Electrostatics f

ELECTROSTATICS

Outline

• Electric Force, Electric fields

• Electric Flux and Gau law

• Electric potential

• Capacitors and dielectric (Electric storage)

The physics of charged objects

• Study of electricity aims to understand the interaction between different charged objects.

+ -

The physics of charged objects

• Study of electricity aims to understand the interaction between different charged objects.

+ +

- -

Structure of Matter

• Fundamental building blocks of the matter are atoms.

++

+

++

+ + -

-

-

-

-

--

Structure of Matter

• Neutral atom – electron = Positive ion

++

+

++

+ + -

-

-

-

-

- -

C101.602charge electron -191

Structure of Matter

• Fundamental building blocks of the matter are atoms.

++

+

++

+ + -

-

-

-

-

--

Structure of Matter

• Neutral atom + electron = negative ion.

++

+

++

+ + -

-

-

-

-

--

-

ELECTRICALLY CHARGING OBJECTS

+ - +

+- - +

+ - +

-

-

-

-

+

+

-

+


+ - +

+- - +

+ - +

-

-

-

-

+

+

-

+


+ - +

+- - +

+ - +

-

-

-

-

+

+

-

+-

• In metals outer atomic electrons are not bound to any atoms (electron see).

Charging by Induction

++

+ +

+ +

+

++

+


• In metals outer atomic electrons are not bound to any atoms (electron see).

++

+ +

+ +

+

++

+

-


• Same atoms have weakly bound electrons.

Electric Polarization

++

+

++

+ +

-

-- -

-

-

-


• Same atoms have weakly bound electrons.

++

+

++

+ +

-

-

-

-

-

-

- +


The Electric Force

Coulomb’s Law

• Quantifies the electric force between two charges.

baba QQF

Coulomb’s Law


2

1

baba r

F

Coulomb’s Law


baba

baba

ba

baba r

r

QQkr

r

QQF ˆˆ

4

122

0

229

212

/10988.8

/10854.8

CNmk

and

NmC

where

0

Electric Force Field

• Gravitational force field:


+Q


+Q+q


• Definition of Electric field:

q

FE qQ


• Definition of Electric field:

qQrr

kQE

qQ

ˆ2


1r 5r

2r

+

+

+

+

+

3r

4r


• The electric field due to a number of source charges is given by the expression

N

ii

i

i

N

iii

rr

qk

rEE

12

1

ˆ

)(


Electric Force Field(Linear distribution of charge)

dL

dL

dq density charge Linear

?dEr

2r

dqkdE


dL

dL


?dE

2r

dLkdE


dL

dL


?dE

2r

dLkE

Electric Force Field(Surface distribution of charge)

da

dqondistributi charge Surface

r?dE

da 2r

dqkdE


da


r?dE

da2r

dakdE


da


r?dE

dasurface r

dakE

2

Electric Dipole

• Electric dipole consists of a pair of point

charges with equal size but opposite sign

separated by a distance d.

d+ -

dqp

Electric Dipole




d+ -

Electric Dipole




d+ -

p

Electric Dipole




d+ -

dqp

Electric Dipole

• Water molecules are

electric dipoles

+ +

-waterp

Exercise 1

A point charge q = -8.0 nC is located at the

origin. Find the electric field vector at the point

x = 1.2 m, y = -1.6 m

m 2.1

-

m 6.1mr 0.2

m 2.1

-

m 6.1mr 0.2

jEiEE yxˆˆ

m 2.1

-

m 6.1mr 0.2

)ˆsinˆ(cos jiEE

m 2.1

-

m 6.1mr 0.2

jCN iCN E ˆ/14ˆ/11

Exercise 2

An electric dipole consists of a positive

charge q and negative charge –q

separated by a distance 2a, as shown in

the figure below. Find the electric field due

to these charges along the axis at the

point P, which is the distance y from the

origin. Assume that y>>a.

