Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
1
Electric charge
“Static electricity's tiny sparks”
Electric charge is a fundamental conserved property of some subatomic particles, which
determines their electromagnetic interaction. Electric charge is quantized when expressed as a
multiple of the elementary charge e. Electrons have a charge of -1, while protons have the
opposite charge of +1.
Coulomb's law
Coulomb's law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may
be stated in scalar form as follows:
The magnitude of the electrostatic force between two point electric charges is directly
proportional to the product of the magnitudes of each charge and inversely proportional to the
square of the distance between the charges.
where is the separation of the charges and is the electric constant. A positive force implies a
repulsive interaction, while a negative force implies an attractive interaction.The prefactor,
termed the electrostatic constant, or Coulomb's constant ( ), is:
Nm2C−2 (also mF−1).
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
2
Electric field
In physics, the space surrounding an electric charge or in the presence of a time-varying
magnetic field there is a property called an electric field (electric flux density).
The electric field is a vector field with SI units of newtons per coulomb (N C−1) or, equivalently,
volts per meter (Vm−1). The strength of the field at a given point is defined as the force that
would be exerted on a positive test charge of +1 coulomb placed at that point; the direction of
the field is given by the direction of that force.
where, is the electric force experienced by the particle, q is its charge,
wherein the particle is located.
is the electric field
Electric Permittivity
To understand how electric permittivity works, let's start by looking at a capacitor. A basic
capacitor has two charged plates that are separated by some substance. One of the plates gathers
positive charge on it, and the other gathers negative charge. This creates an electric field through
the substance between them.
Electric permittivity is what happens to the substance between the capacitor plates when it is
placed in an electric field. That substance is made out of atoms, which bind together to form
molecules. When these atoms form into molecules, they often form dipole moments. This means
that positive charge is at one end of the molecule and negative charge at the other.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
3
These molecules normally align randomly with each other in a substance, but when an external
electric field is introduced, they align themselves in such a way that the electric field of the
dipole moments resists the external electric field. In other words, the electric field created by the
dipole moments acts in the opposite direction of the external electric field. The better the
molecules align, the more they resist the external electric field.
Electric permittivity is a measure of how well the molecules of substance align, a.k.a.
polarization under an electric field. The higher the electric permittivity the more the molecules
polarize and that the substance resists the external electric field.
Permittivity is a fundamental material property that describes how a material will affect, and be
affected by, a time-varying electromagnetic field. Specifically, the real and imaginary parameters
defined within the complex permittivity equation describe how a material will store
electromagnetic energy and dissipate that energy as heat. A material with high permittivity
polarizes more in response to an applied electric field than a material with low permittivity,
thereby storing more energy in the electric field. In electrostatics, the permittivity plays an
important role in determining the capacitance of a capacitor.



ε′ is the real part of the permittivity;
ε″ is the imaginary part of the permittivity;
δ is the loss angle.
The processes that influence the response of a material to a time-varying electromagnetic field
are frequency dependent and are generally classified as either ionic, dipolar, vibrational, or
electronic in nature. These processes are highlighted as a function of frequency in Figure. Ionic
processes refer to the general case of a charged ion moving back and forth in response a timevarying electric field, whilst dipolar processes correspond to the ‘flipping’ and ‘twisting’ of
molecules, which have a permanent electric dipole moment such as that seen with a water
molecule in a microwave oven. Examples of vibrational processes include molecular vibrations
(e.g. symmetric and asymmetric) and associated vibrational-rotation states that are Infrared (IR)
active. Electronic processes include optical and ultra-violet (UV) absorption and scattering
phenomenon seen across the UV-visible range.
The most common relationship scientists that have with permittivity is through the concept of
relative permittivity: the permittivity of a material relative to vacuum permittivity. Also known
as the dielectric constant, the relative permittivity (εr) is given by,
εr = ε / ε0
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
4
where ε is the permittivity of the substance and ε0 is the permittivity of a vacuum
(ε0 = 8.85 x 10-12 Farads/m).
Magnetic Permeability
The best thing magnetic permeability can be compared to is conductivity. Conductivity allows
electricity to pass through them. For example, copper is a much better conductor than rubber.
Magnetic permeability is a similar concept to this, but here it is magnetic flux instead of
electricity. The higher the magnetic permeability, the better the material allows for magnetic flux
to pass through it.
The permeability of free space (μ0) we talked about earlier is actually just the magnetic
permeability associated with a vacuum (i.e., the permeability of a magnetic field in the absence
of any material). This is an important quantity in physics, and it shows up in many relations. As
you can see here, examples of this include the speed of light in a vacuum, Ampere's law, and
magnetic inductance.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
5
Magnetic permeability of the vacuum is denoted by; µo and has value;
µo = 4π.10-7 Wb./Amps.m
We find the permeability of the matter by following formula;
µ=B/H
where; H is the magnetic field strength and B is the flux density. Relative permeability is the
ratio of a specific medium permeability to the permeability of vacuum.
µr = µ/µo
Diamagnetic matters: If the relative permeability of the matter is a little bit lower than 1 then
we say these matters are diamagnetic.
Paramagnetic matters: If the relative permeability of the matter is a little bit higher than 1 then
we say these matters are paramagnetic.
Ferromagnetic matters: If the relative permeability of the matter is higher than 1 with respect
to paramagnetic matters then we say these matters are ferromagnetic matters.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
6
Electric potential
The energy required to bring unit electric charge from infinity to the point in an electric field
at which the potential is being specified. It is typically measured in volts. Electric potential may
be conceived of as "electric pressure". The electrical potential difference is defined as the amount
of work done in carrying a unit charge from one point to another in an electric field.
Since the work done is measured in joules and charge in coulombs, the unit of electric potential
is joules /coulombs, or volts.
Electric potential at a point:
Consider a positive point charge of Q coulombs placed in air. At a point x metres from it, the
Q
force on one coulomb positive charge is
. Suppose this one coulomb charge is moved
4 0 x 2
towards Q through a small distance dx. Then work done is
dW 
Q
4 0 x 2
  dx 
The negative sign is taken because dx is considered along the negative direction of x. The total
work done in bringing this coulomb of positive charge from infinity to any point D (r metres
from Q) is given by
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
x r
W 

