Geometrical Optics

It  is a branch of physics that studies light and its interaction with matter, including phenomena like reflection, refraction, and dispersion. It also explores the use of devices like lenses and prisms to manipulate light, playing a key role in technologies such as eyeglasses, cameras, microscopes, and telescopes."

1.0Condition for Rectilinear Propagation of Light

1.The medium is isotropic.

2.The obstacle or opening through which light passes is not very small.

3.Bending is negligible if $\frac{D\lambda }{a}\ll aora\gg \sqrt{D\lambda }$ . If this condition is fulfilled, light is said to move rectilinearly.                              

Beam : A bundle or bunch of rays is called a beam. It is of following three types

(a) Convergent beam:Diameter of beam decreases in the direction of ray.                           

(b) Divergent beam: Diameter of beam goes on increasing as the rays proceed forward.                                          

(c) Parallel beam:The beam moves parallel to each other and the diameter of the beam remains the same.

2.0Properties Of Light

(1) Speed of light in vacuum, denoted by $c=3\times 10^8\mathrm{m}/\mathrm{s}$ approximately.

(2) Light is an electromagnetic wave .It consists of varying electric fields and magnetic fields.

(3) Light carries energy and momentum.

(4) The formula $v=f\lambda$ is applicable to light.

(5) When light is reflected within the same medium, its frequency, speed, and wavelength remain unchanged.

(6) The frequency of light stays the same during reflection or refraction.

3.0Reflection of Light

When light rays hit the boundary between two media, such as air and glass, some of the light is reflected back into the original medium.

(a) Regular Reflection:"Regular reflection occurs when light reflects off a smooth, flat surface."

(b) Diffused Reflection:When light reflects off a rough surface, scattering in multiple directions.

Laws of Reflection                                

(a) The incident ray, reflected ray, and the normal at the point of incidence all lie in the same plane, known as the plane of incidence (or plane of reflection).

This condition can be expressed mathematically as $\hat{e}\cdot (\hat{n}\times \hat{r})=0$

(b)The angle of incidence (the angle between the normal and the incident ray) is equal to the angle of reflection (the angle between the reflected ray and the normal), i.e., i.e. $\angle i=\angle r$

In Vector form $\hat{r}=\hat{e}-2(\hat{e}.\hat{n})\hat{n}$

4.0Deviation by Plane Mirror

$\delta =180^{\circ }-2i$

${\delta }_{\max }=180^{\circ }$

Deviation by two plane mirrors

$\delta =360-2\theta$

Rotation of reflected rays

(1) If the incident ray's direction remains constant and the mirror is rotated by an angle around an axis in its plane.

As the mirror turns through , the reflected ray turns through 2.

(2) If the mirror is kept fixed and the incident ray is rotated then the reflected ray will rotate in the opposite direction by the same angle.

Note: Intersection point of converging reflected/refracted rays is called real image and that of diverging reflected/refracted rays is called virtual image.

5.0Plane Mirror

• A plane mirror is formed by polishing one surface of a plane thin glass plate. It is also said to be silver on one side.
• A beam of parallel light rays incident on a plane mirror will reflect as a parallel beam of rays.

Properties of Image Formed by a Plane Mirror

Point object

1. The object distance equals the image distance from the mirror.
2. Incident rays from a point object converge at a single point after reflection, forming an image.
3. The line joining the object and its image is normal to the reflecting surface.
4. A real object forms a virtual image, and a virtual object forms a real image.
5. The field of view is the area where the observer must be to see the image.

Extended object:

• An extended object refers to an object that occupies a finite amount of space and consists of multiple points, each of which can produce its own image. This contrasts with a point object, which has no dimensions. 

6.0Relation Between Velocity of Object and Image         

For x-Axis $v_{iG}-v_{mG}=-\left(v_{oG}-v_{mG}\right)$

for y axis and z axis $v_{iG}-v_{mG}=\left(v_{oG}-v_{mG}\right)or{v}_{iG}=v_{oG}$

$v_{iG}$ == velocity of image with respect to ground.

Number of images formed by two inclined mirrors

(1) If $\frac{360^{\circ }}{\theta }=evennumber;numberofimage=\frac{360^{\circ }}{\theta }-1$

(2)If $\frac{360^{\circ }}{\theta }=\text{odd number; number of image}=\frac{360^{\circ }}{\theta }-1$ , if the object is placed on the angle bisector.