q q

r

aa

r

Vector Flux

Vector Flux

Vector Flux

• Definition of flux:

Av

Electric Flux

Electric Flux

AdEE

Gau ’s Law

Surface Enclosed

EnclosedQAdE

0

Gau ’s Law

dA EAdEd E

+r

Gau ’s Law

dA EE

+r

Gau ’s Law

dAr

kQE 2

+r

Gau ’s Law

0

2

2

0

44

Qr

r

QE

+r

Exercise 3dE

Solution

Coulomb’s Law

22 R

dAk

R

kdqdE

Solution

Infinitesimal area of disk

rdrdA 2

Solution

Infinitesimal area of disk

2

2

R

rdrkdE

Solution

Y-component of E-field element

cos2

cos2R

rdrkdEdEy

Solution

R

Lcos

Solution


R

L

R

rdrkdEy 2

2

Solution


04

1k

Solution


2/322

0 )(2 Lr

rdrLdEy

Solution


0

0

2/322

0

2/322

0

2

1

)(

)(2

y

y

E

LLr

rdr

Lr

rdrLE

identity the Using

Two Oppositely charged Parallel Plates (Capacitor)

Two Oppositely charged Parallel Plates (Capacitor)

?E

Exercise 4

00

LQAdE

Gauss

0

0

2

2

rE

LrLE

Area rL2cylinder a of

Electric Potential

+Q+q

Electric Potential

+Q

Electric Potential

test

test

Q

Q on

oodneighboreh its in source due Ppoint aat potential

)( PWork

Electric

Electric Potential

PSource

P

P

test

rdrr

kQ

rdE

rdFQ

Electric

ˆ

1

2

oodneighboreh its in source due Ppoint aat potential

Electric Potential

r

Q

r

kQ SourceSource

04V(r)

1J/C V 1:Unit

Electric Potential

N

i i

i

r

Q

104

1PV

Electric potential at position P due to a system of N source charges is given by:

Electric Potential

• Potential difference:

Electric Potential

• Potential difference:

Electric Potential

b

a

r

r

absourceab

rdE

rrkQrVrVV

11)()(

Electrostatics

Electric charge

Conservation of charge

Insulators & conductors

Charging objects

Electroscopes

Lightning

Van de Graff generators

Equilibrium problems

Grounding

Static electricity

Coulomb’s law

Systems of charges

Electric Charge

• Just as most particles have an attribute known as mass,

many possess another attribute called charge. Charge and

mass are intrinsic properties, defining properties that particles

possess by their very nature.

• Unlike mass, there are two different kinds of charge: positive

and negative.

• Particles with a unlike charges attract, while those with like

charges repel.

• Most everyday objects are comprised of billions of charged,

but usually there are about the same number of positive

charges as negative, leaving the object as a whole neutral.

• A charged object is an object that has an excess of one type

of charge, e.g., more positive than negative. The amount of

excess charge is the charge we assign to that object.

Conservation of ChargeCharged particles can be transferred from one object to another, but

the total amount of charge is conserved. Experiments have shown

that whenever subatomic particles are transferred between objects or

interact to produce other subatomic particles, the total charge before

and after is the same (along with the total energy and momentum).

Example: An object with 5 excess units of positive charge and

another with 2 units of excess negative charge are released from rest

and attract each other. (By Newton’s 3rd law, the forces are equal

strength, opposite directions, but their accelerations depend on their

masses too.) Since there is no net force on the system, their center of

mass does not accelerate, and they collide there. As they “fall” toward

each other, electric potential energy is converted to kinetic energy.

When contact is made charge may be exchanged but they total

amount before and after must be the same. After the collision the total

momentum must still be zero.

+5 -2 +1.5 +1.5

Before After

Total charge: +3 Total charge: +3

Conservation of Charge: β-decay

• The stability of the nucleus of an atom depends on its size

and its proton-neutron ratio. This instability sometimes results

in a radioactive process known as β-decay.

