+» Light is a form of energy, which is propagated as electromagnetic wave. (The waves of
energy are called electromagnetic (EM) because they have oscillating electric and
magnetic fields.)
+» Light has a dual nature. It behaves as wave and particle both.
* Acc. to wave theory: light consist of electromagnetic waves, which do not require
material medium to travel.
+ Acc. to particle theory: light is composed of particles which travels in a straight line at a
very high speed.
* For e.g: The phenomenon of diffraction, interference and polarization of light can only be
explained if light is considered to be of wave nature. The particle theory of light cannot
explain this phenomenon.
+ The phenomenon of reflection and refraction of light, and casting of shadows of objects
by the light can be explained only if it is thought to be made up of particles. Wave theory
of light cannot explain this phenomenon.
Speed of light was first measured by Roemer.
It is the radiation which make our eyes able to 'see the object.
Its speed is 3 x 108 m/s. It is the form of energy. It is a
transverse wave.
It takes 8 min 19s to reach on the earth from the sun and the
light reflected from moon takes 1.28s to reach earth.
Primary Colours- Blue, Red, Green
Secondary Colours- The coloured produced my mixing any |
A Yellow Magenta
two primary colors. E.g. Yellow, Magenta and Cyan(Peacock Kr
Blue) 4
Complementary Colours- Any two colours when added creén
produce white light. Red+ Cyan= White, Blue+Yellow=White |
Red
Cyan Blue
PRISM: VIBGYOR
Phenomenon of 7 colors through prism is called
Dispersion of Light.
Wavelength of violet colour is minimum and red
colour is maximum.
Deviation of light in prism depends on
wavelength and its refractive index.
Violet colour shows maximum deviation on
passing white light through prism.
Red Colour has longest wavelength and low
frequency and low refractive index.
White Light
Eye forms the real image of the objects on the retina.
Ciliary muscles change the focal length of eye lens. This power of eye is called
Accommodation of eye.
Least distance of distinct vision is 25 cm.
Myopia or short sightedness- far objects cannot see clear. Concave lens are used.
Hyperopia or hypermetropia or Long-sightedness- Near objects cannot see clear.
Convex lens are used.
Presbyopia- in elder person, both far and near cannot see clear. To correct this
defect, a person is prescribed bifocal lens.
Astigmatism- Person cannot focus on horizontal and vertical lines at the same
distance at the same time. Cylinderical lens are used.
Cataract- White membrane is developed on cornea due to which a person loses
power of vision partially or completely. Cured by Surgery.
Light is a form of energy which helps us to see objects.
When light falls on objects, it reflects the light and when the reflected
light reaches our eyes then we see the objects.
Light travels in straight line.
The common phenomenon of light are formation of shadows formation of
images by mirrors and lenses, bending of light by a medium, twinkling of
stars, formation of rainbow etc.
Speed of light in vacuum is 3x108 m/s but different in different media.
Value of speed of light is high in air > water > Glass
=)
* Luminous Objects : Those objects which emit its own light called luminous
objects. E.g. Sun, Stars etc.
* Non-Luminous Objects : Those objects which do not emit its own light but
become visible due to reflection of light falling on them are called non
luminous objects. E.g. Moon, Table.
Reflection of Light
When light falls on a highly polished surface like a mirror, most of the light
is sent back into the same medium. This process is called reflection of
light.
(i) The angle of incidence is equal to the angle of reflection, and
(ii) The incident ray, the normal to the mirror at the point of incidence and
the reflected ray, all lie in the same plane.
These laws of reflection are applicable to all types of reflecting surfaces
including spherical surfaces.
Reflection of Light
| Reflected ray
Incident ray
Incident Ray Reflected Ray
Miri
PLANE MIRROR
Properties of image formed by a plain mirror
* Image formed by a plain mirror is always virtual and erect.
* The size of the image is equal to that of the object.
* The image formed is as far behind the mirror as the object is in front of it.
* The image is laterally inverted.
+ If an object moves towards a plain mirror with speed v, relative to the
object the moves towards it with a speed 2v.
* To see his full image in a plane mirror, a person required a mirror of at
least half of his height.
oan) Properties of image formed by a plain mirror
When two plane mirrors are kept
facing each other at an angle theta
and an object is placed between
then:
Number of images formed by two inctined mirrors
360°
If —¿— = even number.