(3) If $\frac{360^{\circ }}{\theta }=\text{odd number; number of image }=\frac{360^{\circ }}{\theta }$ , if the object is not placed on the angle bisector.

(4) If $\frac{360^{\circ }}{\theta }\ne$ Integer ,then count the number of images as explained above.

7.0Spherical Mirrors

• A curved mirror is part of a hollow sphere. If reflection takes place from the inner surface then the mirror is called concave and if its outer surface acts as reflector it is convex.   

Important terms related with spherical mirrors

(a) Center of Curvature (C) : The center of curvature is the center of the sphere from which a spherical mirror is made.

(b) Pole (P) :The center of the mirror, known as the pole, is represented by point P on the mirror APB.

(c) Principal Axis :The principal axis is a line perpendicular to the mirror's plane, passing through the pole.

(d) Aperture (A) :The aperture is the part of the mirror that reflects light; in the figure, APB represents the aperture.

(e) Focal Length:When a parallel beam of light strikes a concave mirror, reflected rays converge at the principal focus (F). For a convex mirror, the rays appear to diverge from F. If the light strikes at an angle, the rays converge or diverge from a point in the focal plane, perpendicular to the principal axis.

Mirror formulae:

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

8.0Magnification

1. Lateral Magnification

$m=\frac{h_i}{h_0}=-\frac{v}{u}$

2. Longitudinal Magnification

If one dimensional object is placed with its length along the principal axis then linear magnification is called longitudinal magnification.

$m_L=\frac{\text{ Length of Image }}{\text{ Length of Object }}=\left|\frac{v_2-v_1}{u_2-u_1}\right|$

For small objects only:

$m_L=-\frac{dv}{du}$

$m_L=-\frac{dv}{du}={\left[\frac{v}{u}\right]}^2=m^2$

3. Superficial Magnification

If a 2D object is placed perpendicular to the principal axis, its magnification is called superficial magnification.

$m_s=\frac{\text{ Area of Image }}{\text{ Area of Object }}=\frac{(ma)\times (mb)}{(a\times b)}=m^2$

4. Volume Magnification

For small cubical objects only, all dimensions will be magnified equally because all dimensions are almost at the same distance from the mirror, hence the final image is also a cube. 

$m_V=\frac{\text{ Volume of Image }}{\text{ Volume of Object }}=\frac{(ma)\times (mb)}{(a\times b)}=m^4$

9.0Velocity of Image of Moving Object (Spherical Mirror)

(a) Velocity component along axis (Longitudinal velocity)

${\vec{v}}_{ix}=-\frac{v^2}{u^2}{\vec{v}}_{ax}=-m^2{\vec{v}}_{ax}$

${\vec{v}}_{ix}=\frac{dv}{dt}=$ velocity of image along principal axis

${\vec{v}}_{ax}=\frac{du}{dt}=$velocity of object along principal axis

(b) Velocity component perpendicular to  axis (Transverse velocity)

$\frac{d{h}_i}{dt}=\left(\frac{f}{f-u}\right)\frac{d{h}_0}{dt}+\frac{f{h}_0}{(f-u{)}^2}\frac{du}{dt}$

${\vec{v}}_y=\left[m{\vec{v}}_{oy}+\frac{m^2{h}_0}{f}{\vec{v}}_{ox}\right]$

$\frac{d{h}_i}{dt}=\text{ velocity of image }\perp \text{ to principal axis }$

$\frac{d{h}_0}{dt}=\text{ velocity of object }\perp \text{ to principal axis }$

Optical power of a mirror (in Dioptre)$=-\frac{1}{f}$

f = focal length with sign and in meters.

Newton's Formula: 

$XY=f^2$

10.0Refraction of Light

Refraction occurs when light changes mediums, causing a shift in speed and direction. If light strikes at an angle (0° < i < 90°), ( 0^{\circ}<\mathrm{i}<90^{\circ} ) it bends due to this speed change. Light incident normally passes straight but is still refracted. Refraction without reflection is impossible, and as the angle of incidence increases, more energy is reflected. The refractive index is the ratio of light speed in vacuum to its speed in the medium.

• The refractive index of a medium is the ratio by which the speed of light decreases compared to its speed in a vacuum.

$\mu =\frac{c}{v}=\frac{\text{ Speed of light in vacuum }}{\text{ Speed of light in medium }}$

• A higher refractive index indicates slower light speed in the medium, making it optically denser, while a lower refractive index means the medium is optically rarer.

