Prove that: a μ w × w μ g × g μ a = 1 where the symbols have their usual meanings.

The refractive index of air with respect to glass is expressed as a μ w × w μ g × g μ a = sin i sin r When ∠ r = 90 ° and sin 90 ° = 1 (in the air). ∠ i in the medium is called critical angle. If refractive angle, r = 90 ° , the corresponding angle of incidence, ' i ' Will be equal to the critical angle. If the angle of incidence exceeds the value of 'i' obtained in part (i.e., critical angle), the total internal reflection will occur.

The radius of curvature of a plano-convex lens is 20 cm . Refractive index of glass is 1 . 5 . Find the focal length of the lens.

Radius of curvature of plane surface R 1 = ∞ Radius of curvature of curved convex surface R 2 = − 20 cm Refractive index of material of lens μ = 1 . 5 Using lens maker formula : f 1 = ( μ − 1 ) 1 R 1 − 1 R 2 ∴ 1 f = ( 1 . 5 − 1 ) 1 ∞ − 1 − 20 ⟹ f = 40 cm Hence, required focal length is 40 cm .

A 5 cm long object is placed in front of a concave mirror of radius of curvature 40 cm . The distance of object from mirror is 40 cm . Find the distance and size of the image.

Focal length f = R 2 = - 40 2 = − 20 cm , using sign convention for concave mirror Object distance u = − 5 cm Image distance v = ? Using mirror formula 1 f = 1 v + 1 u − 1 20 = 1 v − 1 5 v = 6 . 67 cm Magnification(M) = h i h o = − v u = - 20 - 3 × 5 = 1 . 33 ⇒ h i = 20 3 = 6 . 67 cm Hence.the position of image is at 6 . 67 cm behind the mirror and size of the image is 6 . 67 cm .

Derive the expression for combined focal length of two thin lenses kept in contact.

Let two thin lenses L 1 and L 2 of focal lengths f 1 and f 2 be put in contact. O is a point object at a distance u from the lens L 1 Its image is formed at I after refraction through the two lenses at a distance v from the combination. The lens L1 forms the image of O at I ' , I ' then serves as a virtual object for the lens L 2 which forms a real image at I. Now, we know that by lens formula 1 v - 1 u = 1 f Applying this relation for refraction at the lens L 1 of focal length f 1 can written as 1 v ' - 1 u = 1 f 1 . . . . . . ( 1 ) For refraction at the lens L 2 u = v ' , v = v For this lens 1 v - 1 v ' = 1 f 2 . . . . ( 2 ) And if is the focal length of the combination, then 1 v - 1 u = 1 F . . . . . . ( 3 ) Adding (1) and (2), we 1 v ' - 1 u + 1 v - 1 v ' = 1 f 1 + 1 f 2 1 F = 1 f 1 + 1 f 2

Write the conditions for obtaining total internal reflection. Write its any two applications.

When a ray of light travels from an optically denser medium to an optically rarer medium and is incident at an angle greater than the critical angle for the two media, the ray totally reflects back into the medium, obeying the laws of reflection. This phenomenon is known as the total internal reflection. For total internal reflection to take place (i) light must travel from a denser medium to a rarer medium and (ii) the angle of incidence inside the denser medium must be greater than the critical angle. i.e., ∠ i > ∠ c .

The magnifying power of objective lens of compound microscope is 06 and eye lens is 05 . Find the net magnifying of compound microscope.

Magnification of objective lens = 06 Magnification of eye lens = 05 Magnification of compound microscope = Magnification of objective lens × Magnification of eye lens = 6 × 5 = 30 Magnification of eye lens = 30

Draw a labelled diagram of simple microscope and establish an expression for the magnifying power of simple microscope, when the image formed at ii Least distance of distinct vision.

Expression of magnifying power of a compound microscope is given by: m = V 0 u 0 1 D f e Where V 0 is the image distance from the objective lens D is the least distance of distinct vision f e is the focal length of eye lens.

Draw a labelled diagram of simple microscope and establish an expression for the magnifying power of simple microscope, when the image formed at i Infinity

A simple microscope has a limited maximum magnification (≤ 9) for realistic focal lengths. For much larger magnifications, one uses two lenses, one compounding the effect of the other. This is known as a compound microscope. A schematic diagram of a compound microscope is shown in above figure.The lens nearest the object, called the objective, forms a real, inverted, magnified image of the object. This serves as the object for the second lens, the eyepiece, which functions essentially like a simple microscope or magnifier, produces the final image, which is enlarged and virtual.The first inverted image is thus near (at or within) the focal plane of the eyepiece, at a distance appropriate for final image formationat infinity, or a little closer for image formationat the near point. Clearly, the final image is inverted with respect to the original object. We now obtain the magnification due to a compound microscope. The ray diagram given in figure shows that the (linear) magnification m o due to the objective, namely h ' / h , equals m 0 = h ' h = L f 0 To get magnification m o due to objective as given above, we have used the properties of similar triangles appearing between objective lens and first image as given below. tanβ = h f 0 = h ' L Here h′ is the size of the first image, the object size being h and fo being the focal length of the objective. The first image is formed near the focal point of the eyepiece. The distance L, i.e., the distance between the second focal point of the objective and the first focal point of the eyepiece(focal length f e ) is called the tube length of the compound microscope. As the first inverted image is near the focal point of the eyepiece, we use the magnification expression for the simple microscope to obtain the (angular) magnification m e . When the final image is formed at the near point, is m e = 1 + D f e When the final image is formed at infinity, the angular magnification due to the eyepiece is m e = D f e Thus, the total magnification, when the image is formed at infinity, is m = m o m e = L f o D f e Clearly, to achieve a large magnification of a small object (hence the name microscope), the objective and eyepiece should have small focal lengths.