• A neutron can turn into a proton, but in the process an

electron (beta particle) is ejected at high speed from the

nucleus to conserve charge.

• A proton can turn into a neutron. In this case the beta

particle is an positron (an antielectron: same mass as an

electron but a positive charge) to make up for the loss of

positive charge of the proton.

• In either case, charge, momentum, and energy are

conserved.

SI unit of Charge: the Coulomb

• Just as we have an SI unit for mass, the kilogram, we

have one for charge as well. It’s called the coulomb, and

its symbol is C.

• It’s named after a French physicist, Charles Coulomb,

who did research on charges in the mid and late 1700’s.

• A coulomb is a fairly large amount of charge, so

sometimes we measure small amounts of charge in μC

(mircocoloumbs).

• An electron has a charge of -1.6 10-19 C.

• A proton has a charge of +1.6 10-19 C.

• In a wire, if one coulomb of charge flows past a point in

one second, we say the current in the wire is one ampere.

Elementary Charge

• Charges come in small, discrete bundles. Another way to

say this is that charge is quantized. This means an object

can possess charge in incremental, rather than continuous,

amounts.

• Imagine the graph of a linear function buy when you zoom

in very close you see that it really is a step function with very

small steps.

• The smallest amount of charge that can be added or

removed from an object is the elementary charge, e = 1.6

10-19 C.

• The charge of a proton is +e, an electron -e.

• The charge of an object, Q, is always a multiple of this

elementary charge: Q = Ne, where N is an integer.

• How many excess protons are required for an object to

Insulators vs. Conductors • A conductor is a material in which excess charge freely

flows. Metals are typically excellent conductors because the

valence (outer shell) electrons in metal atoms are not

confined to any one atom. Rather, they roam freely about a

metal object. Metal are excellent conductors of electricity (and

heat) for this reason.

• An insulator is a material in which excess charge, for the

most part, resides where it is deposited. That is, once placed,

it does not move. Most nonmetallic material are good

insulators. Valence electrons are much more tightly bound to

the atoms and are not free to roam about. Insulators are

useful for studying electrostatics (the study of charge that can

be localized and contained).

• Semi-conductors, like silicon used in computer chips, have

electrical conductivity between that of conductors and

insulators.

Details on Conductors, Semiconductors, and Insulators

http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/comp/conduct/conduct.htm

Electrons and Chemical Bonds

All chemical bonding is due to forces between electrostatic

charges.

Covalent bonding: A pair of electrons is shared between two

nonmetal atoms, allowing each atom to have access to enough

electrons to fill its outer shell. Except for hydrogen, this usually

means 8 electrons in the outer shell (octet rule).

Ionic bonding: One or more valence electrons of a metal atom

are “stolen” by a nonmetal atom, leaving a positive metal ion

and a negative nonmetal ion, which then attract one another.

Metallic bonding: Valence electrons of metals flow freely

throughout a metal object. These delocalized electrons are

attracted to the nuclei of the atoms through which they are

moving about. This produces a strong binding force that holds

the atoms together. In an iron bar, for example, there is no

covalent or ionic bonding. Metallic bonding hold the metal

together.

Charging up Objects

Charging up an object does not mean creating new charges.

Charging implies either adding electrons to an object, removing

electrons from an object, or separating out positive and

negative charges within an object. This can be accomplish in 3

different ways:

• Friction: Rubbing two materials together can rub electrons off

of one and onto the other.

• Conduction: Touching an object to a charged object could

lead to a flow of charge between them.

• Induction: If a charged object is brought near (but not

touching) a second object, the charged object could attract or

repel electrons (depending on its charge) in the second object.

This yields a separation charge in the second object, an

induced charge separation.