Relative refractive index

$1{\mu }_2=\frac{{\mu }_2}{{\mu }_1}=\frac{\left(\frac{c}{v_2}\right)}{\left(\frac{c}{v_1}\right)}=\frac{v_1}{v_2}$

Laws of Refraction

(a) "The incident ray, the normal at the point of incidence, and the refracted ray all lie in the same plane, known as the plane of incidence."

(b) $\frac{\mathrm{Sin}i}{\mathrm{Sin}r}=\text{ Constant }\Rightarrow \text{ Snell’s Law }$

$\Rightarrow \frac{\mathrm{Sin}i}{\mathrm{Sin}r}=\frac{n_2}{n_1}=\frac{v_1}{v_2}=\frac{{\lambda }_1}{{\lambda }_2}$

$n_1\mathrm{Sin}i=n_2\mathrm{Sin}r$

$\frac{n_2}{n_1}=1n_2==$ Refractive Index of the second medium with respect to the first medium.

Special Cases

1. When Normal incidence : i = 0 ; from Snell’s law : r = 0                         

2. When light moves from an optically denser to an optically rarer medium, it bends away from the normal.                 

4. When light moves from an optically rarer to an optically denser medium, it bends towards the normal.             

Refraction through Multiple Parallel Slabs

$n_1\mathrm{Sin}{\theta }_1=n_m\mathrm{Sin}{\theta }_m$

Where $n_m$ is the refractive index of the $m^{th}$ slab, and ${\theta }_m$ is the angle of incidence at the $m^{th}$ slab.

Principle of Reversibility of Light Rays:

(a) A ray following the reflected path will reflect back along the incident ray’s path.
(b) A reversed refracted ray will refract along the incident ray’s path, showing that incident and refracted rays are reversible.

11.0Apparent Depth and Shift of Submerged Object

$d^{\prime }=\frac{d}{\left(\frac{n_i}{n_r}\right)}$

$n_{rel}=\frac{n_i}{n_r}$

$d^{\prime }=\frac{d}{n_{rel}}\text{ and }{v}^{\prime }=\frac{v}{n_{\text{relative }}}$

Apparent Shift= $d\left(1-\frac{1}{n_{rel}}\right)$

Refraction through  a Parallel Slab

"When light passes through a parallel slab with the same medium on both sides, the emergent ray is parallel to the incident ray. If the mediums differ, the emergent ray won't be parallel."

(b) Light is shifted laterally, given by 

$d=\frac{t\sin (i-r)}{\cos r}$

t=thickness slab

Apparent Shift:

(a) for converging rays:

When a slab of thickness t and refractive index $\mu$ is placed in the path of a convergent beam, then the point of convergence is shifted by         

$S=\left(1-\frac{1}{\mu }\right)t$

(b) for diverging rays:

When the same slab is placed in the path of divergent beam, then the point of divergence is shifted by,

$S=\left(1-\frac{1}{\mu }\right)t$

Note:

(1) The shift ‘S’ is always in the direction of light.

(2) If the slab is made of air and surrounding medium is of refractive index $\mu$. Then the apparent shift would be $S=t(\mu -1)$

(3) If n number of slabs with different thickness and refractive index are placed between the observer and the object, then the total apparent shift is equal to the summation of the individual shifts.

$S=S_1+S_2+\dots \dots \dots \dots S_n$

$S=t_1\left(1-\frac{1}{{\mu }_1}\right)+t_2\left(1-\frac{1}{{\mu }_2}\right)+\dots \dots \dots .t_n\left(1-\frac{1}{{\mu }_n}\right)={\sum }_{i=1}^n{t}_i\left(1-\frac{1}{{\mu }_i}\right)$

(4) If there are n number of slabs with different thickness and refractive index, one over the other then

$d_{AC}=t_1+t_2+\dots ..t_n$

$d_{AP}=\frac{t_1}{{\mu }_1}+\frac{t_2}{{\mu }_2}+\dots \dots \dots ..\frac{t_n}{{\mu }_n}$

$\mu =\frac{d_{AC}}{d_{AP}}=\frac{t_1+t_2+\dots ..t_n}{\frac{t_1}{{\mu }_1}+\frac{t_2}{{\mu }_2}+\dots \dots \dots ......\frac{t_n}{{\mu }_n}}=\frac{\Sigma t_t}{\sum \left(\frac{t_i}{u_i}\right)}$

12.0Critical Angle And Total Internal Reflection ( T. I. R.)