x 
Q
Q.
dx
4 0 x
2

Q
4 0
r
7
dx
x
2

r
1
1 Q
joules



4 0 x  4 0 r
Concept of flux
Concept of solid angle:
The steradian (symbol: sr) or square radian is the unit of solid angle in the International System
of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which
quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length on
the circumference, a solid angle in steradians, projected onto a sphere, gives an area on the
surface.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
8
Gauss' law
Consider a point charge q enclosed by a surface S. Consider a general surface element dS which
lies a distance r from q. the E-field at dS is
and the flux through dS is hence given by outward flux across dS
However the term (dScos)/r2 is simply the solid angle d subtended by dS at q. Hence
Outward flux across dS
And total flux across closed surface S is
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
9
Hence final, result which applies to a collection of charges Q within S, is
(A)
where the RHS represents the algebraic sum of all charges enclosed by the surface S divided by
0.
Equation (A) gives the integral form of Gauss's law for E-fields. In words it states:
'the outward flux of E over any closed surface is equal to the algebraic sum of the charges
enclosed by the surface divided by 0.
Differential form of Gauss's Law
The integral form is
We now consider the general case where the charge contained within the surface S is
continuously distributed with a volume density  (which may be depend upon spatial position).
In this case the total charge within S is given by a suitable volume integral of .
where  is the volume enclosed by the surface S.
Hence Gauss's law is now
We now apply Gauss's divergence theorem to convert the surface integral on the LHS to a
volume integral.
Gauss's theorem states that 'if a closed surface S encloses a volume  then the surface integral of
any vector A over S is equal to the volume integral of the divergence of A over .'
Mathematically Gauss's theorem can be written as
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
10
where A is the divergence of the vector A.
Applying the divergence theorem to Gauss's law we have
The two volume integrals can now be combined as they are both evaluated over the same volume
As this is true for any closed surface the result requires that at every point in space
This is Gauss's law for electric fields in differential form. It is the first of Maxwell's equations.
In words Gauss’s law states that 'at any point in space the divergence of the E-field is equal to
the charge density at that point divided by 0.
As the divergence can be thought of as giving the number of field lines starting (if positive) or
terminating (if negative) at a given point, the above equation states that ‘the number of field lines
starting or terminating at a given point is proportional to the charge density at that point’.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
11
Gauss’s law and Coulomb’s law
Let us consider a spherical Gaussian surface of radius r centered on point charge q. Both E and
ds at any point on the Gaussian surface are directed radially outward. The angle between them is
zero and the quantity E.ds simply becomes E ds . Now Gauss’s law can be written as
q
 E.ds   E ds  
0
Since E is constant for all points on the sphere
 0 E  ds  q
 0 E  4 r 2   q
E
1
q
4 0 r 2
Let us now put a second point charge q0 at the point at which E is calculated. The magnitude of
the force that acts on it is
F  Eq0
so that
F
1 qq0
4 0 r 2
Which is precisely Coulomb’s law.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
12
Application of Gauss's Law to find the E-field
Electric Field due to Infinite Wire
Consider an infinitely long wire with linear charge density λ and length l. To calculate electric
field, we assume a cylindrical Gaussian surface due to the symmetry of wire. As the electric field
E is radial in direction; flux through the end of the cylindrical surface will be zero, as electric
field and area vector are perpendicular to each other. The only flowing electric flux will be
through the curved Gaussian surface. As the electric field is perpendicular to every point of the
curved surface, its magnitude will be constant.
Fig: a cylindrical Gaussian surface of radius r and length l
The surface area of the curved cylindrical surface will be 2πrl. The electric flux through the
curve will be
E × 2πrl
And according to Gauss’s Law
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
13
Vectorially, the above relation is
where
is radial unit vector pointing the direction of electric field.
