Timetable
Tuesday
11:10 – 12:50
(Lecture)
KA & OCH
2/5/2025 11:07:34 AM
Wednesday
16:40 – 18:00
(Tutorials)
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Friday
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Test 1/
Assignment 1
Test 2/
Assignment 2
Test 3/
Assignment 3
2/5/2025 11:07:34 AM
Electric Field and Gauss's Law, 07 March 2025
Electric Potential Energy and 04 April 2025
Difference and Capacitors and
Dielectrics,
Electric Current, Direct current 09 May 2025
and Electromagnetism
Chapter 1. Electric Fields
Chapter 2. Gauss' Law
Chapter 3. Electric Potential
Chapter 4. Capacitance and Dielectrics
Chapter 5. Current and Resistance
Chapter 6. Direct Current Circuits
Chapter 7. Magnetic Fields
2/5/2025 11:07:34 AM
Revision
❖ ELECTROSTATICS:
▪ Is a branch of physics that deals with the
phenomena and properties of stationary or
slowly moving electric charges with no
acceleration.
OR
▪ Is a situation when electric charges are
stationary, or moving very slowly, such that
there are no magnetic forces between
them.
Properties of electric charge
• There are two types of electric charges, called positive
and negative. Benjamin Franklin (1706 – 1790).
• Electrons are identified as having negative charge,
while protons are positively charged.
• Like charges repel, and unlike charges attract.
• Charge is conserved. This implies that a charge cannot
be created or destroyed, it can only be transferred from
one location to another and the total charge on an object
is the sum of all the individual charges carried by the
object.
• Charge is quantized, meaning it comes in discrete
amounts. (All protons carry the same amount of charges
+e, and all electrons carry a charge –e.
The electromagnetic force between charged particles
is one of the fundamental forces of nature.
✓ Basic Properties of electromagnetic forces
✓ Coulomb’s law
✓ Electric field
✓ Methods used to calculate
✓ Uniform electric field
CHAPTER 01
ELECTRIC FIELD
An electric field is a physical field that surrounds electrically
charged particles
Properties of electric charge
Example
• A glass rod is rubbed with silk.
• Electrons are transferred from the
glass to the silk.
• Each electron adds a negative
charge to the silk.
• An equal positive charge is left on
the rod.
Properties of electric charge
• Charge transfers from one type of material to another.
• Rubbing the two materials together serves to increase the
area of contact, facilitating the transfer process.
• An important characteristic of charge is that electric charge
is always conserved.
• Charge is not created when two neutral objects are rubbed
together.
• The objects become charged because negative charge is
transferred from one object to the other.
• One object gains a negative charge while the other loses
equal amount of negative charge.
Properties of electric charge
• A rubber rod that has been
rubbed with fur is suspended by
a piece of string.
• When the glass rod that has
been rubbed with silk is brought
near the rubber rod, the rubber
rod is attracted towards the
glass rod.
• Hence the force of attraction
exist between the two rods.
Figure (a) illustrates the interaction of the
two charges.
Properties of electric charge
• If two charged rubber rods (or two
charged glass rods) are brought
near each other, the force
between them is repulsive.
• These observations may be
explained by assuming that the
rubber and glass rods have
acquired different kinds of excess
charge.
• Charges of the same sign repel
one another and charges with Figure (a) illustrates the interaction of
opposite signs attract one the two charges.
another.
Properties of electric charge
• Q and q are the standard symbols used for charge.
• A charge Q or q is measured in Coulombs (C) in honor
of the French Physicist Charles de Coulomb (1736-1806)
• The charge on a single electron is:
electron charge = - e = -1.6 x 10-19 C
• and the charge carried by a single proton is:
proton charge = + e = +1.6 x 10-19 C
• Thus this indicates that:
Electron: q = - e and Proton: q = + e
Charging process
Charging means gaining or losing electron
❑ Types of charging:
1. charging by friction.
2. charging by conduction.
3. charging by induction.
Charging by friction
• When you rub one material to another, they
are charged by friction.
• Material losing electron is positively charged
and material gaining electron is negatively
charged.
• Amount of gained and lost electron is equal
to each other. In other words, we can say that
charges of the system are conserved.
• When you rub glass rod to a silk, glass rod
lose electron and becomes positively
charged, while silk gain electron and
becomes negatively charged.
Charging by conduction
• A negatively charged metal rod touches to the neutral metal
sphere and some of the electrons pass to the sphere. As a
result neutral sphere is negatively charged by contact.
