Transcript the electric fields of point charges

THE ELECTRIC FIELD
• ELECTRIC CHARGE
• COULOMB’S LAW
• THE ELECTRIC FIELD
Written by Dr. John K. Dayton
ELECTRIC CHARGE:
Electric charge is a fundamental quantity. The smallest possible charge
that can be isolated is given by:
e  1.6  1019 C
The unit of electric charge is the coulomb, abbreviated C. There are two
types of electric charge. While they may be designated in any manner, it
is most convenient to designate them8 as positive, +, and negative, -.
Thus the smallest possible charges are +e and -e. Any electric charge,
usually designated as q, is composed of collections of +e or -e. Thus
q = (+/-)ne where n is a positive integer. Due to this nature we say
electric charge is quantized. This is very important when n is small.
When n is large the quantized nature of electric charge is not important
and is usually ignored.
Benjamin Franklin
January 17, 1706 – April 17, 1790
Benjamin Franklin was one of the
Founding Fathers of the United States and
in many ways was "the First American". A
renowned polymath, Franklin was a
leading author, printer, political theorist,
politician, postmaster, scientist, inventor,
civic activist, statesman, and diplomat. As
a scientist, he was a major figure in the
American Enlightenment and the history
of physics for his discoveries and theories
regarding electricity. As an inventor, he is
known for the lightning rod, bifocals, and
the Franklin stove, among other
inventions. He facilitated many civic
organizations, including Philadelphia's fire
department and a university.
ELECTRICALLY CHARGE OBJECTS:
There are many occasions when we encounter electrically charge objects.
When an object is electrically neutral it contains exactly the same
number of electrons and protons. Thus the same number of -e charges
and +e charges. Electrons furthest from the nuclei in atoms are held in
place most weakly and are often lost to other materials. The objects
gaining these lost electrons now have more negative charge than positive
charge and we say they are negatively charged. Objects whose atoms
lost electrons also have an imbalance of charge due to a deficit of
negative charge. We say such objects are positively charged. Protons,
which each have +e charge, do not transfer between objects due to the
fact they are tightly bound within atomic nuclei. Electrically charged
objects that are positively charged have a deficit of electrons, not an
excess of protons.
THE FUNDAMENTAL PRINCIPLE OF ELECTROSTATICS:
There is a force of interaction between electric charges that obeys the
following rule:
Like charges repel each other and
Unlike charges attract each other.
+
+
+
-
BASIC ATOMIC STRUCTURE:
For our purposes it is convenient to use the Bohr model of the atom. This
consists of a dense, massive central core composed of protons and
neutrons. About the core are electrons in well defined orbits.
THE PROTON:
Electric Charge:
Mass:
+e
1.67  1027 kg
THE NEUTRON:
Electric Charge:
Mass:
0
1.67  1027 kg
THE ELECTRON:
Electric Charge:
Mass:
-e
v
9.11  1031 kg
+
-
F
Niels Henrik David Bohr
7 October 1885 – 18 November 1962
Niels Bohr was a Danish physicist
who made foundational
contributions to understanding
atomic structure and quantum
theory, for which he received the
Nobel Prize in Physics in 1922.
Bohr was also a philosopher and a
promoter of scientific research.
THE COULOMB INTERACTION, COULOMB’S LAW
q1q2
Fk 2
r
q1 and q2 are two interacting electric charges.
r is the distance between the two charges.
k is a constant of proportionality.
k  8.99  109
F is the magnitude of the force between q1 and q2
To find the direction of F, use the Fundamental
Principle of Electrostatics.
Nm2
C2
Charles-Augustin de Coulomb
14 June 1736 – 23 August 1806
He was best known for developing
Coulomb's law, the definition of the
electrostatic force of attraction and
repulsion, but also did important
work on friction. The SI unit of
electric charge, the coulomb, was
named after him.
His name is one of the 72 names
inscribed on the Eiffel Tower.
Example:
Charge q1 = 3.0 mC is placed at x = 0 cm. Charge q2 =
7.0mC is placed at x = 30.0 cm. Where between q1 and q2 should a third
charge, q, be placed so that the net force on it is zero?
0cm
3rd charge (unknown q)
Click
F For Answer
+
d
q1 = 3 0 mC
30 cm
+
x-d
q2 = 7.0 mC
Charge q is placed a distance d from q1. Its distance from q2
will be x – d where x is 30cm. The resulting forces on q,
from the left and right, will be equal and opposite when d is
the correct distance. You must solve for d.
Solution is continued on next slide.
kqq1
kqq2