Electric Field due to thin spherical shell
Consider a thin spherical shell of surface charge density σ and radius “R”. By observation, it’s
obvious that shell has spherical symmetry. The electric field due to the spherical shell can be
evaluated in two different positions:
1.
Electric Field Outside the Spherical Shell
To find electric field outside the spherical shell, we take a point P outside the shell at a distance r
from the center of the spherical shell. By symmetry, we take Gaussian spherical surface with
radius r and center O. The Gaussian surface will pass through P, and experience a constant
electric field all around as all points is equally distanced “r’’ from the center of the sphere. Then,
according to Gauss’s Law
The enclosed charge inside the Gaussian surface P will be σ × 4πR2. The total electric flux
through the Gaussian surface will be
Φ = E × 4 πr2
Then by Gauss’s Law, we can write
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
14
Putting the value of surface charge density σ as q/4 πR2, we can rewrite the electric field as
In vector form, electric field is
where
is radius vector, depicting the direction of electric field.
Note: If the surface charge density σ is negative, the direction of the electric field will be radially
inward.
2. Electric Field Inside the Spherical Shell
To evaluate electric field inside the spherical shell, let’s take a point P inside the spherical shell.
By symmetry, we again take a spherical Gaussian surface passing through P, centered at O and
with radius r. Now according to Gauss’s Law
The net electric flux will be E × 4 π r2. But the enclosed charge q will be zero, as we know that
surface charge density is dispersed outside the surface, therefore there is no charge inside the
spherical shell. Then by Gauss’s Law
Note: There is no electric field inside spherical shell because of absence of enclosed charge.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
15
Capacitance
The property of a capacitor to store electricity is called capacitance. A capacitor consists of two
conducting surfaces separated by a layer of an insulating medium called dielectric. The purpose
of a capacitor is to store electrical energy by electrostatic stress in the dielectric.
The capacitance of a capacitor is defined as “the amount of charge required to create a unit
potential difference between its plates”. Suppose Q coulomb of charge is given to one of the two
plates of the capacitor and a potential difference of V volts is established between the two, the its
capacitance is
C
Q
V
The unit of capacitance is coulomb/volt which is also called farad. One farad is defined as the
capacitance of a capacitor which requires a charge of one coulomb to establish a potential
difference of one volt between its plates.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
16
Capacitance of a parallel plate capacitor
Fig shows a parallel plate capacitor formed by two parallel conducting plates of area A separated
by a distance d. If each plate is connected to the terminal of a battery, a charge of +q will appear
on one plate and a charge of –q on the other. If d is small compared with the plate dimensions,
the electric field strength E between the plates will be uniform. The dashed line in the fig. shows
a Gaussian surface of height h. From Gauss’s law
q
E  EA 
0
q   0 EA   0E
The work required to carry a test charge q0 from one plate to the other can be expressed as q0 V
or as the product of a force q0 E times a distance d or q0 Ed. Therefore
V  Ed
Substituting the value of q and V into the equation C 
C
 0 EA
Ed

q
V
0 A
d
we get .
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
17
Cylindrical capacitor
A cylindrical capacitor consists of two coaxial cylinders of radius a and b and length l. As the
Gaussian surface construct a coaxial cylinder of radius r and length l, closed by plane caps.
Gauss’s law
q
 E.ds  
Gives
0
 0 E  2 r l   q
The flux being entirely through the cylindrical surface and not through the end caps. Solving for
q
E yields
E
2 0 rl
The potential difference between the plates [E and dl (=dr) point in the opposite directions]
b
b
a
a
b
dr
2 0l r
a
V   E. dl   E. dr  
q
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific

The capacitance is given by
C
q
2 0l
ln
18
b
a
q 2 0l

V
b
ln  
a
Capacitors connected in series
For capacitors connected in series, the magnitude q of the charge on each plate must be the same.
Now applying the relation q = CV to each capacitor yields
V1 
q
;
C1
V2 
q
q
; and V3 
C2
C3
The potential difference for the series combination is
V  V1  V2  V3
 1
1
1 
 q 
 
