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Condition for Total internal reflection are

(i) Incident ray is in denser medium.

(ii) Angle of incident should be larger than critical angle

\(n_{12}=\frac{1}{sin\,C}\)

where, C is the critical angle

(i) \(m=\frac{f_o}{f_e}\)

By increasing f_o / decreasing f_e

(ii) Any two

(a) No chromatic aberration

(b) No spherical aberration

(c) Mechanical advantage – low weight, easier to support

(d) Mirrors are easy to prepare

(e) More economical

The geometric centre of a spherical mirror is called its pole while that of a spherical lens is called its optical centre. The line joining the pole and the centre of curvature of the spherical mirror is known as the principal axis.

It works on the principle of total internal reflection.  

P = P₁ + P₂ + P₃ + ……………  

Mirage, sparkling of diamond, total internal reflecting prisms, optical fibers. 

(a) A ray of light should travel from denser to rarer medium. 

(b) Angle of incidence must be greater than the critical angle. 

To disappear the lens the refractive index of liquid should be same i.e. 1.5.

When a ray of light travels from denser to rarer medium and if the angle of incidence is greater than the critical angle then the light gets totally internally reflected to the same medium. This phenomena is called total internal reflection. 

I^st law:  the incident ray the reflected ray and the normal drawn at the point of incidence all lie in the same plane. 

II^nd law:- angle of incidence is equal to angle of reflection.

It is the angle subtended by an object or image at the eye. It decreases with increasing distance of the object or image from the eye.

The two aberrations that objectives of refracting telescope suffer from are

(i) Spherical aberrations: Because of the surface geometry of the lens, sharp point image of star is difficult to obtain on a point.

In reflecting telescope, we use parabolic mirror to remove this aberration.

(ii) Chromatic aberrations: Different colours of light have different refractive index with respect to glass. Hence different colour would focus at different points. hence image of white object would appear as different colour point images. This is known as chromatic aberrations.

in reflecting telescope, image is formed with reflected rays hence this aberration is removed.

(a) All the distances are measured from the pole or optical center of the lens. 

(b) The distances measured along the direction of incident light are taken as positive and negative in a direction opposite to it. 

(c) The heights measured upwards with respect to X-axis are positive and negative downwards.

From the lens maker formula, it is clear that n₂₁ decreases then focal length increase.

\(\frac {1}{f} = (n_{21} - 1)(\frac{1}{r_1}-\frac{1}{r_2}) (n_{21} =\frac {n_2}{n_1})\)

Here refractive index of the glass with respect to surrounding material decreases. Hence, focal length increases which will also increase the image distance.

For lens L₁

u = −15cm

f = +20 cm

using lens formula

\(\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\)

\(\frac{1}{v}=\frac{1}{u}+\frac{1}{f}\)

\(\frac{1}{v}=-\frac{1}{15}+\frac{1}{20}=-\frac{1}{60}\)

v = −60 cm

This image will act as object for lens L₂ lens

u = −20 − 60 = −80cm

v = + 80 cm

\(\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\)

\(\frac{1}{80}+\frac{1}{80}=\frac{1}{f}\)

\(\frac{1}{f}=\frac{1}{40}orf=40cm\)

(i) \(λ=\frac{589\,nm}{1.33}=442.8\,nm\)

Frequency \(v=\frac{3\times10^8ms^{-1}}{589\,nm}=5.09\times10^{14}Hz\)

Speed \(v=\frac{3\times10^8}{1.33}m/s=2.25\times10^8m/s\)

(ii) \(\frac{1}{f}=[\frac{μ_{ga}}{u_1}-1][\frac{1}{R_1}-\frac{1}{R_2}]\)

\(∴\frac{1}{20}=[\frac{1.55}{1}-1]\frac{2}{R}\)

∴ R = (20 × 1.10) cm = 22 cm

(i) \(\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\) ( is negative)

\(u=-f⇒\frac{1}{v}=0⇒v=∞\)

\(u=-2f⇒\frac{1}{v}=\frac{-1}{2f}⇒v=-2f\)

Hence if -2f < u < -f

⇒ -2f < u < ∞

(ii) \(\frac{1}{f}=\frac{1}{v}+\frac{1}{u}\)

Using sign convention, for convex mirror, we have

f > 0, u < 0

From the formula

\(\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\)

∵ f is positive and u is negative,

⇒ v is always positive, hence image is always virtual.