360°
number of image = 2¿— —1,
(i) If = odd number;
6
number of image = — 1, if the object is placed
on the angle bisector.
360°
(iti) If 2
= odd number;
number of image = ——., if the object is not placed
8
on the angle bisector.
* A spherical mirror, whose reflecting surface is curved inwards, that is,
faces towards the centre of the sphere, is called a concave mirror. A
spherical mirror whose reflecting surface is curved outwards, is called a
convex mirror.
Concave mirror
+ It is a spherical mirror whose reflecting
surface is curved inwards. Rays of light are
parallel to the principal axis after reflection
from a concave mirror, meet at a point
(converge) on the principal axis. F
* Also called converging mirror.
* Image formed by a concave mirror is
generally real and inverted.
Convex mirror
* A spherical mirror whose reflecting
surface is curved outwards. Rays of light
parallel to the principal axis after
reflection from a convex mirror get
diverged and appear to come from a
point behind the mirror.
* Also called diverging mirror.
* Image formed by a convex mirror is
always virtual, erect and diminished.
Mirror’s Formula
1/u+1/v=1/f
Where f= focal length of mirror, u= distance of object from mirror, v=
distance of image from mirror
Concave mirrors are commonly used in torches, search-lights and vehicles
headlights to get powerful parallel beams of light.
They are often used as shaving mirrors to see a larger image of the face.
The dentists use concave mirrors to see large images of the teeth of
patients. Large concave mirrors are used to concentrate sunlight to
produce heat in solar furnaces.
= (i) As a shaving mirror
As a reflector for the head lights of a vehicle, search light
(ii
= (iii) In ophthalmoscope to examine eye, ear, nose by doctors.
= (iv) In solar cookers.
Uses of Convex Mirrors
* Convex mirrors are commonly used as rear-view (wing) mirrors in vehicles.
* These mirrors are fitted on the sides of the vehicle, enabling the driver to
see traffic behind him/her to facilitate safe driving. Convex mirrors are
preferred because they always give an erect, though diminished, image.
* Also, they have a wider field of view as they are curved outwards. Thus,
convex mirrors enable the driver to view much larger area than would be
possible with a plane mirror.
= (i) As a rear-view mirror in vehicle because it provides the maximum
rear field of view and image formed is always erect.
= (ii) In sodium reflector lamp.
The height of the object is taken to be positive as the object is
usually placed above the principal axis.
The height of the image should be taken as positive for virtual
images.
It is to be taken as negative for real images, a negative sign in the
value of the magnification indicates that the image is real.
A positive sign in the value of the magnification indicates that the
image is virtual.
oan) Terminology in study of spherical mirrors
* Center of curvature: It is the centre of the
sphere of which the mirror is a part(C).
* Radius of curvature(R): It is the radius of the
sphere of which the mirror is a part.
* Pole: Is the centre of the spherical mirror. RC
* Principal axis: It is the straight line passing
through the centre of curvature and the pole.
+ Focal length: It is the distance between the
pole and principal focus(f). in a spherical
mirror, the radius of curvature is twice the | C-centre of curvature CP - radius of curvature
focal length. P-pole XY - principal axis
F - principal focus PF - focal length
+ R=2forf=R/2
Image Formation by a Concave and Convex
Mirror for Different Positions Of the Object
C - centre of curvature
P-pole
F - principal focus
CP - radius of curvature
XY - principal axis
PF - focal length
Summary of the Images Formed by a Concave and Convex Mirror
At infinity
Position
of Image
At Focus F
Nature of
Image
Real and inverted
Between F and
Beyond C a Diminished Real and inverted
AtC AtC Same size Real and inverted
Between F and C Beyond € Enlarged Real and inverted
AtF At infinity Highly enlarged Real and inverted
Between PandF Behind the Enlarged Virtual and erect
mirror
Convex Mirror
At infinity Atfocus | Highly diminished | Virtual point size
Anywhere on | Between pole | biminished Virtual and erect
Principal axis & focus
When light travels obliquely from one
transparent medium into another it
gets bent. This bending of light is called
refraction of light.
When light travels from a rarer medium
to a denser medium, it bends towards
the normal.
When light travels from a denser
medium to a rarer medium, it bends
away from the normal.
The cause of refraction is that the speed
of light is different in different media.