• A positively charged metal rod touches a neutral metal sphere,
then free electrons in the neutral sphere are attracted to the
positively charged rod. Some of the free electrons will pass
over to the positively charged rod. Then, neutral sphere is now
missing some of its electrons and it is now positively charged.
Charging by induction
• Charging
by
induction
requires no contact with the
object inducing the charge.
• If we consider a neutral
metallic sphere in figure (a)
that has the same number of
positive and negative charges
Charging by induction
• A charged rubber rod is placed near
the sphere.
• It does not touch the sphere.
• The electrons in the neutral sphere
are redistributed.
• If the sphere is grounded fig (c).
• Some electrons can leave the
sphere through the ground wire.
Charging by induction
• The ground wire is removed.
• There will now be more positive charges.
• The charges are not uniformly distributed.
• The rod is removed.
• The electrons remaining
redistribute themselves.
on
the
sphere
• There is still a net positive charge on the sphere.
• The charge is now uniformly distributed
Different types of materials
•
•
•
Insulators
Conductors
Semiconductors
Insulators
• Electrical insulators are materials in which all of the
electrons are bound to atoms.
• These electrons can not move relatively freely through the
material.
• Examples of good insulators include materials like glass,
rubber and wood.
• When a good insulator is charged in a small region, only
that area becomes charged and charge is unable to move
to other regions of the material.
Conductors
• Electrical conductors are materials in which some of the
electrons are free electrons.
• Free electrons are not bound to the atoms.
• These electrons can move relatively freely through the
material.
• Examples of good conductors include metals like copper,
aluminum and silver.
• When a good conductor is charged in a small region, the
charge readily distributes itself over the entire surface of
the material.
Semiconductors
• The electrical properties of semiconductors are somewhere
between those of insulators and conductors.
• Examples of semiconductor materials include silicon and
germanium commonly used for the fabrication of a variety of
electronic devices.
• The electric properties of semiconductors are changed over
many orders of magnitude by adding controlled amounts
foreign atoms: Doping.
Charles Coulomb
• Charles Augustin de Coulomb (1736-1806)
• French physicist, made major contributions in
areas of electrostatics and magnetism.
COULOMB’S LAW
• If the charges are at rest then the force between them is known as the
electrostatic force (Sometimes called the electric force).
• The electrostatic force was first studied in detail by Charles Augustin
de Coulomb around 1784.
• The electrostatic force between charges increases when the
magnitude of the charges increases or the distance between the
charges decreases.
• Through his observations he was able to show that the magnitude of
the electrostatic force between two point-like charges is inversely
proportional to the square of the distance between the charges.
COULOMB’S LAW
• He also discovered that the magnitude of the force is
proportional to the product of the charges.
• That is:
• Where q1 and q2 are the magnitude of the two charges
respectively and r is the distance between them.
• Coulomb’s law states that the magnitude of the
electrostatic force between two point charges is directly
proportional to the product of the magnitudes of the
charges and inversely proportional to the square of the
distance between their centres.
COULOMB’S LAW
• Mathematically,
F12 =
k q1q2
r12
2
• The SI unit of charge is the coulomb (C).
• k is called the Coulomb constant.
➢0 = 8.8 x 10-12 N-1.m-2.C2 is the permittivity of free space
COULOMB’S LAW
• The electric force is attractive if the charges are of opposite
sign.
• The electric force is repulsive if the charges are of like sign.
• The electric force is a conservative force.
• Remember the charges need to be in coulombs.
➢ e is the smallest unit of charge, its value is e = 1.6 x 10-19 C.
➢ So 1 C needs 6.24 x 1018 electrons or protons.
VECTOR NATURE OF ELECTRIC FORCE
• In vector form,
→
F 12 =
k q1q2
r12
•
2
r̂12is a unit vector directed
from q1 to q2
• The like charges produce
a repulsive force between
them
rˆ12
VECTOR NATURE OF ELECTRIC FORCE
• Electrical forces obey Newton’s Third Law.
• The force on q1 is equal in magnitude and
opposite in direction to the force on q2.
F21 = −F12
• With like signs for the charges, the product q1q2 is
positive and the force is repulsive.
VECTOR NATURE OF ELECTRIC FORCE
• Two point charges are separated by a
distance r.
• The unlike charges produce
attractive force between them.
an
• With unlike signs for the charges, the
product q1q2 is negative and the force
is attractive
• The sign of the product of q1q2 gives the relative
direction of the force between q1 and q2.
