PHY 2049 Lecture Notes
Chapter 22: Page 1 of 8
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
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 2 of 8
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 ≈ 1.6⋅⋅10-19 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:
r q1 ⋅ q2
q1 ⋅ q 2
1
⋅
F =
k
=
R 2 4πε 0
R2
k ≈ 8.99x109 N⋅⋅m2/C2 ≈ 9x109 N⋅⋅m2/C2
k = 1/(4πε
πε 0), where ε 0 ≈ 8.85x10-12 C2/(N⋅⋅m2)
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 3 of 8
Other "Charges" and Forces in Nature - I
Gravitational force:
responsible for attraction of planets to the Sun, for an apple falling
down, etc., etc.
F=
m1 ⋅ m2
G
2
R
• 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 ≈ 9.1⋅⋅10-31 kg and charge qe = -e ≈ -1.6⋅⋅10-19 C
• G = 6.67x10-11 Nm2/kg2, k = 9x109 Nm2/C2
me ⋅ me
G
R2
qe ⋅ qe
e2
FE =
k= 2k
R2
R
m2
FG / FE = 2 ⋅ G / k ≈ 2 ⋅ 10 − 43
e
-43
10 : Incomprehensibly small number!!!!
FG =
→ 1 sec and age of Universe: 1 s / (15⋅⋅109 years × 3⋅⋅107 s/year ) ≈ 10-18
→ (smallest distance we can resolve) / (observable universe) ~ (10-16 m)/(1022 m)
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 4 of 8
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
π-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
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 5 of 8
Electrostatic Force is a Vector
Electrostatic Force is a vector (as any other force).
Q1
R1
F2
q
R2
Q2
F3
F1
R3
Q3
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:
Fi =
qQ i
k
2
Ri
• The net force acting on charge q is the vector sum of all these
three forces:
F = F1 + F2 +F3
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 6 of 8
Vectors & Vector Addition
Graphical Addition of Vectors:
C=A+B
y-axis
C
B
A
x-axis
Breaking a Vectors into x- and y-components:
y-axis
Ay =A sin θ
A
θ
Ax =A cos θ
x-axis
To add vectors you add the components of the vectors as follows:
r
A = Axiˆ + Ay ˆj + Az kˆ
r
B = Bxiˆ + By ˆj + Bz kˆ
r r r
C = A + B = ( Ax + Bx )iˆ + ( Ay + By ) ˆj + ( Az + Bz )kˆ
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 7 of 8
Useful Approximations
For any small ε (|εε|<<1), the following expressions
can be approximated as:
(1 + ε )
p
≈ 1 + pε
eε ≈ 1 + ε
sin ε ≈ ε
tan ε ≈ ε
ε2
cos ε ≈ 1 −
2
A. Korytov
PHY 2049 Lecture Notes
Chapter 22: Page 8 of 8
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
d
F ≈
x
-Q
(Qd ) q
k
3
x
2. Same, but with the dipole oriented along x-axis.
+Q
-Q
q
F ≈2
x
d
(Qd ) q
k
3
x
3. Find the force between two dipoles oriented as shown:
+Q
-Q
+Q
-Q
x
d
d
(Qd ) 2
F ≈6
k
4
x
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.
A. Korytov