Andres La Rosa
Portland State University
Lecture Notes
PH-212
ELECTRIC FIELD
P qo
If an electrical charge qo located at a
position P experiences a force, we say
that there exist an electric field in a
region around P.
The electric field is produced by the
system of charges. The electric field
characterizes the system of charges.
system of
charges
Operational procedure to calculate the ELECTRIC FIELD produced a
given system of charges at the point P
i) Place a test charge qo at the point P.
ii) Find the electrical force F that such system of
charges exerts on qo.
iii) The ELECTRIC FIELD at P will be given by
P
qo
E
P
The ELECTRIC FIELD
is a vector
=E
First, some basic concepts about vectors
Next, we show that
the electric field produced by a positive charge Q
is independent of the sign of the test charge used to evaluate it.
Positive test
charge qo
P
E
qo (+)
Q
Q
Negative test
charge q1
q1 (-)
F
Q
E
Force
Q
Now we show that the electric field produced by a negative charge q
is independent of the sign of the test charge used to evaluate it.
P
Positive test P
q3 (+)
charge q3
q
Negative test
charge q4
E
F
q
Force
P
F
q4 (-)
E
q
q
Q
E
E
q
Example. Indicate the electric filed at the point P established
by the two positive charges shown in the figure
Notice, there
is no charge at P
P
First, let's place a positive test charge qo at the position indicated by P, and
evaluate the total electric force acting on that charge.
F
qoP
Next, we simply calculate the ratio
Notice, there
is no charge at P
Force vector
F
E
qo
P
( Since qo is positive, then E and F will be
oriented in the same direction)
E
Electric field
vector
Had we have chosen a negative test charge qo, we would have obtained
the following
Force
vector
F
qo
( Since qo is negative, then E and F will be
oriented in the opposite direction)
P
E
System of charges
Conclusion
E does not depend on the "test" charge qo.
Exercise. Find the electric field at point P(0,4) and at the point T(0,-2)
q1 = 10 μC
Y
(cm)
q2 = - 10 μC
q1
q2
X (cm)
System of charges
Solution
E01 = 9x109 (10x10-6) / (5x10-2)2 = 3.6 x 107 N/C
Unit vector u1 = (3/5, 4/5) = ( COS 530, SIN 530 )
E02 = 3.6 x 107 N/C
Unit vector u2 = (3/5, - 4/5)
E (at P) = 2 x 3.6 x 107 N/C x (3/5, -0 ) =
(4.32 x 107 N/C , 0 )
Does the electric field exist at only one point?
Does the electric field depend on the magnitude or sign of
the test charge qo?
How can we mathematically verify that the electric field of a
system of charges is independent of the test charge qo alluded
in the experimental check
procedure given at the beginning of this
chapter?
System of charges
(Charges on a
stick bar)
Summary
P
E
System of charges
E
E
If a charge q (positive or negative) were placed at the position P shown in the figure above
(and provided that such charge does not disturb the position of the system of charges)
then q will experience a force given by,
F =
q
E
ELECTRIC FIELD
(Continuation and Review)
+q
electric field
+q
E
How to calculate the electric field E at a point P located a distance "r"
from a positive point-charge of magnitude e ?
E
E (at P) =
+e
Units
e:
How to calculate the electric field E at a point S located a distance "r " from
a negative point-charge of magnitude e ?
E (at S) =
-e
rr
E
S
Example: Electric field produced by a positive point-charge q
Example: Electric filed produced by a negative point-charge Q
-
What is a DIPOLE ?
Definition of the electric dipole moment:
is a vector that point from the negative charge towards
the positive charge
Microscopic description of non-conductivity, in terms of the electricdipole concept
-+
Charge-neutral
molecule
Charge induction inside the
molecule results in the creation
of a dipole.
The dipole orients in response to
the external electric field
-Q
non-conducting
material
The dielectric breakdown
There exists a maximum value for the external electric field beyond which
the atoms inside the material become ionized.
The dislodged electrons (now free) give rise to a high current, which heats
up, and eventually destroy, the material.
This phenomenom is called dielctric breakdown.
Dielectric breakdown occurs in air at E = 3x 106 N/C .
The water molecule
H
Oxygen
H
Negative
side
Positive
side
Accordingly, the molecule of
water can be treated as an
electric dipole
p = 6.2 x 10-30 Cm
For a neutral water molecule in
its vapor state
Real structure
10 protons (+)
10 electrons (-)
Simplified model
p = Qd d = 6.2 x 10-30 Cm
= (10e) dd
d=
d = 0.04 Angstroms
Electric field established by a dipole
P
E
+Q
T
E
-Q
E
S
Electric field at different locations
(P, T , S) produced by the charges
+Q and - Q
P
T
W
S
Charges inside an electric field
Charge moving inside a uniform electric field
Y
X
E
Negative charge
of mass m
E = 1.4 x 106 N/C
Question: What would be trajectory followed by the
charged particle?
Does the acceleration of the particle
change during its the motion inside the field?
Calculate:
i) The acceleration of the particle
ii) How much does the particle deflects
(vertically) when it travels a horizontal
distance of 1.6 cm from the instant it enters
the region of the electric field?
Electric-dipole inside an electric field
Order of magnitude of the electric force
Repulsive force between 2 protons inside a nucleus
Attractive force between a proton in the nucleus of an atom and an
electron flying around the nucleus
Electric field established by a proton at a distance 1 Angstrom far away
Electrical Ground: A very large conductor able to supply an unlimited
amount of charge
Review on Dielectric Breakdown
Exploiting symmetry (of the charge distribution in a
system of charges) to calculate electric fields
Example 1. Electric field established by a
uniformly charged circular line.
P
A total amount of positive
charge Q is UNIFORMLY
distributed along a plastic
ring of radius R
Question: What is the direction of the electric field at
point P?
Solution
First
Second
Third: Apply symmetry
For each small segment containing a charge
dq1, there exists another segment
symmetrically located containing a charge
dq2 (with dq2=dq1)such that the HORIZONTAL
components of the electric field cancel out.
That is,only the VERTICAL components make
a net contribution to the TOTAL FIELD.
Thus, based on the grounds of symmetry,
we calculate the VERTICAL component of
the electric field produced by a charge dq
contained in a segment of length ds, and
then ADD UP all those vertical
components correspondent to each
charge on the circular line
Example 2. Electric field established by a
uniformly charged disk.
P
A=πR2
Question: What is the direction of the electric field at
point P?
Ring of radius r and thickness dr
How much charge (dq) does this
ring contains?
dA = 2πr dr
σ
E(z) =
-2
-2
R2
z
R