• The absolute direction is determined by the actual
location of the charges.
THE SUPERPOSITION PRINCIPLE
• The resultant force on any one charge equals the vector
sum of the forces exerted by the other individual charges
that are present.
➢ Remember to add the forces as vectors.
• The resultant force on q1 is the vector sum of all the forces
exerted on it by other charges:
EXAMPLE 1
Find the resultant force acting on the +5μC charge.
→
F ab =
k qa qb
rab
2
rˆ
EXAMPLE 2
→
F ab =
k qa qb
rab
Find the net force on the q1 charge
2
rˆ
Exercise 1
Find the net force on the charge q= -15μC
q= +20 µC q= -15 µC
X= 0 m
X= 2 m
q= +30 µC
X= 3.5 m
THE SUPERPOSITION PRINCIPLE
Consider three point charges located at the corners of a triangle
as in the diagram,
What is the resultant force exerted on q3 ?
• The force exerted by q1 on q3
is 𝐹Ԧ13
• The force exerted by q2 on q3 is
𝐹Ԧ23
• The resultant force exerted on
q3 is the vector sum of F13 and
F23
→
→
→
F 3 = F 13 + F 23
EXAMPLE 3
Consider three point charges located at the
corners of a triangle as in the diagram, where
q1 =q3= 5C, q2= -2 C, and a = 0.1 m. Find
the resultant force exerted on q3.
EXAMPLE 4
Find the resultant force on charge q2 in the diagram if
q1 = 13.0 C, q2 = 4.00 C, q3 = 5.00 C. r13 = 0.500
m, r23 = 0.800 m. Also find the magnitude and the
direction of the resultant force.
The resultant force on charge q2 is
found as
EXAMPLE
Three charges are arranged as shown in the Figure. Find
the force on the charge 𝒒𝟏 = +𝟔. 𝟎 × 𝟏𝟎−𝟔 𝑪, 𝒒𝟐 =
− 𝒒𝟏 = −𝟔. 𝟎 × 𝟏𝟎−𝟔 𝑪,𝒒𝟑 = +𝟑. 𝟎 × 𝟏𝟎−𝟔 𝑪 and 𝒂 =
𝟐. 𝟎 × 𝟏𝟎−𝟐 𝒎
EXAMPLE
In an xy-plane, charge q1 = +1.20 nC is located
+210.0 cm from the origin on the x-axis,
charge q2 = +1.20 nC is located ‒210.0 cm
from the origin on the x-axis. Calculate the
magnitude and direction of the net force these
charges exert on charge q3 = +4.43 C located
at +607.0 cm on the y-axis. Also draw a
diagram.
EXAMPLE 5
• Two identical small charged spheres,
each having a mass of 3.0 ×10-2 kg,
hang in equilibrium as shown. The
length of each string is 0.15 m and the
angle  is 5o. Find the magnitude of
the charge.
• The spheres are in equilibrium
• The spheres are separated and they
exert a repulsive force on each other
since they are like charges.
ELECTRIC FIELD
• We have seen in the previous section that point
charges exert forces on each other even when
they are far apart and not touching each other.
• How do the charges know about existence of
other charges around them?
• A charged body or particle creates an electric field
in the space around it.
• The electric field is the region of space in which an
electric charge will experience a force.
ELECTRIC FIELD
If we consider the interaction between two
positively charged particles (A and B).
r
B
A
• Body B experiences force 𝐹Ԧ0 because of
body A.
• If we remove B and label that as point P,
Then point P will experience electric field
caused by A.
r
A
p
All charged objects create an electric field in the
space that surrounds it. The charge alters that space,
causing any other charged object that enters the
space to be affected by this field.
ELECTRIC FIELD
• The direction of the electric field represents the
direction of the force a positive test charge would
experience if placed in the electric field.
• In other words, the direction of an electric field at a
point in space is the same as direction in which a
positive test charge would move if placed at that
point.
• The strength and the direction of an electric field at
a point can be represented by electric field lines.
ELECTRIC FIELD
❑The field lines radiate
outward in all directions.
• In three dimensions, the
distribution is spherical.
❑The lines are directed away
from the source charge.
• A positive test charge
would be repelled away
from the positive source
charge
ELECTRIC FIELD
• The field lines radiate
inward in all directions.
• The lines are directed
toward the source charge.
• A positive test charge
would be attracted toward
the
negative
source
charge.
ELECTRIC FIELD
• The direction of 𝐸 is that of the force on a positive
test charge.
• The SI units of 𝐸 are N/C.
