Q29 of 31 Page 1

(a) A point object ‘O’ is kept in a medium of refractive index n1 in front of a convex spherical surface of radius of curvature R which separates the second medium of refractive index n2 from the first one, as shown in the figure. Draw the ray diagram showing the image formation and deduce the relationship between the object distance and the image distance in terms of , and R.

(b) When the image formed above acts as a virtual object for a concave spherical surface separating the medium from ( > ), draw this ray diagram and write the similar (similar to (a)) relation. Hence obtain the expression for the lens maker’s formula.


(a) Let a spherical surface separate a rarer medium of refractive index n1 from second medium of refractive index n2. Let C be the centre of curvature and R = MC be the radius of the surface.


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Consider a point object O lying on the principle axis of the surface. Let a ray starting from O incident normally on the surface along OM and pass straight. Let another ray of light incident on NM along ON and refract along NI.
From M, draw MN perpendicular to OI. ”
The above figure shows the geometry of formation of image I of an object O and the principal axis of a spherical surface with centre of curvature C. and radius of curvature R.



Let us make the following assumptions :
(i) The aperture of the surface is small as compared to the other distance involved.
(ii) Nil will be taken as nearly equal to the length of the perpendicular from the point N on the principal axis.





For ∆NOC i is the exterior angle,



For small angles.


-------- (i)


Similarly



------ (ii)


By Snell's law:



For small angles,


n1i = n2r


Substituting the value of I from (i) and (ii), we obtain



Or, -------- (iii)


From the Cartesian sign conventions, we get


OM=-uMI=+v.MC=+R


Putting this in above equation, we get:


---------- (iv)


(b)


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Now the image l acts as virtual object for the second surface that will form real image at l. As refraction takes place from denser medium to rarer medium


From equation (iv)






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27

(a) Define mutual inductance and write its S.I. units.

(b) Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other.


(c) In an experiment, two coils and are placed close to each other. Find out the expression for the emf induced in the coil c1 due to a change in the current through the coil.


 28

(a) Using Huygens’s construction of secondary wavelets explain how a diffraction pattern is obtained on a screen due to a narrow slit on which a monochromatic beam of light is incident normally.

(b) Show that the angular width of the first diffraction fringe is half that of the central fringe.


(c) Explain why the maxima at become weaker and weaker with


increasing n.


 30

(a) An electric dipole of dipole moment consists of point charges +q and –q separated by a distance 2a apart. Deduce the expression for the electric field due to the dipole at a distance x from the centre of the dipole on its axial line in terms of the dipole moment. Hence show that in the limit

(b) Given the electric field in the region=, find the net electric flux through the cube and the charge enclosed by it.


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 31

(a) Explain, using suitable diagrams, the difference in the behavior of a

(i) Conductor and (ii) dielectric in the presence of external electric field. Define the terms polarization of a dielectric and write its relation with susceptibility.


(b) A thin metallic spherical shell of radius R carries a charge Q on its surface. A point charge Q/2 is placed at its centre C and another charge +2Q is placed outside the shell at a distance x from the centre as shown in the figure. Find (i) the force on the charge at the centre of shell and at the point A, (ii) the electric flux through the shell.