Electroscopes

An electroscope is an apparatus comprised of a metal

sphere and very light metal leaves. A metal rod connects the

leaves to the sphere. The leaves are enclosed in an

insulating, transparent container. When the electroscope is

uncharged the leaves hang vertically. The scope is charged

by placing a charged rod near the sphere. The rod is charged

by friction. If a rubber rod is rubbed

with fur, electrons will be rubbed off the fur

and

onto the rubber rod, leaving the rod

negatively

Electroscopes

uncharge

d

charged. If a glass rod is rubbed with silk,

electrons will be rubbed off the rod onto the

silk, leaving the glass rod positively charged.

Either rod, if brought near, will charge the

scope by induction. Also, either rod, if contact

is made with the sphere, will charge the

scope by conduction.

continued…

Electroscopes (cont.)

+ + + + + + + + + + + + + + +

When a positively charged rod is placed near but not touching

the metal sphere, some of the valence electrons in the metal

leaves are drawn up into the sphere, leaving the sphere

negatively charged and the leaves positively charged. Thus,

the rod has induced a chargeseparation in the scope. The light,

positive leaves repel each other

and separate. The electroscope as

a whole is still electrically neutral,

but it has undergone a charge

separation. As soon as the rod is

removed from the vicinity, the

charge separation will cease to

exist and the leaves the drop.

Note: Only the electron are

mobile; the positives on the leaves

represent missing electrons.

+

+

+ +

+

+

--- -

---

continued…


- - - - - - - - - - - - - - - - - - - -

When a negatively charged rod is placed near but not touching

the metal sphere, some of the valence electrons in the sphere

are repelled down into the metal leaves, leaving the sphere

positively charged and the leaves negatively charged. The rod

has again induced a chargeseparation in the scope. The light,

negative leaves repel each other

as before. Again, the electroscope

as a whole is electrically neutral,

but the charge separation will

remain so long as the rod remains

nearby. Note that this situation is

indistinguishable from the situation

with the positive rod. Since the

effects are the same, how do we

know that the rods really do have

different charges?

-

-

- -

-

-

+ +++

+ +

continued…


- - - - - - - - - - - - - - - -

Now let’s touch the negative rod to the sphere. Some of the

electrons can actually hop onto the sphere and spread

throughout the scope. This is charging by conduction since,

instead of rearranging charges in the scope, new charges have

been added; the scope is no longer neutral. The extra electrons

force the leaves apart, even when the rod is removed. If the

negative rod returns, it charges the leaves further, but this time

by induction (by driving some of electrons on the sphere

down to the leaves). This

causes an increased

separation of the leaves.

When the rod is removed,

the scope will return to the

state on the left. -

-

- -

-

-

- ---

- -

Continued…

extra e- ’s added

--

- -

--

--

--- -

leaf spread

increases

Electroscopes (cont.)The pic on the left shows a scope that has acquired extra

electrons from a negative rod that has since been removed.

Now we bring a positive rod nearby. This has the opposite

effect of bringing the negative rod near. This time some of the

extra electrons in the leaves head to the sphere and the spread

of the leaves diminishes. Note: the scope is still negatively

charged overall, but the presence of thepositive rod means more of

the excess negative charge

will reside in the sphere and

less in the leaves. When the

rod is removed, the scope

return to the state on the

left.

-

-

- -

-

-

- ---

- -

Continued…

extra e- ’s added leaf spread

decreases

-

-

- -

-

-

- ---

- -

+ + + + + + + + + + + + + + +

Grounding an ElectroscopeWhether a scope has charged by conduction, either positively

or negatively, the quickest way to “uncharge” it is by grounding

it. To do this we simply touch the sphere. When a negatively

charged scope is grounded by your hand, the excess electrons

from the scope travel into your body and, from there, into your

surroundings. When a positively charged scope is

grounded, electrons from

your body flow into the

scope until it is neutral. Your

surroundings will replace

the electrons you’ve

donated to the scope. As

always, it’s only the

electrons that move around.

-

-

- -

-

-

- ---

- -

+

+

+ +

+

+

+ +++

+ +

--

- --

-

Electroscope Practice Problem

For the following scenario, try to predict what would happen

after each step. Explain each in terms of electrons and

charging.1. A rod is rubbed with a material that has a greater affinity

for electrons than the rod does.