Parallel-sided:
glass slab
€
incident
ray
A
rer medium)
Normal
N
Glass
(Denser medium)
Refracted
=
A
Incident
ray
Glass i
(Denser medium)
Air =
(Rarer medium) = Refracied
m
Mermal
ray
REFRACTION OF LIGHT
In Refraction, speed of light and its wavelength change but its frequency
remains unchanged, when light travels from one transparent medium to
another.
Examples :
= when a thick glass slab is placed over some printed matter, the letters
appear raised when viewed through the glass slab the bottom of a
tank or a pond containing water appears to be raised seen a pencil
partly immersed in water in a glass tumbler. It appears to be displaced
at the interface of air and water
= A lemon kept in water in a glass tumbler appears to be bigger than its
actual size, when viewed from the sides.
Laws of Refraction
1. The incident ray, the refracted ray and the normal at the point of
incidence all three lie in the same plane.
2. The ratio of sine of angle of incidence to the sine of angle of refraction
remains constant for a pair of media i.e. sin i/sin r = constant
This is called snell’s law and also called refractive index of second medium
with respect of first medium.
Refractive index = The ratio of speed of light in vacuum ( c ) to the speed
of light in any medium (v)
A coin appears at lesser depth in water.
ATMOSPHERIC REFRACTION
The air just above the fire becomes hotter than the air further up. The
hotter air is lighter (less dense) than the cooler air above it, and has a
refractive index slightly less than that of the cooler air.
Since the physical conditions of the refracting medium (air) are not
stationary, the apparent position of the object, as seen through the hot
air, fluctuate.
This wavering is thus an effect of atmospheric refraction (refraction of
light by the earth’s atmosphere).
REFRACTIVE INDEX
The absolute refractive index of a medium is the ratio of the speed
light in air or vacuum to the speed of light in medium.
Refractive index = speed of light in air or vacuum/speed of light in
the medium
Speed of light in air c
m
~ Speed of light in the medium u
The twinkling of a star is due to atmospheric refraction of starlight.
The starlight, on entering the earth’s atmosphere, undergoes refraction
continuously before it reaches the earth. The atmospheric refraction
occurs in a medium of gradually changing refractive index.
Since the atmosphere bends starlight towards the normal, the apparent
position of the star is slightly different from its actual position.
As the path of rays of light coming from the star goes on varying slightly,
the starlight entering the eye flickers — the star sometimes appears
brighter, and at some other time, fainter, which is the twinkling effect.
Advance sunrise and delayed sunset
+ The Sun is visible to us about 2 minutes before the actual sunrise, and
about 2 minutes after the actual sunset because of atmospheric
refraction.
* The time difference between actual sunset and the apparent sunset is
about 2 minutes.
Examples of Refraction
* Bending of a linear object when it is partially dipped in a liquid inclined to
the surface of the liquid.
* Oval shape of the sun in the morning and evening.
* A fish in a pond when viewed from air appears to be at a smaller depth
them actual depth.
* Acoin at the base of a vessel filled with water appears raised.
oan) Dispersion of white light by a glass prism
* The prism has probably split the incident white light into a band of
colours. The sequence of colours VIBGYOR.
* The splitting of light into its component colours is called dispersion.
* Different colours of light bend through different angles with respect to the
incident ray, as they pass through a prism.
* The red light bends the least while the violet the most.
* Thus the rays of each colour emerge along different paths and thus
become distinct. It is the band of distinct colours that we see in a
spectrum.
The dispersion of light is due to different velocities of light of different
colours in a medium. As a result, the refractive index of a medium is
different for different colours of light.
The velocity of light in a medium is maximum for that colour for which
refractive index is minimum. Clearly, the velocity of violet colour or light is
minimum in a medium and refractive index of that medium in a maximum
for violet colour.
Similarly, the velocity of light in a medium is maximum for red colour and
refractive index of that medium in a minimum for red colour.
Rainbow
A rainbow is a natural spectrum appearing in the sky after a rain shower. It
is caused by dispersion of sunlight by tiny water droplets, present in the
atmosphere.
A rainbow is always formed in a direction opposite to that of the Sun.
The water droplets act like small prisms. They refract and disperse the
incident sunlight, then reflect it internally, and finally refract it again when
it comes out of the raindrop.
Due to the dispersion of light and internal reflection, different colours
reach the observer's eye.