• The critical angle is the angle in the denser medium where the refraction in the rarer medium is $90^{\circ }$. If the angle exceeds the critical angle, total internal reflection occurs, and the interface acts like a mirror.  

${\theta }_c={\sin }^{-1}\frac{n_r}{n_d}$

Conditions of T. I. R.

(a) Light is incident on the interface from denser medium.

(b) Angle of incidence should be greater than the critical angle $\left(i>{\theta }_c\right)$

Deviation ($\delta$):

• A light ray travelling from a denser to a rarer medium at an angle $<C\alpha <{\theta }_C$ then 

$\delta =\beta -\alpha ={\mathrm{Sin}}^{-1}(\mu \mathrm{Sin}\alpha )-\alpha$

${\delta }_{\max }=\frac{\pi }{2}-{\theta }_c$

• If light is incident at an angle $\alpha >{\theta }_c$ , Then the angle of deviation is

$\delta =\pi -2\alpha \text{ and }{\delta }_{\max }=\pi -2{\theta }_c$

• Graphically the relation between $\delta$ & $\alpha$ can be shown as

13.0Prism

• "A prism is a homogeneous, solid, transparent refracting medium with two plane surfaces inclined at an angle."                                     

A=Refracting angle or the Angle of Prism( Apex angle)

$\delta$ == Angle of Deviation

Angle of Deviation ($\delta$)

• It is the angle between the emergent and the incident ray. In other words, it is the angle through which the incident ray turns while passing through a prism.

$\delta =i+e-A$

Condition of No Emergence

For face AB is

$\mathrm{r}i=\max =90^{\circ },\text{at face AC}{r}_2>{\theta }_c$

• A ray of light will not emerge out of a prism (what ever be the angle of incidence) if $A>2{\theta }_c$
• $\mu >\mathrm{Cosec}\left(\frac{A}{2}\right)$

Critical Angle

• It is the angle of prism above which incident light ray on first surface will not emerge out from the second surface for all possible values of angle of incidence

$A=2{\theta }_c$

Condition of Grazing Emergence

• If a ray can emerge out of a prism, the value of angle of incidence i for which angle of emergence e=90° is called the condition of grazing emergence.

$r_2={\theta }_c{r}_1=A-r_2=A-{\theta }_c$

$i={\mathrm{Sin}}^{-1}\left[\sqrt{{\mu }^2-1}\mathrm{Sin}A-\mathrm{Cos}A\right]$

Note: The light will emerge out of a given prism only if the angle of incidence is greater than the condition of grazing emergence.

Condition of Maximum Deviation

${\delta }_{\max }=90^{\circ }+{\mathrm{Sin}}^{-1}\left[\mu \mathrm{Sin}\left(A-{\theta }_c\right)-A\right]$

Note: This situation is the reverse of grazing emergence and may also be viewed as deviation at grazing incidence.

Condition for Minimum Deviation

Minimum deviation happens when the angle of incidence equals the angle of emergence.

$i=e,r_1=r_2=r,{\delta }_{\min }=2i-A,r=\frac{A}{2}$

$\mu =\frac{\sin i}{\sin r}=\frac{\sin \left(\frac{{\delta }_{\mathrm{man}}+A}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

Note: In the condition of minimum deviation the light ray passes through the prism symmetrically, i.e., the light. The ray in the prism becomes parallel to its base.

Graphical Representation of angle of deviation         

Thin Prisms

• In thin prisms the distance between the refracting surfaces is negligible and the angle of prism (A) is very small.

$A=r_1+r_2$ ,therefore $r_{1}and{r}_2$ both are small and the same is true for $i_{1}and{i}_2$

$i=\mu r_{1}ande=\mu r_2$

Deviation, $\delta =A(\mu -1)$

Note: The deviation for a small angled prism is independent of the angle of incidence.

Image formation by small angled prism 

If object is real

1. Image is virtual

2. Shifting of image is OI

14.0Dispersion of Light

• Dispersion of light is the angular separation of white light into its components, caused by varying speeds of different wavelengths in a medium. The refractive index  varies with a wavelength, as described by Cauchy’s formula.
• $\mu (\lambda )=a+\frac{b}{{\lambda }^2}$ where a and b are positive constants of a medium.

Note: Such a phenomenon is not presented by sound waves. The angle between the rays of the extreme colors in refracted light is called the angle of dispersion."