2. This rod is brought near a neutral electroscope.

3. This rod touches the electroscope and is removed.

4. A positive rod is alternately brought near and removed.

5. A negative rod is alternately brought near and removed.

6. Finally, you touch the scope with your finger.

Redistributing Charge on Conducting

Spheres

A B

-Q- - - -

B

Two neutral spheres, A & B, are placed side by side, touching. A

negatively charged rod is brought near A, which induces a charge

separation in the “A-B system.” Some of the valence e-’s in A migrate

to B. When the rod is re-moved and A & B are separated, A is +, B is -,

but the system is still neutral.

A

+Q

A is now brought near neutral sphere C, inducing a charge separation

on it. Valence e-’s in C migrate toward A, but since C is being touched

on the positive side, e-’s from the hand will move into C. Interestingly,

C retains a net negative charge after A and the hand are removed

even though no charged object ever made contact with it.

A

+Q

C C

-

Static Electricity: Shocks

If you walk around on carpeting in your stocking feet, especially

in the winter when the air is dry, and then touch something

metal, you may feel a shock. As you walk you can become

negatively charged by friction. When you make contact with a

metal door knob, you discharge rapidly into the metal and feel

a shock at the point of contact. A similar effect occurs in the

winter when you exit a car: if you slide out of your seat and

touch then touch the car door, you might feel a shock.

The reason the effect most often occurs in winter is because

the air is typically drier then. Humidity in the air can rather

quickly rob excess charges from a charged body, thereby

neutralizing it before a rapid, localized discharge (and resulting

shock) can take place.

Care must be taken to prevent static discharges where

sensitive electronics are in use or where volatile substances

are stored.

Static Electricity: Balloons

Pic #1: If you rub a balloon on your hair,

electrons will be rubbed off your hair onto

the balloon (charging by friction).

Pic #2: If you then place the negatively

charged balloon near a neutral wall, the

balloon will repel some of the electrons

near it in the wall. This is inducing a

charge separation in the wall. Now the

wall, while still neutral, has a positive

charge near the balloon. Thus, the balloon

sticks to the wall.

Pick #3: Your hair now might stand up.

This is because it has been left positively

charged. As with the leaves of a charged

electroscope, the light hairs repel each

# 1

- ---- -

--

+-

+-

+-

+-

+-

+-

+-

+-

-+

-+

-+

-+

+-

+-

+-

+-

+-

+-

+-

# 2

# 3

#1

#

2

#

3

You hang two balloons from the

ceiling and rub them on your hair.

When you move out of the way, the negatively charged

balloons repel each other. On each balloon there are three

forces: tension in the string, gravity, and the electric force.

The angle of separation will grow until equilibrium is

achieved (zero net force).

If you move your head close to

either of the balloons, it will move

toward you since your hair remains

positively charged.

Hanging Balloons

Polarization of a Cloud

Detailed Lightning

Diagrams

One mechanism incorporates friction: when moist, warm air rises, it cools

and water droplets form. These droplets collide with ice crystals and

water droplets in a cloud. Electrons are torn off the rising water droplets

by the ice crystals. The positive droplets rise to the top of the cloud, while

the negative ice crystals remain at the bottom.

A second mechanism involves the freezing process: experiments have

shown that when water vapor freezes the central ice crystal becomes

negatively charged, while the water surrounding it becomes positive. If

rising air tears the surrounding water from the ice, the cloud becomes

polarized.

There are other theories as well.