2
2
d
x d
The forces on q from q1 and q2
are equal in magnitude.
q1
q2
 2
2
2
d
x  2 xd  d
2
2
2
q1  x  2 xd  d   q2 d
d
2
 q1  q2   2q1 xd  q1 x
Cancel common factors k and
q and expand denominator.
Cross multiply.
2
0
Render in standard
quadratic form.
6
2
6
6

4

10
C
d

1.8

10
Cm
d

.27

10
Cm   0

 
 
2
Simplify and solve using
4d  1.8d  .27  0
the quadratic formula.
d  0.12m or -.57 m(not a solution)
2
Only d = 0.12m is a valid solution.
CONDUCTORS AND NONCONDUCTORS:
Conductors are materials, including metals, that have large numbers of
electrons capable of moving throughout the material virtually as a free
electron gas.
PROPERTIES OF CONDUCTORS:
• Any net electric charge exists on the outer surface.
• Any net electric charge on a conductor will adjust its position until an
equilibrium condition is established.
• The only mobile charge in a conductor is negative due to mobile
electrons.
Nonconductors have none of these properties.
THE ELECTROSCOPE:
The electroscope is a simple device used to
prove whether or not an object has excess
charge on it. The electroscope has three parts: a
jar, a conducting rod, and a pair of thin metal
strips. The rod sticks through the top of the jar
and holds the two metal strips inside the jar.
Usually the rod has a round metal ball on its
outer end.
When an object with excess charge is brought
near the metal rod, electrons will be moved
from or toward the metal strips. This causes the
metal strips to acquire excess charge, but of the
same type on both strips. In turn this causes the
strips to repel and swing away from each other,
indicating the presence of excess charge.
THE ELECTRIC FIELD:
Surrounding every electric charge is an electric field. Thus at every
point in space surrounding an electric charge there is a vector quantity
called the electric field. The electric field is defined as the force per unit
coulomb and has SI units N/C. If q1 and q2 are two point charges then the
force between2:them is:
q1q2
Fk 2
r
then the magnitude of the electric field of q1 is:
q1
Ek 2
r
By definition the electric field at a point in space is:
E
F
qo
where F is the force on a charge qo placed at that point.
Electric field vectors point away from positive charges and
toward negative charges.
Michael Faraday
22 September 1791 – 25 August 1867
Michael Faraday was an English scientist who
contributed to the fields of electromagnetism and
electrochemistry. His main discoveries include those
of electromagnetic induction, diamagnetism and
electrolysis. Although Faraday received little formal
education, he was one of the most influential
scientists in history. It was by his research on the
magnetic field around a conductor carrying a direct
current that Faraday established the basis for the
concept of the electromagnetic field in physics.
Faraday also established that magnetism could affect
rays of light and that there was an underlying
The concept of an electric field was relationship between the two phenomena. He
similarly discovered the principle of electromagnetic
introduced by Michael Faraday.
induction, diamagnetism, and the laws of electrolysis.
His inventions of electromagnetic rotary devices
formed the foundation of electric motor technology,
and it was largely due to his efforts that electricity
became practical for use in technology.
THE SUPERPOSITION PRINCIPLE:
Every point charge has its own electric field.
When two or more point charges are near each other their individual
electric fields superimpose producing a net, composite electric field.
If EP ,1 is the electric field of q1 at point P and EP ,2 is the electric field
of q2 at point P, then EP is the net electric field at P given by:
EP
EP  EP ,1  EP ,2
EP ,1
EP ,2
P
In general:
ENET   Ei
Note that this is a vector sum.
The net electric field at a point in space is the vector sum of all
individual electric fields produced by point charges.
ELECTRIC FIELD LINES:
Electric field lines are used to graphically represent an electric field. The following set
of rules apply to electric field lines:
1 Field lines originate on positive charges and terminate on negative charges,
otherwise they come in from or go out to infinity.
2 Field lines do not cross other field lines.
3 The number of field lines on a charge is proportional to the charge.
4 The number of field lines in a region of space is proportional to the field strength in
that region. The closer together they are - the stronger the field.
5 The electric field at a point in space is tangential to the field line through that point.
6 Electric field lines are perpendicular to all conducting and equipotential surfaces.
7 The electric field inside a conductor in electrostatic equilibrium is zero.
MORE ABOUT CONDUCTORS:
The following object is a conducting material with a cavity inside.
Inside the cavity is a charge Q (how it got there we don’t know or care,
it’s just there).
conductor
cavity
charge Q
opposite
charge –Q
same
charge Q
When Q is placed inside the cavity, electrons in the conductor
redistribute so that a charge of –Q lines the inside wall of the cavity and
a charge of Q lines the outer surface of the conductor. Thus the net
charge inside the cavity will be 0 and the overall net charge will lie on
the outer surface.
conductor
net charge lies
on surface
region of
no field
charge
placed inside
a cavity
opposite charge
due to mobile
electrons
electric field
between charges
inside cavity
The red lines represent the electric field. Because the cavity contains
charge there is an electric field between these charges. This internal field
does not penetrate the conductor. In the conducting material there is no
field (no electric field vectors, no field lines). A field extends from the
surface due to the surface charge. It is important to note that the net
charged inside the cavity is zero when the conductor is in electrostatic
equilibrium.
THE ELECTRIC FIELDS OF POINT CHARGES:
Essentially, all electric fields are the result of superposition of point charge electric
fields.
q = electric charge producing the field
q
Ek 2
r
k = electrostatic constant,
k  8.99  10
9 Nm2
C2
r = distance field point is from charge
E = magnitude of electric field at the field point
Electric fields point away from positive charges and
toward negative charges. This establishes the direction
of the electric field.
The electric field outside any spherical charge
distribution is the same as that of a point charge with
the same total charge.
The field of a + charge.
EXAMPLE: q1 = -4.0 mC is located at the origin. q2 = +3.0 mC is
located on the y-axis at 10.0 cm. Calculate the net electric field on the xaxis at 15.0 cm.
r2 
.1m   .15m 
q2 = +3mC Click For Answer
+
r2 = .1803m
10cm
-
r1 = .15m
15cm
2
2
 0.1803m
 .1 
o
q  tan    33.69
 .15 