(iii) Using mirror formula

\(\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\)

For concave mirror, f < 0

\(\frac{1}{v}=\frac{1}{f}-\frac{1}{u}\)

For an object placed between focal and pole of mirror; f < u < 0

Hence \(\frac{1}{u}<\frac{1}{f}\) . It means \(\frac{1}{v}\) > 0

Hence image is always virtual.

Also, as f < 0

\(\frac{1}{v}-\frac{1}{u}<0\)

\(\frac{1}{v}-\frac{1}{u}<0\) (v is always positive)

∵ u < v

So, \(m=|\frac{v}{u}|>1,\)

Hence, image is always enlarged.

(i) Convex Mirror.

(ii) Concave mirror.

Using mirror formula

\(\frac{1}{v}+\frac{1}{u}=\frac{1}{f}\)

For plane mirror f = ∞

Hence

\(\frac{1}{v}+\frac{1}{u}=0\)

So, v = -u

It means the image is at equal distance and in opposite side of object. This is the true condition in image formation through plane mirror. Hence spherical mirror formula holds equally to a plane mirror.

The relationship between angle of incidence ‘i’, angle of prism ‘A’ and angle of minimum deviation for triangular prism is A + δ_m = 2i

(i) Refractive index of a medium is the ratio of speed of light (c) in the free space to the speed of light (v) in that medium.

\(μ=\frac{c}{v}\) 

(ii) \(μ=\frac{c}{v}=\frac{1}{sin\,i_c}\) 

\(\frac{3 \times 10^8}{v}=\frac{1}{\frac {30}{50}}\) 

\(v=\frac{30}{50}\times3\times10^8=1.8\times10^8m/s\)

(i) Let the power at the far point be P_f for the normal relaxed eye.

Then

\(P_f=\frac{1}{f}=\frac{1}{0.1}+\frac{1}{0.02}=60\,D\)

With the corrective lens the object distance at the far point is ∞.

The power required is

\(P_f'=\frac{1}{f'}=\frac{1}{∞}+\frac{1}{0.02}=50\,D\)

The effective power of the relaxed eye with glasses is the sum of the eye and that of the glasses P_g.

\(∴P_f'=P_f+P_g\)

∴ P_g = −10 D.

(ii) His power of accommodation is 4 diopters for the normal eye. Let the power of the normal eye for near vision be P_n.

Then

4 = P_n - P_f or P_n = 64 D.
Let his near point be x_n,

Then

\(\frac{1}{x_n}+\frac{1}{0.02}=64\,or\,\frac{1}{x_n}+50=64\)

\(\frac{1}{x_n}=14,\)

\(∴x_n= \frac{1}{14};0.07m\)

(iii) With glasses

P′_n = P′_f + 4 = 54

54 = \(\frac{1}{x_n'}+\frac{1}{0.02}=\frac{1}{x_n'}+50\)

\(\frac{1}{x_n'}=4,\)

\(∴x_n'=\frac{1}{4}=0.25m.\)

To see the final image at D = 25 cm

For eye lens

\(u=-f\) and \(v=-25\) cm

\(m=\frac{v}{u}\)

\(10=\frac{-25}{-f}\)

f = 2.5 cm = 0.025 m

power of the lens = \(\frac{1}{f}=\frac{1}{0.025}=+40D\) 

It is the distance between the principal focus and the pole of the mirror. 

Visible light in Electromagnetic spectrum has wavelength of about 400 nm to 750 nm. The speed of light in vacuum is the highest speed attainable in nature and is equal to c = 3 x 10⁸ m/s. The path of light is called a ray of light, and a bundle of such rays constitutes a beam of light. 

The phenomenon of change in direction of propagation of an obliquely incident ray of light that enters from one medium to the other medium, at the interface of the two media is called refraction of light.

It is the ratio of the height of the image to the height of the object.

\(∵\frac{sin\,i}{sin\,r}=μ\) 

\(∴\frac{sin\,60^º}{sin\,r}=√3\) 

\(∴sin\,r=\frac{1}{2}\) 

r = 30º

The phenomenon of splitting of white light into its component colors is known as dispersion .