$\theta ={\delta }_v-{\delta }_r$

For prism of small ‘A’ and with small ‘i’ 

$\theta ={\delta }_v-{\delta }_r=\left({\mu }_v-{\mu }_r\right)A$   

Deviation of beam (also called mean deviation)

$\delta ={\delta }_y=\left({\mu }_y-1\right)A$

Note: Fig (a) and (c) represents dispersion, whereas in fig. (b) there is no dispersion.

Numerical data reveals that if the average value of $\mu$ is small $\left({\mu }_v-{\mu }_r\right)$ is also small and if the average value of is large $\left({\mu }_v-{\mu }_r\right)$ is also large. Therefore, a greater mean deviation results in a higher angular dispersion.

Dispersive power ($\omega$):

Dispersive power ($\omega$) of the medium of the material of prism is given by: $\omega =\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}$

$\omega$ is the property of a medium.

For small angled prism $\left(A<10^{\circ }\right)$ with light incident at small angle i

$\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}=\frac{{\delta }_v-{\delta }_r}{{\delta }_y}=\frac{\theta }{{\delta }_y}=\frac{\text{ Angular Dispersion }}{\text{ Deviation of Mean Ray(Yellow) }}$

$\left[{\mu }_y=\frac{{\mu }_v+{\mu }_r}{2}\text{, if }{\mu }_y\text{ is not given in the problem }\right]$

$\mu -1$== Refractive index of the medium for the corresponding colour.

Dispersion without deviation (Direct Vision Combination)

$\left[{\mu }_y-1\right]A=\left[{\mu }_y-1\right]A^{\prime }\iff \left[\frac{{\mu }_v+{\mu }_r}{2}-1\right]A=\left[\frac{{\mu }_v^{\prime }+{\mu }_r^{\prime }}{2}-1\right]A^{\prime }$

Deviation without dispersion (Achromatic Combination)

$\left({\mu }_v-{\mu }_r\right)A=\left({\mu }_v^{\prime }-{\mu }_r^{\prime }\right)A^{\prime }$

15.0Law of refraction at spherical surface

• When light passes from a medium of refractive index 1 to a medium of refractive index 2 by a spherical surface of radius of curvature R then the relation between object distance u and image distance v is given by

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

Sign convention

(1) All distances are quantified from the pole (P).

(2) Distances along the course of the incident rays are considered positive.

(3) Distances above the principal axis are taken as positive.       

Terms related to refraction at spherical surfaces

(A) Centre of curvature (C): It is the centre of a sphere of which the surface is a part.

(B) Radius of curvature (R): It is the radius of the sphere of which the surface is a part.

(C) Pole (P) :It is the geometrical centre of the spherical refracting surface.

(D) Principal Axis : Straight line joining the centre of curvature to the pole.

(E) Focus :"When a parallel beam of paraxial rays hits a spherical refracting surface, the rays converge or diverge depending on the surface's curvature and refractive indices. The point of convergence or apparent divergence on the principal axis is called the focus."

Note:(1) It is not always necessary that for a convex boundary the parallel rays always converge. Similarly, for concave boundaries the incident parallel ray may converge or diverge depending upon the refractive index of two media.

(2) Laws of refraction are valid for spherical surfaces also.

(3) Pole, centre of curvature, Radius of curvature, Principal axis etc. are defined as spherical mirrors except for the focus.

Lateral Magnification

$m=\frac{h_i}{h_0}=\frac{\text{ height of image }}{\text{ height of object }}$

$m=-\frac{\frac{v}{{\mu }_2}}{-\frac{u}{{\mu }_1}}=\frac{{\mu }_1}{{\mu }_2}\cdot \frac{v}{u}$

16.0Lens Theory

• A lens is a transparent object with two refracting surfaces, at least one of which is curved, and the refractive index of the material differs from that of its surrounding medium.

Types of Lenses

Depending upon the shape of the refracting surfaces following types of lenses can be formed:

Lens-Maker’s Formula

It relates the focal length of the lens to the relative refractive index  of the lens material and the radii of curvature of the two surfaces.

$\frac{1}{f}=(\mu -1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

$\mu =\frac{{\mu }_2}{{\mu }_1}=\frac{\text{Refractive index of lens}}{\text{Refractive index of surrounding}}$

$R_1$ is the radius of curvature of first surface and $R_2$ is the radius of curvature of the second surface from where light emerges out in the first medium.