Lightning is the discharge of static electricity

on a

massive scale. Before a strike the bottom part

of a

cloud becomes negatively charged and the top

part

positively charged. The exact mechanism by

which this polarization (charge separation)

takes place is uncertain, but this is the

precursor to a lightning strike from cloud to

cloud or cloud to ground.

http://whyfiles.larc.nasa.gov/text/kids/Problem_Board/problems/electricity/static2.html

http://whyfiles.larc.nasa.gov/text/kids/Problem_Board/problems/electricity/static2.html

Lightning Strikes

The negative bottom part of the cloud

induces

a charge separation in the ground below. Air

is normally a very good insulator, but if the

charge separation is big enough, the air

between the cloud and ground can become

ionized (a plasma). This allows some of the

electrons in the cloud to begin to migrate

into the ionized air below. This is called a

“leader.” Positive ions from the ground

migrate up to meet the leader. This is called

a “streamer.” As soon as the leader and

streamer meet, a fully conductive path

exists between the cloud and ground and a

lightning strike occurs. Billions of trillions of

electrons flow into the ground in less than a

millisecond. The strike can be hotter than

the surface of the sun. The heat expands

the surrounding air; which then claps as

thunder.

++ + + + +

+ + +

- - - - - - - --

++

+ ++

++ + +

Lightning Rods and Grounding

Discovered by Ben Franklin, a lightning rod is a long, pointed,

metal pole attached to a building. It may seem crazy to attract

lightning close to a susceptible structure, but a lightning rod

can afford some protection. When positive charges

accumulate beneath a cloud, the accumulation is extremely

high near the tip of the rod. As a result, an electric field is

produced that is much greater surrounding the tip than around

the building. (We’ll study electric fields in the next unit.) This

strong electric field ionizes the air around the tip of the rod and

“encourages” a strike to occur there.

If a strike does occur, the electricity travels down the rod into a

copper cable that connects the lightning rod to a grounding rod

buried in the earth. There the excess charge is grounded, i.e.,

the electrons are dissipated throughout the landscape. By

taking this route, rather than through a building and its wiring,

much loss is prevented.

A Van de Graaff generator consists of a large metal dome attached to a tube,

within which a long rubber belt is turning on rollers. As the belt turns friction

between it and the bottom roller cause the e-’s to move from the belt to the

roller. A metal brush then drains these e-’s away and grounds them. So, as the

belt passes the bottom roller it acquires a positive charge, which is transported

to the top of the device (inside the dome). Here another metal brush facilitates

the transfer of electrons from the dome to the belt, leaving the dome positively

charged.

In short, the belt transports electrons from a metal dome to the ground,

producing a very positively charged dome. No outside source of charge is

required, and the generator could even be powered by a hand crank. A person

touching the dome will have some of her e-’s drained out. So, her lightweight,

positive hair will repel itself. Coming close to the charge dome will produce

sparks when electrons jump from a person to the dome.

Van de Graaff

Generator

Internal workings Detailed explanation

http://www.edm.net/~pmiller/graphics/gen4.jpg

http://images.google.com/imgres?imgurl=www.englib.cornell.edu/displays/stickershock/images/vandegraf2.jpg&imgrefurl=http://www.englib.cornell.edu/displays/stickershock/default.html&h=466&w=460&prev=/images?q=hair+raising&svnum=10&hl=en

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/vandeg.html

http://science.howstuffworks.com/vdg2.htm

Coulomb’s Law

K = 9 109 Nm2 /C2

Coulomb's Law Detailed

ExampleCharges in Motion

F = K q1 q2

r 2

There is an inverse square formula, called Coulomb’s law, for

finding the force on one point charge due to another:

This formula is just like Newton’s law of uniform gravitation with

charges replacing masses and K replacing G. It states that the

electric force on each of the point charges is directly proportional to

each charge and inversely proportional to the square of the distance

between them. The easiest way to use the formula to ignore signs

when entering charges, since we already know that like charges

repel and opposites attract. K is the constant of proportionality. Its

units serve to reduce all units on the right to nothing but newtons.

Forces are equal but opposite.