The phenomenon of splitting of light into its component colours is known as dispersion. 

The pattern of colour components of light is called the spectrum of light. 

There are two types of spherical

(a) Concave mirrors

(b) Convex mirrors

(a) Uses of Concave Mirrors :

1. A concave mirror is used as shaving or make up mirror because it forms a magnified and erect image of the face when it is held closer to the face.

2. Doctors use concave mirrors as head mirror. The mirror is strapped to the doctor’s forehead and light from a lamp after reflection from the mirror is focussed into the throat or ear of the patient.

3. A small concave mirror with a small hole at its centre is used in the doctor’s Ophthalmoscope. The doctor looks through the hole from behind the mirror while a beam of light from a lamp reflected from it is directed into the pupil of patient’s eye which makes the retina visible.

4. Concave mirrors are used as reflectors in headlights of cars, etc. The source is placed at the focus of a concave mirror. The light rays after reflection travel over a large distance as a parallel intense beam.

(b) Uses of Convex Mirrors :

A convex mirror is used as a rear view mirror in automobiles. The reason is that it always forms a small and erect image and it has a larger field of view than that of a plane mirror of the same size.

Refractive index of prism is maximum for violet colour and minimum for red colour.
∴ µ_V = µ_R

sin i_c = \(\frac{1}{\mu}\)
Refractive index µ depends on colour of light, so critical angle also depends on colour of light.

(i) Refractive index =  1/sin C
where C is the critical angle.

(ii) Since, refractive index depends upon the wavelength of light, the critical angle for a given pair of media is different for different wavelengths (colours) of light.

Concave mirror. In concave mirror, parallel rays after reflecting from mirror actually meet at focus.

Convex mirror is preferred over plane mirrors as rear-view mirrors because as it gives diminished images of objects and hence can cover larger field of view.

No

The focal length of small aperture mirror is half of its radius. It does not depend upon the refractive index of the surrounding medium.

The focal length of small aperture mirror is half of its radius. It does not depend upon the refractive index of the surrounding medium.

A concave mirror forms a magnified image of an object when

Case – 1: Object is between centre of curvature (C) and principal focus (F) of the mirror.

Case – 2: Object is between principle focus (F) and pole (P) of the mirror.

Difference:

In first case the image is real and inverted while in case-2, the image would be virtual and erect.

(i) Real,

(ii) magnified.

If the wavelength of light incident on a convex lens is increased, its focal length will also increase.

(a) Myopia or near sightedness 

(b) Hypermetropia or far sightedness 

(c) Astigmatism 

It is object is kept between the optical centre and the focus of the biconvex lens, then the lens behave like a magnifying glass.

Magnifying power is defined as the angle subtended at the eye by the image to the angle subtended (at the unaided eye) by the object.

(Alternatively: Also accept this definition in the form of formula)

\(m=m_o \times m_e = \frac{L}{f_o}\times\frac{D}{f_e}\) 

To increase the magnifying power both the objective and eyepiece must have short focal lengths (as \(m=\frac{L}{f_o}\times \frac{D}{f_e}\))

At sunset and sun rise, sun is at horizon. Sun rays have to pass through larger distance in the atmosphere. Most of the blue and other shorter wavelengths are removed by scattering. The least scattered red light reaches our eyes. Hence sun is red at rise and set. 

Blue has a shorter wavelength than red and is scattered much more strongly than any other color. Violet scatters more than that of blue, but our eyes are more sensitive to blue than violet. Therefore sky appears blue. 

(b) 25 cm and infinite
For a healthy eye, the least distance of distinct vision is 25 cm and maximum distance is infinite.

(b) Concave mirror
Concave mirror forms apparent image of size greater than object.

(c) Sum of the focal distances
Length of a simple astronomical telescope
L = f₀ + f_e

In compound microscope, the aperture of objective lens is smaller than eyepiece, while in telescope the aperture of objective is greater than aperture of eyepiece.

In simple microscope a convex lens of less focal length is used.

(d) The focal length of the lens may be less than 20 cm
The focal length of the lens may be less than 20 cm.