Note:

(1) The Lensmaker’s formula is applicable for thin lenses only. The values of $R_1$ and are to be put in accordance with the cartesian sign convention.

(2) Position of object and image are interchangeable.These positions are called conjugate positions.

Lens formula

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$

$\frac{1}{f}=\left(\frac{{\mu }_2}{{\mu }_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Lateral magnification

$m=\frac{h_i}{h_0}=\frac{v}{u}$

Note:

(1) If converging rays fall, the focus is on the other side of the direction of incidence and for diverging rays focus is on the same region of the direction of incidence.

(2) m has negative and positive values for real virtual pairs.

(3) Use cartesian sign convention with optical centre of lens as origin.

17.0Power of a lens

When focal length is written in metre then $P=\frac{\mu }{f}D$ is known as the power of the lens. Where D is (diopter) unit of power.

Displacement Method 

$f=\frac{D^2-d^2}{4D}$

$D_{\min }=4f$

Lateral magnification

$H=\sqrt{h_1{h}_2}$

Combination of lenses

(A) Net magnification, m=m1 ✕ m2 ✕ m3  ✕ ...............

(B) If thin lenses are kept close together with their principal axis coincide then,

$\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\dots \dots ...$

f = (+) ve gives equivalent converging lens

= (–) ve gives equivalent diverging lens

$\frac{1}{v}$versus $\frac{1}{u}$ Graph:

$u-v\text{Graph}$

18.0Velocity of Image of moving object (In lens)

(a) Velocity component along the principal (Optical) axis

$(V_L{)}_{\parallel }=+\frac{v^2}{u^2}(V_{OL}{)}_{\parallel }$

$(V_L{)}_{\parallel }=\text{Velocity of image w.r.t. lens along the principal axis.}$

$(V_{OL}{)}_{\parallel }=\text{Velocity of object w.r.t. lens along the principal axis.}$

(b) Velocity component perpendicular to the principal axis

$(V_L{)}_{\perp }=+\frac{v}{u}(V_{OL}{)}_{\perp }$

$(V_L{)}_{\perp }=$ Velocity of image w.r.t. lens perpendicular the principal axis.

$(V_{OL}{)}_{\perp }=$ Velocity of object w.r.t. lens perpendicular the principal axis.

19.0Silvering at One Surface of Lens

When one surface of a thin lens is silvered, then the focal length F of the effective lens-mirror combination is expressed as,

$\frac{1}{F}=\sum \frac{1}{f_i}$

Angle of deviation of a ray when it passes a lens

O is the object and I is the image $\delta$ is the angle of deviation

$\delta =\frac{h}{f}$

20.0Microscope

• It is an optical device designed to view extremely small objects, and its magnifying power is expressed by.

$m=\frac{\text{Visual angle with instrument}(\beta )}{\text{angle subtends by object at least distance of distinct vision}(\alpha )}$

1. Simple Microscope

(1) It is a single convex lens of lesser focal length.

(2) Also called magnifying glass or reading lens.

(3) Magnification’s, when final image is formed at

$Dand\infty ,i.e.,m_{0}and{m}_e$

$m_0={\left(1+\frac{D}{f}\right)}_{\text{max}}\text{and}{m}_0={\left(\frac{D}{f}\right)}_{\text{min}}$

Note:

$m_{\text{max}}-m_{\text{min}}=1$

If lens is kept at a s a from the eye then$m_D=1+\frac{D-a}{f}\text{and}{m}_{\infty }=\frac{D-a}{f}$

2. Compound Microscope        

(1) Consist of two converging lenses called objective and eye lens.

(2)

$f_{\text{eye lens}}>f_{\text{objective}}and(\text{Diameter}{)}_{\text{eye lens}}>(\text{Diameter}{)}_{\text{objective}}$

(3) The final image is magnified, virtual and inverted.

Magnification

$m=m_{\text{objective}}\times m_{\text{eye lens}}$

$m_D=\left(\frac{v_0-f_0}{f_0}\right)\left(1+\frac{D}{f_e}\right)$

$m_{\infty }=\left(\frac{v_0-f_0}{f_0}\right)\frac{D}{f_e}$

Length of the tube 

1. When final image is formed at D

$L_D=v_0+u_e=\frac{u_0{f}_0}{u_0-f_0}+\frac{f_{e}D}{f_e+D}$

2. When final images is formed at ∞ 

$L_{\infty }=v_0+f_e=\frac{u_0{f}_0}{u_0-f_0}+f_e$

Note:

$m_{\infty }=\frac{(L_{\infty }-f_0-f_e)D}{f_0{f}_e}$

(1) For maximum magnification both $f_{0}and{f}_e$ must be less.