• We can also say that an electric field exists at a
point if a test charge at that point experiences an
electric force.
Q is the source of the 𝐸
The 𝐸
strength could be
measured by any other charge
placed
somewhere
in
its
surroundings.
Test charge q0– used to
measure the 𝐸 strength.
ELECTRIC FIELD
Remember Coulomb’s law, between the source
and test charges, can be expressed as:
𝑘 𝑄𝑞0
𝐹Ԧ = 2 𝑟Ƹ
𝑟
Electric field strength 𝐸
(1)
𝐹𝑜𝑟𝑐𝑒
𝐹Ԧ
=
,→𝐸 =
𝑇𝑒𝑠𝑡 𝐶ℎ𝑎𝑟𝑔𝑒
𝑞0
From (2) we get that: 𝐹Ԧ = 𝐸𝑞0
(3)
If we substitute (3) into (1) we get that:
𝑘 𝑄𝑞0
𝐸𝑞0 =
𝑟Ƹ
2
𝑟
𝑘 𝑄𝑞0
→𝐸 =
𝑟Ƹ
2
𝑞0 𝑟
𝑘𝑄
Thus: 𝐸 = 2 𝑟Ƹ
𝑟
(4)
(2)
ELECTRIC FIELD
• At any point P, the total electric field due to a group
of source charges equals the vector sum of the
electric fields of all the charges.
qi
E = ke  2 rˆi
i ri
ELECTRIC FIELD
• a) q is positive, the force is
directed away from q.
• b) The direction of the field
is also away from the
positive source charge.
• c) q is negative, the force is
directed toward q.
• d) The field is also toward
the negative source charge.
ELECTRIC FIELD LINES
▪ An electric field line is an imaginary line drawn trough
a region of space so that its tangent at any point is in
the direction of the electric field vector at that point.
ELECTRIC FIELD LINES
• Electric field lines show the direction of E at
each point.
• Their spacing gives an idea of the magnitude
of E at each point.
• If E is strong - the lines are close to each other
& If E is weaker - the lines are farther apart.
• At any particular point, the electric field has a
unique direction.
• Field lines never intersect.
• The field lines are directed away from the
positive charges and toward negative charges.
ELECTRIC FIELD LINES
EXAMPLE 6
𝐸𝑅𝑒𝑠
EXAMPLE 7
Q1 = 6 nC and Q2 = -3 nC. Find the magnitude of the
resultant electric field acting on A from the two charges
and also the angle made with the x-axis and the
resultant electric field.
𝐸𝑟𝑒𝑠
𝐸𝐴1
𝐸𝐴2
EXAMPLE 8
Two electric charges, q1 = + 20.0 nC and q2 = + 10.0
nC, are located on the x -axis at x = 0 m and x = 1.00
m, respectively. What is the magnitude of the electric
field at the point x = 0.50 m, y = 0.50 m ?.
CHARGE IN UNIFORM ELECTRIC FIELD
-
+
Acceleration = constant
Kinematic equations
MOTION OF CHARGED PARTICLE IN
ELECTRIC FIELD
• When a charged particle is placed in an
electric field, it experiences an electrical force.
• If this is the only force on the particle, it must
be the net force.
• The net force will cause the particle to
accelerate according to Newton’s second law
MOTION OF CHARGED PARTICLE IN
ELECTRIC FIELD
• If 𝐸 is uniform, then the acceleration is constant.
• If the particle has a positive charge, its acceleration
is in the direction of the field.
• If the particle has a negative charge, its
acceleration is in the direction opposite the electric
field.
• Since the acceleration is constant, the kinematic
equations can be used.
Kinematic equations
vxf = vxi + axt ,
vxf = vxi + 2ax (x f − xi ),
1 2
x f = xi + vxt + axt ,
2
1
x f = x i + vxi + vxf t
2
2
2
(
)
EXAMPLE 9
(a) What is the acceleration of an electron in a uniform
electric field that has a magnitude of 100 N/C?
EXAMPLE 9
(b) Compute the time it takes for an electron placed at
rest in a uniform electric field that has a magnitude of
100 N/C to reach a speed of 0.01c. (c)How far does the
electron travel in that time?
EXAMPLE 9
A water droplet of mass 5.0 x 10-12 kg is located in the air near
the ground during a stormy day. An atmospheric electric field of
magnitude 3.0 x 103 N/C points vertically downward in the
vicinity of the water droplet. If the droplet remains suspended at
rest in the air. What is the electric charge on the droplet?
