of
the E field at point D?
A. 100 V/cm
B. 50 V/cm
C. 25 V/cm
D. 12.5 V/cm
EOC #36
In which direction does the
E field point at D?
A. up
B. down
C. left
D. right
Capacitors and Energy Storage
• The capacitor stores
energy, which it can
release very rapidly.
•The capacitor stores
separated charges, which
creates a potential
difference (ΔV). Each of
the charges has electric
potential energy due to
their position in the electric
field of the capacitor .
The potential difference between the battery
terminals (ΔVbat) causes the charge to
accumulate on the plates, which results in ΔVc
When ΔVc =ΔVbat the charge stops moving and the
capacitor is fully charged.
19.5 Capacitors and Dielectrics
•The capacitor retains
the potential difference
and the stored charge
after the battery is
disconnected.
•Because of this, the
capacitor can be used
as a “temporary
battery” in a circuit.
•Parallel plate
capacitors are often
rolled into a cylindershaped device.
ΔVc
The potential difference between the plates depends on the
amount of charge separated: V  q
•But how much charge do I need to obtain
a desired ∆V ? This depends on the
geometry of the capacitor.
•The area of the plate determines the
charge density and is proportional to
number of charges needed to maintain a
given ∆V.
•The distance between the plates
separates the unlike charges; and is
inversely proportional to the number of
charges needed:
A
V  q
d
Given the relationship:
The constant of
proportionality is ε0 :
A
 0 V  q
d
And the whole quantity
is defined as C, the
capacitance of the
capacitor:
A
V  q
d
ε0 = 8.85 x 10-12 C2 /(N-m2 )
A
C  0
d
Now we can say:
q  CV
where C is called Capacitance. (“the
charge capacity at a given potential
difference”)
SI Unit of Capacitance: coulomb/volt
= farad (F)
It is customary to set the arbitrary zero
for potential at the negative plate and
use V+ instead of ∆V:
q  CV
19.5 Capacitors and Dielectrics
• Capacitance is a
property of the geometry
of the capacitor, not its
charge or potential.
•For a commercial
capacitor, the geometry
is fixed, and the value of
C is marked on the
outside.
•The capacitance does
not change if the charge
or the potential
difference changes.
For a capacitor without a dielectric:
A
C  0
d
ε0 = 8.85 x 10-12 C2 /(N-m2 )
A parallel plate capacitor is
connected to a battery that
maintains a constant potential
difference between the plates. If
the plates are pulled away from
each other, doubling the
separation (d), by what factor
does the amount of charge on the
plates change ?
A.
B.
C.
D.
q  CV
No change, the separation distance is not a factor
By a factor of 2 (doubles).
By a factor of one half
Impossible to say without knowing the area and original
separation distance of the capacitor.
19.5 Capacitors and Dielectrics
•To increase the energy storage
of a capacitor, we must increase
it’s capacitance.
• An insulator, (“dielectric”)
inserted between the plates
allows more charge storage on
the plates for a given ΔV (which
is usually determined by the
energy source in the circuit).
19.5 Capacitors and Dielectrics
•Therefore a capacitor with a
dielectric can store more charge
at a given potential difference
resulting in greater energy
strorage.
•Since q = C ∆V, increasing
charge while keeping ∆V the
same means the capacitance
must increase.
THE CAPACITANCE OF A PARALLEL PLATE CAPACITOR
WITH A DIELECTRIC
C
 o A
d
C  C0
Where k is the dielectric constant of the
material (found in a table in most texts),
and C0 is the capacitance without
dielectric.
ε0 = 8.85 x 10-12 C2/Nm2
Energy Storage in a Capacitor
and Dielectrics
When a capacitor
stores separated
charge , it also
stores energy:
EPE = ½ q ∆V
Since q = C ∆V,
EPE = ½ C (∆V)2
ΔVc
Remember power?
What is the potential difference between the
plates of a 3.2-F capacitor that stores
sufficient energy to operate a 83-W light
bulb for three minutes?
Recall that power (in units of Watts) is
energy used per unit time (SI units are J/s)
Remember power?
What is the potential difference between the
plates of a 3.2-F capacitor that stores
sufficient energy to operate a 83-W light
bulb for three minutes?
Ans: 96.6 V
Which two, or more, of the following actions would increase the
energy stored in a parallel plate capacitor when a constant
potential difference is applied across the plates (i.e. battery
remains attached)?
1. increase the area of the plates
2. decrease the area of the plates
3. increase the separation between the plates
4. decrease the separation between the plates
5. insert a dielectric between the plates
A. 2,3,5
B. 1,4,5
C. 1,3 D. 2,4
Energy storage with a dielectric
• An empty capacitor is capable of storing
1.0 x 10-4 J of energy when connected to a
certain battery. If the distance between the
plates is halved and then filled with a
dielectric (κ = 2.5), how much energy
could this modified capacitor store when
connected to the same battery?
Energy storage with a dielectric
• An empty capacitor is capable of storing
1.0 x 10-4 J of energy when connected to a
certain battery. If the distance between the
plates is halved and then filled with a
dielectric (κ = 2.5), how much energy
could this modified capacitor store when
connected to the same battery?
• Ans: 5E0 where E0 is 1.0 x 10-4 J