(a) A point-object is placed on the principal axis of a convex spherical surface of radius of curvature R, which separates the two media of refractive indices n₁ and n₂ (n₂ > n₁). Draw the ray diagram and deduce the relation between the distance of the object (u), distance of the image (v) and the radius of curvature (R) for refraction to take place at the convex spherical surface from rarer to denser medium.

(b) Use the above relation to obtain the condition on the position of the object and the radius of curvature in terms of n₁ and n₂ when the real image is formed.

OR

(a) Draw a labelled ray diagram showing the formation of image by a compound microscope in normal adjustment. Derive the expression for its magnifying power.

(b) How does the resolving power of a microscope change when

(i) the diameter of the objective lens is decreased,

(ii) the wavelength of the incident light is increased? Justify your answer in each case.

Three circuits, each consisting of a switch ‘S’ and two capacitors, are initially charged, as shown in the figure. After the switch has been closed, in which circuit will the charge on the left-hand capacitor (i) increase, (ii) decrease and (iii) remain same? Give reasons.

Sunita and her friends visited an exhibition. The policeman asked them to pass through a metal detector. Sunita’s friends were initially scared of it. Sunita, however, explained to them the purpose and working of the metal detector. Answer the following questions:

(a) On what principle does a metal detector work?

(b) Why does the detector emit sound when a person carrying any metallic object walks through it?

(c) State any two qualities which Sunita displayed while explaining the purpose of walking through the detector.

(a) State Faraday’s law of electromagnetic induction.

(b) Explain, with the help of a suitable example, how we can show that Lenz’s law is a consequence of the principle of conservation of energy.

(c) Use the expression for Lorentz force acting on the charge carriers of a conductor to obtain the expression for the induced emf across the conductor of length l moving with velocity v through a magnetic field B acting perpendicular to its length.

OR

(a) Using phasor diagram, derive the expression for the current flowing in an ideal inductor connected to an A.C. source of voltage, v = V_o sin ωt. Hence plot graphs showing variation of (i) applied voltage and (ii) the current as a function of ωt.

(b) Derive an expression for the average power dissipated in a series LCR circuit.

(a) State Gauss’s law in electrostatics. Show, with the help of a suitable example along with the figure, that the outward flux due to a point charge ‘q’, in vacuum within a closed surface, is independent of its size or shape and is given by q/ε_o.

(b) Two parallel uniformly charged infinite plane sheets, ‘1’ and ‘2’, have charge densities + and – respectively. Give the magnitude and direction of the net electric field at a point

(i) in between the two sheets and

(ii) outside near the sheet ‘1’.

OR

(a) Define electrostatic potential at a point. Write its S.I. unit. Three point charges q₁, q₂ and q₃ are kept respectively at points A, B and C as shown in the figure. Derive the expression for the electrostatic potential energy of the system.

(b) Depict the equipotential surfaces due to

(i) an electric dipole,

(ii) two identical positive charges separated by a distance.