lightning rod electric potential and capacitors
TRANSCRIPT
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
College Physics B - PHY2054C
Electric Potential and Capacitors
09/08/2014
My Office Hours:
Tuesday 10:00 - Noon
206 Keen Building
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 1
Consider the four field patterns shown below. Assumingthere are no charges in the regions shown, which of thepatterns represent(s) a possible electrostatic field:
(a) (b)
(c) (d)
A (a)
B (b)
C (b) and (d)
D (b) and (c)
E None of these.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 1
Consider the four field patterns shown below. Assumingthere are no charges in the regions shown, which of thepatterns represent(s) a possible electrostatic field:
(a) (b)
(c) (d)
A (a)
B (b)
C (b) and (d)
D (b) and (c)
E None of these.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 2
The figure below shows the electric field lines outside several
Gaussian surfaces, but doesn’t show the electric charge inside.
In which cases is the net charge inside positive, negative, or
perhaps zero? How do you know?
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 2
ΦE = 0
The figure below shows the electric field lines outside several
Gaussian surfaces, but doesn’t show the electric charge inside.
In which cases is the net charge inside positive, negative, or
perhaps zero? How do you know?
Gauss′ Law : ΦE = ~E ·~A =
Q
ǫ0
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 2
ΦE = 0 ΦE < 0
A positive
B negative
C zero
The figure below shows the electric field lines outside several
Gaussian surfaces, but doesn’t show the electric charge inside.
In which cases is the net charge inside positive, negative, or
perhaps zero? How do you know?
Gauss′ Law : ΦE = ~E ·~A =
Q
ǫ0
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 2
ΦE = 0 ΦE < 0 ΦE > 0
The figure below shows the electric field lines outside several
Gaussian surfaces, but doesn’t show the electric charge inside.
In which cases is the net charge inside positive, negative, or
perhaps zero? How do you know?
Gauss′ Law : ΦE = ~E ·~A =
Q
ǫ0
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 3
A cylindrical piece of insulating material is placed in anexternal electric field, as shown. The net electric fluxpassing through the surface of the cylinder is
A positive.
B negative.
C zero.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Review Question 3
A cylindrical piece of insulating material is placed in anexternal electric field, as shown. The net electric fluxpassing through the surface of the cylinder is
A positive.
B negative.
C zero.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Outline
1 Electric Potential
Review: Electric Potential Energy
Voltage
2 Examples
Lightning Rod
3 Equipotential Surfaces
4 Capacitors
Capacitors in Series
Capacitors in Parallel
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Potential Energy
A point charge in an electric field experiences a force:
~F = q ~E
Assume the charge moves a distance ∆x : W = F ∆x .
Electric force is conservative: Work done independent of path.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Potential Energy
The change in electric potential energy is
∆PE elec = −W = −F ∆x = −q E ∆x
The change in potential energy depends only on the endpoints
of the motion, but not on the path taken.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Potential Energy
The change in electric potential energy is
∆PE elec = −W = −F ∆x = −q E ∆x
The change in potential energy depends only on the endpoints
of the motion.
A positive amount of energy can be stored in a system that is
composed of the charge and the electric field.
Stored energy can be taken out of the system:
• This energy may show up as an increase in the kinetic
energy of the particle.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Question 4
A point charge Q is held fixed at the origin while an external
force ~F ext is used to move a second charge q from point A
to point B along the three different paths shown in the figure.
The charge q begins from rest and has zero velocity when
it reaches B. Is the work done by the external force largest
along
A Path I
B Path II
C Path III, or
D is the work done by ~F ext the
same for all three paths?
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Question 4
A point charge Q is held fixed at the origin while an external
force ~F ext is used to move a second charge q from point A
to point B along the three different paths shown in the figure.
The charge q begins from rest and has zero velocity when
it reaches B. Is the work done by the external force largest
along
A Path I
B Path II
C Path III, or
D is the work done by ~F ext the
same for all three paths?
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Two Point Charges
From Coulomb’s Law:
F =k q1q2
r2
The electric potential energy is given by:
PE elec =k q1q2
r=
q1q2
4πǫ0 r
Note that PE elec varies as 1/r while the
force varies as 1/r2.
• PE elec approaches zero when the
two charges are very far apart.
