Lightning Rod Electric Potential and Capacitors

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

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Tuesday 10:00 - Noon

206 Keen Building

CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



Review Question 2

ΦE = 0





Gauss′ Law : ΦE = ~E ·~A =

Q

ǫ0

CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



Review Question 2

ΦE = 0 ΦE < 0

A positive

B negative

C zero






Q

ǫ0

CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



Review Question 2

ΦE = 0 ΦE < 0 ΦE > 0






Q

ǫ0

CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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


Capacitors



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


Capacitors



Outline

1 Electric Potential

Review: Electric Potential Energy

Voltage

2 Examples

Lightning Rod

3 Equipotential Surfaces

4 Capacitors



CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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

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Capacitors




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


Capacitors




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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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

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Accelerating a Charge






W = q E L.


∆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


Capacitors



Accelerating a Charge






W = q E L.


∆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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



Outline



Voltage

2 Examples

Lightning Rod


4 Capacitors



CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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

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Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



Outline



Voltage

2 Examples

Lightning Rod


4 Capacitors



CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



Outline



Voltage

2 Examples

Lightning Rod


4 Capacitors



CollegePhysics B

ElectricPotential

Review: Electric

Potential Energy

Voltage

Examples

Lightning Rod


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors



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


Capacitors




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


Capacitors




When dealing with multiple capacitors, equivalent capacitance

is useful (V = Q/C):

Q total = Q1 + Q2

C equiv. = C1 + C2

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Lightning Rod Electric Potential and Capacitors