Theory of Colours
Colour is the sensation perceived by the rods in the eye due to light.
Primary Colours : The spectral colours blue, red and green are called
primary colours because all the colours can be produced by mixing these
in proper proportion.
Blue + Red + Green = White
Secondary Colours : The colour produced by mixing any two primary
colours is called secondary colour. There are 3 secondary colours yellow,
magenta and cyan as
Green + Red = Yellow Red + Blue = Magenta Blue + Green = Cyan
Complementary Colours
The complementary primary-secondary
combinations are red-cyan, green-magenta,
and blue-yellow. In the RGB color model,
the light of two complementary colors, such
as red and cyan, combined at full intensity,
will make white light, since
two complementary colors contain light
with the full range of the spectrum.
In coloured television, the three primary
colours are used.
Red + Cyan = White ; Blue + Yellow = White
etc.
Red
Yellow Y Magenta
a!
White
/ |
Green Cyan ——— Blue
Coloured Television
Light coming from the scene to be televised is split up in red, blue and
green primary colour components with the help of dichromatic mirrors
fitted in TV camera.
Now, light of different colours are incident on three separate camera tubes
which converts them into electrical signals.
These signals are transmitted to distant places. At the receiving end, these
signals are received and the picture tube of a TV set produces the same
scene on the screen.
Mixed coloured Pigments
The pigments in common use are of impure colours. Therefore, on mixing
paints of different colours the resultant colour of the paint is not obtained
as given by colour triangle.
When blue and yellow mixed together, they produce green inspite of
white paint.
Colour of Bodies
The colour of a body is the colour of light which it reflects or transmits. An
object is white, if it reflects all the components of white light and it is
black if it absorbs all the light incident over it.
This is why red rose appears red when viewed in a white or red light but
appears black when viewed in blue or green light.
SCATTERING OF LIGHT
When light passes through a medium in which |
particles are suspended whose sizes are of the
order of wavelength of light, then light on striking pate
these particles, deviated in different directions. NS taj sate (anewadengh,
This is called scattering of light. amma
Seater ght NS fama sate (tent
+ According to lord Rayleigh, the intensity of wavelength es abundant)
scattered light is fourth power of wavelength.
Therefore, red colour of light is scattered least and Y,
violet colour of light is scattered most. =
+ The blue colour of the sky, colour of water in deep Sample
sea, the reddening of the sun at sunrise and the |
sunset. |
Why is the colour of the clear Sky Blue?
* The red light has a wavelength about 1.8 times greater than blue light.
Thus, when sunlight passes through the atmosphere, the fine particles in
air scatter the blue colour (shorter wavelengths) more strongly than red.
The scattered blue light enters our eyes.
+ If the earth had no atmosphere, there would not have been any
scattering.
* Then, the sky would have looked dark. The sky appears dark to passengers
flying at very high altitudes, as scattering is not prominent at such heights.
Danger signals are of red colours because red colours light scatters least
and therefore these signals can be seen from far away.
Sun appears reddish at the time of sunrise or sunset, because the sun is
near horizon and sunlight travels maximum distance in earth’s
atmosphere.
Clouds appear white because clouds are consists of large size of droplets
of water and dust particles which scatter all colours of light equally. The
mixture of all seven colours of sunlight is white.
Sky appears black to astronauts from space because there is no
atmosphere in space and therefore, no scattering of light take place.
Critical angle
* Critical angle : The angle of incidence in a denser
medium for which the angle of refraction in rarer
medium becomes 90 degree, is called Critical
Angle (C).
* Refractive index of denser medium =
1/sin Critical Angle
Less dense
medium
Total Internal Reflection
+ When a light ray travelling from a denser medium towards a rarer medium
is incident at an angle of incidence greater than critical angle, then light
rays reflected back into denser medium. This phenomenon is called TIR.
N
Total internal
reflection
1
'
1
1
i
1
Critical angle
1
1
i
1
For total internal reflection,
A) Light must be propagating from denser to rarer medium.
B) Angle of incidence must exceeds the critical angle.