• The electric force also approaches
zero in this limit.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Two Point Charges
From Coulomb’s Law:
F =k q1q2
r2
The electric potential energy is given by:
PE elec =k q1q2
r=
q1q2
4πǫ0 r
The changes in the potential energy are
important:
∆PE elec = PE elec, f − PE elec, i
=k q1q2
rf
−
k q1q2
ri
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Potential: Voltage
Electric potential energy is a property of a system of charges
or of a charge in an electric field, it is not a property of a single
charge alone.
Electric potential energy can be treated in terms of a test
charge, similar to the treatment of the electric field produced
by a charge:
V =PE elec
q
• Units are the Volt or [V]: 1 V = 1 J/C.
(Named in honor of Alessandro Volta)
• The unit of the electric field can also be given in terms of
the Volt: 1 N/m = 1 V/m.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Potential: Voltage
The diagram shows the electric potential in a television picture
tube: Two parallel plates are charged and form an “accelerator”:
1 The electric force on the test charge is F = q E and the
charge moves a distance L.
2 The work done on the test charge by the electric field is
W = q E L.
3 There is a potential difference between the plates of:
∆V =∆PE elec
q= −E L
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Accelerating a Charge
The diagram shows the electric potential in a television picture
tube: Two parallel plates are charged and form an “accelerator”:
1 The electric force on the test charge is F = q E and the
charge moves a distance L.
2 The work done on the test charge by the electric field is
W = q E L.
3 There is a potential difference between the plates of:
∆V =∆PE elec
q= −E L
4 Conservation of Energy:
KE i + PE elec, i
= KE f + PE elec, f
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Accelerating a Charge
The diagram shows the electric potential in a television picture
tube: Two parallel plates are charged and form an “accelerator”:
1 The electric force on the test charge is F = q E and the
charge moves a distance L.
2 The work done on the test charge by the electric field is
W = q E L.
3 There is a potential difference between the plates of:
∆V =∆PE elec
q= −E L
4 Conservation of Energy: (v i = 0 for an electron)
KE f = −∆PE elec = −q ∆V = e∆V = 1/2 m v2f
v f =
√
2e∆V
m
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Field and Potential
The electric field may vary with position.
The magnitude and direction of the electric field are related to
how the electric potential changes with position:
∆V = −E ∆x
E = −∆V/∆x
It is convenient to define a unit
of energy called the electron-volt
(eV). One electron volt is defined
as the amount of energy gained
or lost when an electron travels
through a potential difference of
1 V:
1 eV = 1.60 × 10−19 J
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Question 5
In the figure, an electron moves from the left to the right
through a region in which the electric potential changed by
∆V = 15, 000 V. The corresponding change in the potential
energy is ∆PE elec = −e∆V = −15, 000 eV. Suppose now
that the electron is replaced by an oxygen ion O2−. What is
the change in potential energy of the oxygen ion as measured
in electron-volts?
A −7, 500 eV
B −15, 000 eV
C −30, 000 eV
D +30, 000 eV
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Question 5
In the figure, an electron moves from the left to the right
through a region in which the electric potential changed by
∆V = 15, 000 V. The corresponding change in the potential
energy is ∆PE elec = −e∆V = −15, 000 eV. Suppose now
that the electron is replaced by an oxygen ion O2−. What is
the change in potential energy of the oxygen ion as measured
in electron-volts?
A −7, 500 eV
B −15, 000 eV
C −30, 000 eV
D +30, 000 eV
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Outline
1 Electric Potential
Review: Electric Potential Energy
Voltage
2 Examples
Lightning Rod
3 Equipotential Surfaces
4 Capacitors
Capacitors in Series
Capacitors in Parallel
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Point Charge
Only changes in potential
(and potential energy) are
important.
Reference point (usually):
V = 0 at r = ∞
Sometimes:
Earth may be V = 0.
The electric potential at a distance r away from a single point
charge q is given by:
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Field Near a Metal
The Fig. A shows a solid metal sphere
carrying an excess positive charge, q:
• The excess charge resides on the
surface.
• The field inside the metal is zero.
• The field outside any spherical
ball of charge is given by:
E =kq
r2=
q
4πǫ0 r2,
where r is the distance from the
center of the ball.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Electric Field Near a Metal
The Fig. A shows a solid metal sphere
carrying an excess positive charge, q:
• The excess charge resides on the
surface.