+ -q1 q2

rF F

http://physics.bu.edu/py106/notes/Coulomb.html

http://physics.bu.edu/py106/notes/Coulomb.html

http://www.colorado.edu/physics/2000/waves_particles/wavpart2.html

Electric Force vs. Gravitational Force

K = 9 109 N m2 / C2FE = K q1 q2

r 2

G = 6.67 10-11 N m2 / kg2FG = G m1 m2

r 2

Gravity is the dominant force when it comes to shaping galaxies

and the like, but notice that K is about 20 orders of magnitude

greater than G. Technically, they can’t be directly compared, since

they have different units. The point is, though, that a whole lot of

mass is required to produce a significant force, but a relatively

small amount of charge can overcome this, explaining how the

electric force on a balloon can easily match the balloon’s weight.

When dealing with high-charge, low-mass objects, such as protons

& electrons, the force of gravity is negligible.

Electric Force Example

+ +15 μm

A proton and an electron are separated by 15 μm. They are released

from rest. Our goal is to find the acceleration each undergoes at the

instant of release. 1. Find the electric force on each particle.

2. Find the gravitational force on each particle. A proton’s

mass is 1.67 10-27 kg, and an electron’s mass is 9.11

10-31 kg.

3. Find the net force on each and round appropriately. Note

that the gravitational force is inconsequential here.

4. Find the acceleration on each particle.

5. Why couldn’t we use kinematics to find the time it would

take the particles to collide?

1.024 10-18 N

4.51 10-58 N

1.024 10-18 N

e-: 1.124 1012 m/s2, p+: 6.13 × 108

m/s2

r changes, so F changes, so achanges.

System of 3 Charges

17

cm

14 cm

115

º

+3

μC

-5 μC

+2 μC

A

CB

In a system of three point charges, each charge exerts a forces

on the other two. So, here we’ve got a vector net force problem.

Find the net force on charge B. Steps:

1. Find the distance in meters between A and B

using the law of cosines.

2. Find angle B in the triangle using the law of

sines.

3. Find FBA (the magnitude of the force on

charge

B due to charge A).

4. Find FBC.

5. Break up the forces on B into components

and find the net horiz. & vertical forces.

0.261947 m

36.027932 º

0.786981 N

4.591836 N

3.78 N (right) , 1.25 N (up)

3.98 N at

18.3 º N of E

System of 4 Charges

-16 µC

+9

µC

-7 µC

+25 µC

3

cm

4 cm

A

B

C

D

Here four fixed charges are arranged in a rectangle.

Find Fnet on charge D.

Solution:

Link

767.2 N at 59.6 º N of W

http://www.phys.ufl.edu/~phy3054/elecstat/efield/polygon4/Welcome.html

Hanging Charge Problem

q, m q, m

LL

mg

T

FE

Two objects of equal charge and mass are

hung from the same point on a ceiling

with equally long strings. They repel each

other forming an angle between the strings.

Find q as a function of m, L and .

Solution: Draw a f.b.d. on one of the

objects, break T into components, and

write net vertical and horiz. equations:

T sin( /2) = FE , T cos( /2) = mg.

Dividing equations and using Coulomb’s law yields:

mg tan( /2) = FE = Kq2 / r 2, where r = 2Lsin( /2). Thus,

q = 4L2 mg tan( /2) sin2( /2)

K

Point of Equilibrium

+2q

d

x = ?

+q

Clearly, half way between two equal charges is a point of

equilibrium, P, as shown on the left. (This means there is zero

net force on any charge placed at P.) At no other point in space,

even points equidistant between the two charges, will

equilibrium occur.

Depicted on the right are two positive point charges, one with

twice the charge of the other, separated by a distance d. In this

case, P must be closer to q than 2q since in order for their

forces to be the same, we must be closer to the smaller charge.

Since Coulomb’s formula is nonlinear, we can’t assume that P is

twice as close to the smaller charge. We’ll call this distance x

and calculate it in terms of d.

+q+q P P

Continued…

Point of Equilibrium (cont.)