Examples :
a) mirage — Desert e.g. Hotter air is less dense, and has smaller refractive
index than the cooler air. On hot summer days, the air near the ground
becomes hotter than the air at higher levels noticed that while moving in a
bus or a car during a hot summer day, a distant patch of road, especially
ona highway, appears to be wet. This is also due to mirage.
b) Diamonds - Their brilliance is mainly due to the total internal reflection
of light inside them.
c) Optical fibers too make use of the phenomenon of total internal
reflection. Light undergoes repeated total internal reflections along the
length of the fiber there is no appreciable loss in the intensity of the light
signal.
Optical Fibers are used in Endoscopy, in decorative table lamps, in
telecommunications and networking.
Sparkling of diamond, mirage and looming, shinning of air bubble in water
and optical Fiber are examples of total internal reflection.
Shining of a smoked ball or a metal ball on which lamp soot is deposited
when dipped in a water.
Interference of light
* Interference is the superposition of two or more waves resulting in a new
wave pattern.
* It refers to the interference of waves which are correlated or coherent
with each other, either because they come from the same source or
because they have the same or nearly the same frequency.
* Two non-monochromatic waves are only fully coherent with each other if
they both have exactly the same range of wavelengths and the same
phase differences at each of the constituent wavelengths.
* The modification in the intensity of light in the region of superposition is
called interference of light.
S
STUDY )
I
Constructive Interference: If the crest of one
wave falls on the crest of another wave, then
the amplitude and resultant intensity is
maximum. This is constructive interference.
Here both the waves have the same
displacement and the waves are in phase.
Destructive interference: If the crest of one
wave falls on the trough of another wave,
then the amplitude and resultant intensity is
minimum. This is destructive interference.
Here the waves do not have the same
displacement and the waves are out of phase.
(ns enc
‘Destructive Interference
Monochromatic (dah)
Light Source
Constructive netas
Gt)
Destructive interference
Two Slits a
Constructive Enterfarence
‘A two-point soure interference pattern creates an alternating pattern
fright and dark ines when itis projected onto screen,
It is the bending of light around the corner of an
obstacle. Reflected light produces fridges of light,
dark or colored bands.
The spreading of light energy beyond the limit
prescribed by rectilinear propagation of light is
called diffraction of light.
At times diffraction of sunlight in clouds
produces a multitude of colors.
Example of diffraction in nature is diamond rays
in the solar eclipse.
A soap bubble of oil film on water appears coloured in a white light due to
interference of light reflected from upper and lower surfaces of soap
bubble or oil film.
In interference patterns of all bright and dark fringes are of same width
but in the diffraction the central fringe is brightest an widest, and
remaining are of gradually decreasing intensities.
The difference between interference and diffraction is that the
interference is superposition of light coming from two different sources
and in the diffraction it is the superposition between the waves coming
from single wavefront.
Polarisation
* It is the only phenomenon which proves that light is a transverse wave.
* Natural sunlight and most forms of artificial illumination transmit light
waves whose electric field vectors vibrate in all perpendicular planes with
respect to the direction of propagation.
* The visible effect of light is only due to electric field vector.
* Polarisation can take place only in the transverse waves.
* Polarised are used for the polarisation of light.
Application of Polaroid :
Used in sun glasses and protect the eye from the glare.
The wind shield of automobile is made up of polaroid. They protect the
eyes of the driver of automobile from dazzling light of vehicle.
Used in window panes of train and especially of a aeroplane.
The picture taken by stereoscopic camera, when seen with the help of
polaroid spectacles, create three dimensional effect.
Tyndall Effect
Tyndall effect is the scattering of light as a light beam passes through a
colloid. The individual suspension particles scatter and reflect light,
making the beam visible. The amount of scattering depends on the
frequency of the light and density of the particles.
The earth’s atmosphere is a heterogeneous mixture of minute particles
like smoke, tiny water droplets, suspended particles of dust and molecules
of air. When a beam of light strikes such fine particles, the path of the
beam becomes visible.
When a fine beam of sunlight enters a smoke-filled room through a small
hole. Tyndall effect can also be observed when sunlight passes through a
canopy of a dense forest.
The color of the scattered light depends on the size of the scattering
particles. Very fine particles scatter mainly blue light while particles of
larger size scatter light of longer wavelengths.
Image Formation by Lenses
* Lens is a transparent medium which is formed by joining two pieces of
spherical glass. There are two types of lenses.
= (i) Convex Lens or Convergent Lens : It is a lens which is thicker at the
centre and thinner at the edges.
= (ii) Concave Lens or Divergent Lens : It is a lens which is thinner at the
centre and thicker at the edges.
