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 Charge Charged 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 = N e, where N is an integer.

• How many excess protons are required for an object to have 1 C

of charge?

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

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

uncharged

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 charge

separation 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…

Electroscopes (cont.)

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

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 charge

separation 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…

Electroscopes (cont.)

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

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 the

positive 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 Electroscope Whether 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 other.

# 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.

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

Coulomb’s Law

K = 9 109 N m2 / C2

Coulomb's Law Detailed Example Charges 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

r F F

Electric Force vs. Gravitational Force

K = 9 109 N m2 / C2 FE =

K q1 q2

r 2

G = 6.67 10-11 N m2 / kg2 FG =

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 a changes.

System of 3 Charges

17 cm

14 cm

115º

+3 μC

-5 μC

+2 μC

A

C B

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.

6. Determine Fnet on B.

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

Hanging Charge Problem

q, m q, m

L L

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 = Kq 2

/ r 2, where r = 2 L sin( / 2). Thus,

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

K

Point of Equilibrium

+2 q

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 2 q 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.)

+2 q

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

2 q. We compute each of these forces via

Coulomb’s law:

K q q0

x 2

K (2 q) 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 = 2 x 2 d

2 - 2 x d + x 2 = 2 x

2

x 2 + 2 x d - d

2 = 0.

Point of Equilibrium (cont.)

From x 2 + 2 x d - d

2 = 0, the quadratic formula yields:

+2 q

d

x

+q P

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

2 )

2 (1)

-2 d 8 d 2

2 =

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

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

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

started with (d - x)2 = x 2 d

2 - 2 x d + x 2 = x

2

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

predicted. This serves as a check on our reasoning.

Equilibrium with Several Charges Several 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

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