+2q

d

x

+q P

Since P is the equilibrium point, no

matter what charge is placed at P, there

should be zero electric on it. Thus an

arbitrary “test charge” q0 (any size any

sign) at P will feel a force due to q and

an equal force due to

2q. We compute each of these forces

via Coulomb’s law:K q q0

x2

K (2q)q0

(d - x)2=

The K’s, q’s, and q0’s cancel, the

latter showing that the location of P

is independent of the charge placed

there. Cross multiplying we obtain:

(d - x)2 = 2x2 d2 - 2xd + x2 = 2x2

x2 + 2xd - d2 = 0.

Point of Equilibrium (cont.)

From x2 + 2xd - d2 = 0,

the quadratic formula

yields:

+2q

d

x

+q P

x = -2d (2d)2 - 4(1)(-d2)

2(1)

-2d 8d2

2=

= -d d 2 Since x is a distance, we choose the positive

root:x = d ( 2 - 1) 0.41d. Note that x < 0.5d, as predicted.

Note that if the two charges had been the same, we would

have started with (d - x)2 = x2 d2 - 2xd + x2 = x2

d2 - 2xd = 0 d (d - 2x) = 0 x = d/2, as

predicted. This serves as a check on our reasoning.

Equilibrium with Several ChargesSeveral equal point charges are to be arranged in a plane so that another

point charge with non-negligible mass can be suspended above the plane.

How might this be done?

Answer: Arrange the charges in a circle, spaced evenly, and fix them in

place. Place another charge of the same sign above the center of the

circle. If placed at the right distance above the plane, the charge could

hover. This arrangement works because of symmetry. The electric force

vectors on the hovering charge are shown. Each vector is the same

magnitude and they lie in a cone. Each vector has a vertical component and

a component in the plane. The planar components cancel out, but the

vertical components add to negate

the weight vector. Continued…

Equilibrium with Several Charges (cont.)Note that the charges in the plane are fixed. That is, they are attached

somehow in the plane. They could, for example, be attached to an

insulating ring, which is then set on a table. Regardless, how could the

arrangement of charges in the plane be modified so as to maintain

equilibrium of the hovering charge but allow it to hover at a different height?

Answer: If the charges in the plane are arranged in a circle with a large

radius, the electric force vectors would be more horizontal, thereby working

together less and canceling each other more. The hovering charge would

lower. Since its weight doesn’t change, it must be closer to the plane in

order to increase the forces to compensate for their partial cancellation. If

the charges in the plane were arranged in a small circle, the vectors would

be more vertical, thereby working together more and canceling each other

less. The hovering charge would rise and the vectors would decrease in

magnitude. To maximize the height of the hovering charge, all the charges

in the plane should be brought to a single point. Continued…

www.phys.ufl.edu/~phy3054/elecstat/efield/twopoint/Welcome.html

www.phys.ufl.edu/~phy3054/elecstat/efield/polygon4/Welcome.html

www.eskimo.com/~billb/emotor/belt.html

207.10.97.102/chemzone/lessons/03bonding/mleebonding.htm

chem.ch.huji.ac.il/~eugeniik/instruments/archaic/electroscopes.html

www.physicsclassroom.com/mmedia/estatics/gep.html

www.cutescience.com/files/collegephysics/movies/GroundPositiveRodA.html

Credits

http://www.phys.ufl.edu/~phy3054/elecstat/efield/twopoint/Welcome.html

http://www.phys.ufl.edu/~phy3054/elecstat/efield/polygon4/Welcome.html

http://www.eskimo.com/~billb/emotor/belt.html

http://207.10.97.102/chemzone/lessons/03bonding/mleebonding.htm

http://chem.ch.huji.ac.il/~eugeniik/instruments/archaic/electroscopes.html

http://www.physicsclassroom.com/mmedia/estatics/gep.html

http://www.cutescience.com/files/collegephysics/movies/GroundPositiveRodA.html

Documents

Electrostatics f