Ray Optics And Optical Instruments

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Ray Optics And Optical Instruments

- A quick revision of all the important concepts

RAY OPTICS

Refraction at a spherical surface

\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}

This equation gives us a relation between object and image distance in terms of the refractive index of the medium and the radius of curvature of the curved spherical surface. It holds for any curved spherical surface.

Refraction by a lens

This equation is known as the lens maker's formula.

$\frac{1}{f} = (n_{2} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

This equation is the formula for a thin lens.

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

There are two foci, F and F'. The focus on the side of the source of light is called the first focal point, whereas the other is called the second focal point.

Ray Diagrams

A ray emanating from the object parallel to the principal axis of the lens after refraction passes through the second principal focus F' (in a convex lens) or appears to diverge (in a concave lens) from the first principal focus F.
A ray of light, passing through the optical centre of the lens, emerges without any deviation after refraction.
A ray of light passing through the first principal focus (for a convex lens) or appearing to meet at it (for a concave lens) emerges parallel to the principal axis after refraction.

Sign convention for the lens is the same as mirror.
Magnification:

m = \frac{h ^{'}}{h} = \frac{v}{u}

For erect (and virtual) images formed by a convex or concave lens, m is positive, while for an inverted (and real) image, m is negative.

Power of a lens

The power P of a lens is defined as the tangent of the angle by which it converges or diverges a beam of light falling at a unit distant from the optical centre.

$P = \frac{1}{f}$

The SI unit for power of a lens is dioptre (D): $1 D = 1 m^{- 1}$
Power is positive for a converging lens(convex lens) and negative for a diverging lens (concave).

Combination of thin lenses in contact
If several thin lenses of focal length

f_{1}, f_{2}, f_{3}

,... are in contact, the effective focal length of their combination is given by

\frac{1}{f} = \frac{1}{f _{1}} + \frac{1}{f _{2}} + \frac{1}{f _{3}} + \dots

.
The net power of the lens combination is

P = P_{1} + P_{2} + P_{3} + P_{4} + . . . . . .

Refraction through a prism

A prism has two plane surfaces AB and AC inclined to each other as shown in figure. $∠ A$ is called the angle of prism or refracting angle of prism.
Deviation $δ$ is the angle between incident ray and emergent ray.

δ = (i_{1} + i_{2}) - A

The angle of deviation depends on the angle of incidence.
The refractive index of the prism is

μ = \frac{s i n ( \frac{A + δ _{m}}{2} )}{s i n \frac{A}{2}}

Dispersion by a Prism

The phenomenon of splitting of light into different colours is known as dispersion.The pattern of colour components of light is called the spectrum of light (VIBGYOR).
In the visible spectrum, red light is at the long wavelength end (~700 nm) while the violet light is at the short wavelength end (~ 400 nm).
Dispersion takes place because the refractive index of medium for different wavelengths is different.
Vacuum is a non-dispersive medium in which all colours travel with the same speed.

Rainbow

Rainbow is a phenomenon due to the combined effect of dispersion, refraction and reflection of sunlight by spherical water droplets of rain.
Sunlight is first refracted as it enters a raindrop, which causes the different wavelengths (colours) of white light to separate.Longer wavelengths of light (red) are bent the least while the shorter wavelength (violet) are bent the most.
These component rays strike the inner surface and get internally reflected.The reflected light is refracted again when it comes out of the drop.
A secondary rainbow is formed when two internal reflections take place inside a rainbow.

Scattering of Light

The amount of scattering is inversely proportional to the fourth power of the wavelength. This is known as Rayleigh scattering.
Light of shorter wavelengths is scattered much more than light of longer wavelengths.

OPTICAL INSTRUMENTS

Angular Magnification

Angular Magnification = \frac{Visual angle of the image}{Visual angle of the object placed at image position}

Microscope

Simple

A simple magnifier or microscope is a converging lens of small focal length.

Magnifying power of a simple microscope is given by

m = (1 + \frac{D}{f})

where D = 25 cm is the least distance of distinct vision and f is the focal length of the convex lens.

If the image is at infinity,

m = \frac{D}{f}

Compound

Magnifying power of a compound microscope is given by $m = m_{o} \times m_{e}$

where

m_{e}

is the magnification due to eyepiece and

m_{o}

is the magnification due to object.

m_{e} = (1 + \frac{D}{f _{e}})

The total magnification when the image is formed at infinity, is

m = m_{o} \times m_{e} = (\frac{L}{f _{o}}) (\frac{D}{f _{e}})

Resolving Power
The reciprocal of the limit of resolution is called the resolving power.

R . P . = \frac{1}{d} = \frac{2 ( N . A )}{1 . 2 2 λ}

;

N . A = μ S i n α

Telescope
Magnifying power m of a telescope is the ratio of the angle $β$ subtended at the eye by the image to the angle

α

subtended at the eye by the object.

m \approx \frac{β}{α} = \frac{h}{f _{e}} . \frac{f _{o}}{h} = \frac{f _{o}}{f _{e}}

Refracting Telescope

Refracting telescopes can be used both for terrestrial and astronomical observations.
These telescopes have, in addition, a pair of inverting lenses to make the final image erect.

Reflecting Telescope

Telescopes with mirror objectives are calledreflecting telescopes.

No chromatic aberration in a mirror.
Magnifying power is

M = \frac{f _{o}}{f _{e}} = \frac{R / 2}{f _{e}}

Resolving Power
It is the ability to form separate images of two distant point objects situated nearly

R . P = \frac{1}{d θ} = \frac{f}{r} = \frac{a}{1 . 2 2 λ}