 C1 C2 C3 
The equivalent capacitance
C
q
1

1
1
1
V


C1 C2 C3
1 1
1
1
 

C C1 C2 C3
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
19
The equivalent series capacitance is always less than the smallest capacitance in the chain.
Capacitors connected in parallel
The potential difference across each capacitor will be the same. Now applying the relation
CV to each capacitor yields
q1  C1V ;
q2  C2V ;
q=
and q3  C3V
The total charge q on the combination is
q  q1  q2  q3
=  C1  C2  C3 V
The equivalent capacitance C is C 
q
 C1  C2  C3
V
Energy storage in the electric field:
Suppose that at a time t a charge q'(t) has been transferred from one plate to the other. The
potential difference V(t) between the plates at that moment will be q'(t)/C. If an extra increment
of charge dq' is transferred, the small amount of additional work needed will be
If this process is continued until a total charge q has been transferred, the total work will be
found from
From the relation q = CV, we get
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
20
In a parallel plate capacitor, the energy density u, which is the stored energy per unit volume, is
given by
But in a parallel plate capacitor,
Therefore
Or
Electrolytic Capacitors
Electrolytic capacitors are those in which electrolyte serves as a dielectric substance. It is made
up of aluminium and tantalum. The reason of using these materials is that these materials form
oxides which possess extremely high dielectric strength. Thus, two aluminium foils are taken, in
one of the foil the layer of oxide is formed by process of “Forming”.
In this process of “Forming,” the oxide layer is grown on it by applying voltage on the foil. The
other foil provides negative connection to the capacitor. The electrolyte is soaked in a piece of
paper. This serves as dielectric to the capacitor.
Ceramic Capacitors
The Ceramic Capacitors are made up of disc or plate which is coated with a metal such as silver
or copper, on both the sides of plate or disc. The leads are made up of tin. The entire capacitor
structure is packed into a plastic casing to prevent it from external environmental conditions.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
21
Mica Capacitors
The Mica capacitor is formed by sandwiching the layer of Mica between the layers of Metal. The
complete structure is then enclosed a plastic package. These types of capacitors possess very
small leakage current because the leakage resistance is very high in case of Mica Capacitors. The
range in which mica capacitors are available commercially varies from 1 pF to 0.1 pF.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
22
Ampère's law
Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated magnetic
field around a closed loop to the electric current passing through the loop.
Gauss' Law for Magnetism states that the net magnetic flux through a closed surface is zero as
there are no such things as monopoles. Since each magnet has both a north and a south pole, any
magnet enclosed within a closed surface would have the same number of flux lines both exiting
and entering the surface - the net flux would be zero.
Ampere's Law says that if we replace the closed surface integral with a closed line integral then
the magnetic field multiplied by the length of the curve will equal the sum of the enclosed
currents times the permeability of free space, µo = 4π  10-7 N/A2.
Remember that a dot product (
) reflects that we must look for the component
of B along the path; that is, we are interested in choosing an Amperian path that flows in parallel
to the magnetic field B so that θ equals either 0º or 180º.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
23
Magnetic fields outside a current-carrying wire
Now let's use Ampere's Law to determine the strength of the magnetic field at a distance r from
a wire carrying a current I.
Now we need to investigate the strength of the magnetic field inside of this wire.
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
24
So, inside the wire the magnetic field is proportional to r, while outside it's proportional to 1/r.
Magnetic fields within a solenoid
Consider a solenoid of length L having N turns. The diameter of the solenoid is assumed to be
much smaller when compared to its length and the coil is wound very closely.
In order to calculate the magnetic field at any point inside the solenoid, we use Ampere’s
circuital law. Consider a rectangular loop abcd as shown in Figure. Then from Ampère’s
circuital law,
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
25
The left hand side of the equation is
Since the elemental lengths along bc and da are perpendicular to the magnetic field which is
along the axis of the solenoid, the integrals
Since the magnetic field outside the solenoid is zero, the integral
along ab, the integral is
for the path
where the length of the loop ab as shown in the Figure is h. But the choice of length of the loop
ab is arbitrary. We can take very large loop such that it is equal to the length of the solenoid L.
Therefore, the integral is
Let I be the current passing through the solenoid of N turns, then
The number of turns per unit length is given by N/L = n, Then
Md. Anisur Rahman
Assistant Professor of Physics
University of Asia Pacific
26
Since n is a constant for a given solenoid and μ0 is also constant. For a fixed current I, the
magnetic field inside the solenoid is also a constant.
Magnetic field of a toroidal solenoid