Optical Centre of Lens : It is the
centre of the lens through which
light can pass without any deviation
Principal Axis : It is the line passing
through optical centre and is
perpendicular to the line joining its
edges.
Principal Focus : It is a point on the
principal axis where all light rays
which are parallel to principal axis
either converge or appear to
diverge from, after refraction.
r
Image by Convex lens Mi.
Object Position of image Nature of image
At infinity At focus Real, diminished, inverted
At 2F At 2F Real, inverted and of same size
Between 2F and F Beyond 2F Real, enlarged, inverted
AtF At infinity Real, enlarged, inverted
Between optical centre On the same side as the object Virtual, enlarged and upright
and F
7 E SS
Uses of Convex Lens
= Used in magnifying
glass.
=" Used to correct the
vision of long-sighted
persons.
su) Images by Concave lens
Object Position of image Nature of image
At infinity At focus Virtual, diminished, upright
Between infinity and F
Between the object and the Virtual, diminished, upright
lens
+ Uses of Concave Lens
= Used in spy holes on
doors to check the
visitors.
=" Used to correct the
vision of short-sighted
persons
oan) Two Types of Images
Fi #
object lens
Ae :
object mirro virtual image
Image: a reproduction derived from light
Real Image: light rays actually pass through image, really exists in space (or on
a screen for example) whether you are looking or not
Virtual Image: no light rays actually pass through image. Only appear to be
coming from image. Image only exists when rays are traced back to perceived
location of source ae
Real Image is an image which can be
projected onto a screen.
Real images are always located in front
of the mirror and behind the lens.
Real images are always inverted and
never upright.
Concave lenses, Convex mirrors and
plane mirrors always form virtual
images and can never form real
images.
Virtual images
Virtual Image is an image which cannot be projected onto a screen.
Virtual images are always located in behind the mirror and in front
of the lens.
Virtual images are always upright and never inverted.
=) Lens formula for spherical lenses
+ The lens formula for spherical lenses is the relationship between
the object distance(u), image distance(v) and focal length (f).
L 4
i
vou =“f
Magnification by lens
Magnification for spherical lens is the ratio of the height of the
image to the height of the object.
Magnification = Height of the image/ Height of the object
Height of the Image h’
Height of the object h
m=
The magnification is also related to the object distance and image
distance. It can be expressed as:
Magnification (m) = H/h=v/u
+ The SI unit of power of a lens whose focal length is 1 metre is one
dioptre. The power of a convex lens is positive (+ve) and the power
of a concave lens is negative (-ve).
Camera ( f-number)
The f-number represents the size of aperture,
f-number = Focal length of the lens/ Diameter of the lens (D)
Generally 2,2.8,4,5.6 ,8,11,22,32 are f-numbers.
The amount of Light (L) entering the camera is directly proportional to the
area (A) of the aperture
ie. LX À D2
Brightness of image « D2/F2
Exposure time is the time for which light is incident on the photographic
film.
This is simply a convex lens of small focal
length. The object to be enlarged is
placed within the focus of lens.
Magnifying power = M= 1 + D/f
Where D= 25cm, f = focal length of lens
When final image is formed at infinity,
then M = D/f
Objective lens is a convex lens of large
aperture and large focal length while eye
piece is a convex lens of small aperture
and small focal length.
Compound Microscope
* It consists of two convex lenses coaxially
fitted in a hollow tube. The lens facing the Eye piece
object is called objective and the lens
towards the eye is called eye piece. eng cu Body tube
+ The aperture of objective is smaller than | fine adjustment
that of eye piece. LE z Revco,
* Both the lenses are of smaller focal Objective
lengths. This increases the magnifying o Condenser
power of instrument. Inden EA Minor
Base
Telescope
They are used to view distant objects which
are not visible to naked eye.
Astronomical Telescope consists of two convex
lenses placed coaxially in a hollow tube. The
lens facing the object is called objective lens
and lens towards the eye is called eye piece.
The objective has large aperture so that the
rays from the object can be easily collected.
The focal length of objective is larger than that
of eye piece.
i ré <— vipangoyal13
o
Vipan Goyal Dr Vipan Goyal
GENERAL STUDIES TEACHER, Study IQ GS faculty @ study Iq
ducation Telegram link : t.me/DrVipanGoyal