E P  E P ,1  E P , 2
1
P
q = 33.69o
This diagram is the
first step in a well
planned solution.
q1 = -4mC
Solution continues on next slide.
kq1
EP ,1  2 
r1
EP ,2
kq2
 2 
r2
8.99  10
9 Nm 2
C2
6

4

10
C

.15m 
8.99  10
9 Nm 2
C2
2
6

3

10
C

.1803m 
2
 1.5982  106
N
C
 8.2964  105
N
C
The direction in standard form for Ep,1 is 180o and for Ep,2 it is
326.31o.
E p ,1  1.5982  106 at 180o
E p ,2  8.2964  105 at 326.31o
E p ,1 
1.5982  106 cos 180o  iˆ

1.5982  106 sin 180o  ˆj
E p ,2  8.2964  105 cos  326.31o  iˆ  8.2964  105 sin  326.31o  ˆj
Ep 
9.0790  105 iˆ

4.6020  105 ˆj
Solution continues on next slide.
EP 
 9.0790  10

5 2
  4.6020  10

5 2
 1.02  106
N
C
5

1 4.6020  10 
o
q  tan 

180

206.9
5 
 9.0790  10 
EP  1.02  10
6 N
C
at 206.9
o
Final Answer
The most common error made is ignoring the vector nature of the electric
field. This problem can only be solved correctly as a vector problem.
THE ELECTRIC DIPOLE FIELD:
EP 
2kqa
x
2
a
2