• The field inside the metal is zero.
• Since the electric field outside the
sphere is the same as that of a
point charge, the potential is also
the same:
V =kq
rfor r > r sphere
• The potential is constant inside
the metal: V = k q/r sphere.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Lightning Rod
Field lines are perpendicular to the surface of the metal rod.
The field lines are largest near the sharp tip of the rod and
smaller near the flat side.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Lightning Rod
A lightning rod can be modeled as two metal spheres, which
are connected by a metal wire.
The smaller sphere represents the tip and the larger sphere
represents the flatter body:
V1 =k Q 1
r1= V2 =
k Q 2
r2
Q 1
r1=
Q 2
r2
Because the spheres are connected by
a wire, they are a single piece of metal.
They must have the same potential.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Lightning Rod
A lightning rod can be modeled as two metal spheres, which
are connected by a metal wire.
The smaller sphere represents the tip and the larger sphere
represents the flatter body:
V1 =k Q 1
r1= V2 =
k Q 2
r2
Q 1
r1=
Q 2
r2
E1
E2=
r2
r1
The field is large near the surface of the
smaller sphere. This means the field
is largest near the sharp edges of the
lightning rod.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Outline
1 Electric Potential
Review: Electric Potential Energy
Voltage
2 Examples
Lightning Rod
3 Equipotential Surfaces
4 Capacitors
Capacitors in Series
Capacitors in Parallel
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Equipotential Surfaces
A useful way to visualize electric fields is through plots of
equipotential surfaces:
• Contours where the electric potential is constant.
• Equipotential lines are in two-dimensions.
The equipotential surfaces are always perpendicular to the
direction of the electric field.
• For motion parallel to an equipotential surface, V is
constant and ∆V = 0.
• Electric field component parallel to the surface is zero.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Example: Point Charge
The electric field lines emanate
radially outward from the charge.
• The equipotential surfaces
are perpendicular to the field.
• The equipotentials are a
series of concentric spheres.
• Different spheres correspond
to different values of V .
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Example: Dipole
The dipole consists of charge
+q and −q.
• Field lines are plotted in
blue.
• Equipotential lines are
plotted in orange.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Outline
1 Electric Potential
Review: Electric Potential Energy
Voltage
2 Examples
Lightning Rod
3 Equipotential Surfaces
4 Capacitors
Capacitors in Series
Capacitors in Parallel
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitors
A capacitor can be used to store
charge and energy.
1 Each plate produces a field:
E =Q
2ǫ0 A
2 In the region between the
plates, the fields from the
two plates add, giving:
E =Q
ǫ0 A=
∆V
d,
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitors
A capacitor can be used to store
charge and energy.
1 Each plate produces a field:
E =Q
2ǫ0 A
2 In the region between the
plates, the fields from the
two plates add, giving:
E =Q
ǫ0 A=
∆V
d,
where d is the distance
between the plates.
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitance
Electric Potential of a capacitor:
E =Q
ǫ0 A=
∆V
d
∆V =Q d
ǫ0 A= E d
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitance
Electric Potential of a capacitor:
E =Q
ǫ0 A=
∆V
d
∆V =Q d
ǫ0 A= E d
Capacitance C is defined as:
∆V =Q
C
C =ǫ0 A
d
”Parallel-Plate Capacitor”
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitance
Capacitance C is defined as:
∆V =Q
C
C =ǫ0 A
d
• Units are the Farad or [F]:
1 F = 1 C / V.
(in honor of Michael Faraday)
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitance
The total energy corresponds to area under ∆V − Q graph:
PE cap =1
2Q (∆V ) =
1
2C (∆V )2 =
1
2
Q 2
C
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitors in Series
When dealing with multiple capacitors, equivalent capacitance
is useful (V = Q/C):
∆V total = ∆V top + ∆V bottom
1
C equiv.
=1
C1+
1
C2
CollegePhysics B
ElectricPotential
Review: Electric
Potential Energy
Voltage
Examples
Lightning Rod
EquipotentialSurfaces
Capacitors
Capacitors in Series
Capacitors in Parallel
Capacitors in Parallel
When dealing with multiple capacitors, equivalent capacitance
is useful (V = Q/C):
Q total = Q1 + Q2
C equiv. = C1 + C2