PHY 2049 Lecture Notes Chapter 22: Page 1 of 8

A. Korytov

Electric Charge and Electrostatic Force

Contemporary vision: all forces of nature can be viewed as interaction between "charges", specific fundamental properties of matter. Electrostatic force: By rubbing amber against fur, one can discover that both amber

and fur acquire some new properties that cause them attract each other. The new property that is responsible for this force is called electric charge q.

What is interesting is that if one splits this way charged amber

piece apart, the smaller pieces repel each other. Both facts can be explained if one assumes that

there are two kinds of charges: positive and negative we chose to call them positive and negative; we could chose "sour" and "sweet", "left" and "right", "day-like" and "night-like" and this would do just as well

same-kind charges repel each other opposite kinds attract each other

PHY 2049 Lecture Notes Chapter 22: Page 2 of 8

A. Korytov

Electric Charge and Electrostatic Force SI Units for charge:

C, Coulomb we will discuss later how this unit was chosen--it was derived from units of current

1 C is a very large charge (just try to hold two 1 C charges in your hands!)

More on electric charges:

total electric charge is conserved, i.e. the net charge in any closed system never changes

Milliken: there is a smallest unit of charge e 1.610-19 C Coulomb's Law for point-like charges:

q1 q2

R

the force is directed along the line connecting the charges two point charges repel or attract each other

(same sign charges repel, opposite sign--attract) the magnitude of the force is as follows:

02

212

21

41

pe

=

=

Rqq

kR

qqFr

k 8.99x109 Nm2/C2 9x109 Nm2/C2

k = 1/(4pepe0), where ee0 8.85x10-12 C2/(Nm2)

PHY 2049 Lecture Notes Chapter 22: Page 3 of 8

A. Korytov

Other "Charges" and Forces in Nature - I

Gravitational force: responsible for attraction of planets to the Sun, for an apple falling down, etc., etc.

GR

mmF 2

21 =

Here masses m1 and m2 are gravitational "charges". There is only one kind of gravitational charges--one may want to

call them "positive" (any other name would be as good). As far as we know mass does not quantize, i.e. there is no smallest

quantum of mass Mass is not conserved, it can be converted into energy: E=mc2 Gravitational force is very weak, incomprehensibly weaker than

electrostatic force (in the world of elementary particles):

Take example of two electrons: electrons have mass me 9.110-31 kg and charge qe = -e -1.610-19 C G = 6.67x10-11 Nm2/kg2, k = 9x109 Nm2/C2

432

2

2

2

2

2

102// -=

=

=

=

kGem

FF

kRe

kR

qqF

GR

mmF

EG

eeE

eeG

10-43: Incomprehensibly small number!!!!

1 sec and age of Universe: 1 s / (15109 years 3107 s/year ) 10-18 (smallest distance we can resolve) / (observable universe) ~ (10-16 m)/(1022 m)

PHY 2049 Lecture Notes Chapter 22: Page 4 of 8

A. Korytov

Other "Charges" and Forces in Nature - II

Strong force: responsible for holding protons and neutrons inside an atom nucleus (protons repel each other, while gravitational force is too week to hold them together). There is six kinds of strong force charges--we chose to call them

"green", "red", "blue", "anti-green", "anti-red", "anti-blue") For example: protons, although color-neutral themselves, consist

of three quarks that carry these charges: proton anti-proton pp-meson What about magnetic force?: Once thought to be one of the fundamental forces. Now we know it is due to the same electric charges set in motion

PHY 2049 Lecture Notes Chapter 22: Page 5 of 8

A. Korytov

Electrostatic Force is a Vector

Electrostatic Force is a vector (as any other force). Q1

q

F1

Q2

Q3

F2

F3R3

R2

R1

Here are a few tips how to draw the vector forces (accurate drawing is the key to handling vector forces): remember that forces act on charges to figure out the force acting on charge q in presence of other

charges, one needs to jump on charge q and count all charges around (three in the example above: Q1, Q2, Q3)

Each of these external charges will exert a force on charge q according to the Coulomb 's Law and you draw all three vectors of the forces Fi, experienced by the charge q

a) starting from the point corresponding to charge q; b) along the line connecting q and Qi; c) in direction of attraction/repulsion according to signs of charges q and QI; d) and with magnitude calculated according to Coulomb's Law:

kRqQ

Fi

ii 2=

The net force acting on charge q is the vector sum of all these three forces:

F = F1 + F2 +F3

PHY 2049 Lecture Notes Chapter 22: Page 6 of 8

A. Korytov

Vectors & Vector Addition

Graphical Addition of Vectors:

C = A + B

x-axis

y-axis

A

C

B

Breaking a Vectors into x- and y-components:

x-axis

y-axis

qq

A

Ax =A cos qq

Ay =A sin qq

To add vectors you add the components of the vectors as follows:

kBAjBAiBABAC

kBjBiBB

kAjAiAA

zzyyxx

zyx

zyx

)()()(

+++++=+=++=++=

rrr

r

r

PHY 2049 Lecture Notes Chapter 22: Page 7 of 8

A. Korytov

Useful Approximations

For any small ee (|ee|

PHY 2049 Lecture Notes Chapter 22: Page 8 of 8

A. Korytov

Electric Dipole -Q +Q

d

An electric dipole is two equal and opposite point charges separated by a distance d. It is an electrically neutral system.

The "dipole moment" p is defined to be the charge Q times the separation d, i.e., p = Qd.

Example Problems:

1. A dipole with charge Q and separation d is located on the y-axis with its midpoint at the origin. A charge q is on the x-axis a distance x from the midpoint of the dipole. What is the electric force on q due to the dipole (assume x >>d)?

+Q

-Q

dq

x

k

x

qQdF

3

)(

2. Same, but with the dipole oriented along x-axis.

+Q -Q

d

q

x

kx

qQdF 3

)(2

3. Find the force between two dipoles oriented as shown: +Q -Q

dx

+Q -Q

dk

xQd

F 42)(

6

Note that despite the fact that both dipoles are neutral, there remains a residual week force between them (~1/x4). Does it contradict to Coulomb's Law? No, the law is formulated for point-like charges, while dipoles are clearly not point-like and have internal structure. This problem is intended to help understand how neutral atoms can attract each other to make molecules and form solid objects.