(2) $(m=m_{\text{objective}}\times m_{\text{eye lens}})$

(3) If objective and eye lens are interchanged, practically there is no change in magnification.

(3) Resolving limit and resolving power : The minimum separation between two lines at which they are still distinguishable is known as the Resolving Limit (RL), and its reciprocal is referred to as the Resolving Power (RP).

$R.L=\frac{\lambda }{2\mu \sin \theta }\text{and}R.P=\frac{2\mu \sin \theta }{\lambda }⟹R.P\propto \frac{1}{\lambda }$

21.0Telescope 

A telescope allows distant objects to be viewed.

1. Astronomical Telescope

(1) Used to see heavenly bodies.

(2) $f_{objective}>f_{eyelens}and{d}_{objective}>d_{eyelens}$

(3) The intermediate image is real, inverted and small.

(4) The final image is virtual, inverted and small.

(5) $Magnification{m}_D=\frac{f_0}{f_e}\left(1+\frac{f_e}{D}\right)and{m}_{\infty }=-\frac{f_0}{f_e}$

(6) $Length{L}_D=f_0+u_e=f_0+\frac{f_{e}D}{f_e+D}and{L}_{\infty }=f_0+f_e$

2. Terrestrial Telescope

(1) Used to see far off objects on the earth.

(2) It consists of three converging lenses : objective, eye lens and erecting lens.

(3) It’s final image is virtually erect and smaller.

(4) $Magnification:m_D=\frac{f_0}{f_e}\left(1+\frac{f_e}{D}\right)and{m}_{\infty }=\left|\frac{f_0}{f_e}\right|$

(5) $Length{L}_D=f_0+4f+u_e=f_0+4f+\frac{f_{e}D}{f_e+D}and{L}_{\infty }=f_0+4f+f_e$

3. Galilean Telescope

(1) It is also a terrestrial telescope but of much smaller field of view.

(2) The object is a converging lens while the eye lens is a diverging lens.

(3) $Magnification{m}_D=\frac{f_0}{f_e}\left(1-\frac{f_e}{D}\right)and{m}_{\infty }=\frac{f_0}{f_e}$

(4) $Length{L}_D=f_0-u_{e}and{L}_{\infty }=f_0-f_e$

Note:

Least distance(d) between objects, so they can just resolved by a telescope is $d=\frac{r}{R.P}$ where r = separation of objects from telescope.

22.0Sample Questions On Optics

Q-1.For what angle of incidence ,the lateral shift produced by a parallel sided glass plate is maximum?

Solution:

$d=\frac{t}{\cos r}\sin (90^{\circ }-r)=\frac{t}{\cos r}\cos r\Rightarrow d=t$

Lateral shift is maximum

Q-2.A substance has critical angle of 45° For yellow light what is its refractive index?

Solution:

$\mu =\frac{1}{\sin C}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2}$

Q-3. Although the surface of sunglasses are curved it does not have any power.Why?

Solution:

The two surfaces of the sunglasses lens  are parallel i.e. one surface convex and other concave thus the power of the two surfaces is equal but of opposite sign.

$P=P_1+P_2=P+(-P)=0$

Q-4.Prove that the limiting value of the prism angle is twice the critical angle of the material.

Solution:

Angle of Prism $A=r1+r2r_1+r_2$

For Limiting $A_{\max }=(r_1{)}_{\max }+(r_2{)}_{\max }$

Value of angle of prism for ${\left(r_1\right)}_{\text{max}}means(i=90^o)$

But When $i=90^{\circ },{\left(r_1\right)}_{\text{max}}=C$

$A_{\text{max}}=C+C=2C$

$A_{\text{max}}=2C$

Q-5.A reflecting telescope uses a concave mirror with a radius of curvature of 120 cm. Find the focal length of the eyepiece to obtain a magnification of 20?

Solution:

M=20

R=120 cm (for concave reflector)

$f_0=\frac{R}{2}=\frac{-120}{2}=-60\text{ cm}$

$M=\frac{f_0}{f_e}\Rightarrow \frac{-60}{f_e}=-20$

$f_e=3\text{ cm}$