3
2
x  a, then
2kqa
Ep  3
x
if
DERIVATION OF THE FIELD ALONG THE PERPENDICULAR
BISECTOR OF AN ELECTRIC DIPOLE.
kq
E 2
r
E P , q  E P , q
E p  2 E y  2 E sin  q
 kq  a 2kqa
Ep  2  2   3
r
r r
Refer to the diagram
on the previous slide.
substitutions used:
a
sin  q  
r
r  x  a
2
Ep 
2kqa
x
2
a
2

3
2
2

1
2
THE UNIFORM ELECTRIC FIELD:
The electric field strength of an infinite, uniform plane of charge, such as
on a flat, non-conducting sheet is:
E

2òo
 = surface charge density ( charge per unit area)
òo = permittivity of free space = 8.85  1012
C2
N m2
The electric field strength of the charge on an infinite conducting plane
is:

E
òo
k
1
4 òo
In both cases the electric fields are uniform fields that point
perpendicularly outward from positive charge and inward toward
negative charge.
MOTION OF A CHARGED PARTICLE IN AN ELECTRIC
FIELD:
F  qE
qE  ma
qE
a
m
A particle with mass m and charge q is
shown in a uniform electric field E. If the
charge is positive, the force on the charge
will be in the direction of the electric
field. If it is negative, the force will point
in the opposite direction. The charge
shown in the diagram must be positive
since the force on it is in the direction of
the field. The acceleration is always in the
direction of the force.
Only in a uniform electric field is the
acceleration of a charged particle constant.
EXAMPLE: What is the acceleration of a proton in the vicinity of an
infinite plane of charge with surface charge density 2.0 pC/m2?
E

2òo

2  10
12 C
m2
12 Nm 2
C2
2  8.85  10
F eE
a 

m m
a  1.08  107
 1.1299  10
1 N
C
Click For Solution
19
1 N
1.6

10
C

1.1299

10


C
1.67  10
m
s2
27
kg
directed away from the plane of charge
GAUSS’S LAW:
Gauss’s Law is best understood and expressed with calculus but an
algebraic version, applicable to very symmetrical charge distributions
can be used.
qenclosed
 E cos q A  ò   E
o
The left side is the net perpendicular electric field passing through a
closed surface of area A surrounding or enclosing the electric charge
qenclosed. It has units N∙m2/C and is known as electric flux, designated E.
Electric charge outside of this surface does not contribute to the electric
field on the surface.
Johann Carl Friedrich Gauss
30 April 1777 – 23 February 1855
Johann Gauss was a German mathematician, who
contributed significantly to many fields, including
number theory, algebra, statistics, analysis,
differential geometry, geodesy, geophysics,
electrostatics, astronomy, Matrix theory, and optics.
Gauss was the son of poor working-class parents.
His mother was illiterate and never recorded the date
of his birth.
Gauss was a child prodigy. There are many
anecdotes about his precocity while a toddler, and he
made his first ground-breaking mathematical
discoveries while still a teenager. He completed
Disquisitiones Arithmeticae, his magnum opus, in
1798 at the age of 21, though it was not published
until 1801. This work was fundamental in
consolidating number theory as a discipline and has
shaped the field to the present day.
Using Gauss’s Law, the following electric fields can be derived:
Outside a point or spherical charge distribution:
Outside a uniform line or cylindrical charge:
E
kq
r2

E
2 ròo
A uniform plane of charge:
E
Inside a uniformly charged non-conducting sphere:
E

2òo
r
3òo
 = linear charge density,  = surface charge density,
 = volume charge density.
The electric field of a uniformly charged, non-conducting sphere of
charge with charge density  and total charge Q and of radius R, both
inside and outside the sphere.
Einside 
E
r
Eoutside
3òo
kQ
E 2
R
(scale in
units of Emax)
R

Q
3Q

Volume 4 R 3
kQ
 2
r
k
1
4 òo
r (scale in units of R)
End of Presentation