ELECTRIC DIPOLES
▪ An electric dipole is a pair of point charges with
equal magnitude and opposite sign (a positive
charge +q and a negative charge -q) separated by
a distance d.
▪ HCl and H2O are example of an electric dipole.
ELECTRIC DIPOLES
+q
d
-q
An electric dipole consists of two equal and opposite
charges (q and -q ) separated a distance d.
ELECTRIC DIPOLES
+q
d
𝒑
-q
We define the Dipole Moment 𝑝Ԧ
magnitude = qd,
𝑝Ԧ
direction = from -q to +q
ELECTRIC DIPOLES
What is the total force acting on the dipole?
Zero, because the force on the two charges cancel: both
have magnitude q 𝐸 . The center of mass does not
accelerate.
But the charges start to rotate.
Why?
There’s a torque because the forces aren’t colinear.
ELECTRIC DIPOLES
The torque is:
t = (magnitude of force) (moment arm)
t = (qE)(d sin Φ)
and the direction of tԦ is (in this case) into the page
but we have defined : 𝒑 = q d and the direction of 𝒑 is
from -q to +q
Then, the torque can be written as:
tԦ= 𝒑x𝑬 (torque on an electric dipole, in vector form)
𝝉 = 𝒑𝑬𝒔𝒊𝒏∅ (magnitude of the torque on an electric
dipole)
EXAMPLE 10
Point charges q1 = - 4.5 nC and q2 = + 4.5 nC are separated
by a distance of 3.1 mm, forming an electric dipole. (a). What
is the net force on the charges? (b). Find the electric dipole
moment (magnitude and direction). The charges are in a
uniform electric field whose direction makes an angle of 36,90
with the line connecting the charges. (c). What is the
magnitude of this field if the torque exerted on the dipole has
magnitude 7.2 × 10−9 Nm ?
Electric Field of a Dipole
An electric dipole is defined as a positive charge q and a negative charge "q separated by a
distance 2a. For the dipole shown in Figure, find the electric field E at P due to the dipole,
where P is a distance y -- a from the origin.
ELECTRIC FIELD – CONTINUOUS CHARGE
DISTRIBUTION
• We have been dealing with electric fields
due to point charges or collection of point
charges.
• However, distributions of charge also
produce fields, and such fields are very
important in practice.
• We will consider charges that are distributed
uniformly throughout a region in space,
whether a line, a surface or a volume.
Consider the calculation of the electric field at point P due to
the charge distribution shown in the figure.
Procedure:
▪ Divide the charge distribution into
small elements, each of which
contains Δq.
▪ Calculate the electric field due to
one of these elements at point P.
▪ Evaluate the total field by
summing the contributions of all
the charge elements.
The electric field at P due to one
charge element carrying charge Δq
is:
q
E = ke 2 rˆ
r
CHARGE DENSITIES
Volume charge density: when a charge is
distributed evenly throughout a volume.
▪ ρ≡q/ V with units C/m3
𝑑𝑞
Where dq= ρdV and 𝐸 = 𝑘 ‫ ׬‬2 𝑟Ƹ
𝑟
Then the volume distribution is specified by

E = k
dV
rˆ
2

r
object
CHARGE DENSITIES
Surface charge density: when a charge is
distributed evenly over a surface area.
▪σ ≡q/ A with units C/m2
𝑑𝑞
Where dq = σdA and 𝐸 = 𝑘 ‫ ׬‬2 𝑟Ƹ
𝑟
Then the Surface distribution is specified by

dA
E = k  2 rˆ
r
object
CHARGE DENSITIES
Linear charge density: when a charge is
distributed evenly along a line.
▪λ≡q/ ℓ with units C/m
𝑑𝑞
Where dq = λdℓ and 𝐸 = 𝑘 ‫ ׬‬2 𝑟Ƹ
𝑟
Then the Linear distribution is specified by

dl
E = k  2 rˆ
r
object
CHARGE DENSITIES
The electric field at P due to one charge element carrying
q
charge Δq is:
E = k
rˆ
e
r2
Because the number of elements is very large and the
charge distribution is modeled as continuous, the total
field at P in the limit Δqi →0 is
qi
dq
E = ke lim  2 rˆi = ke  2 rˆ
qi →0
ri
r
i
• If the charge is nonuniformly distributed over a volume,
surface, or line, the amount of charge, dq, is given by
– For the volume: dq = ρ dV
– For the surface area: dq = σ dA
– For the length element: dq = λ dℓ