A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness, $$\delta = {d \over N}$$. The dielectric constant of the m^th layer is $${K_m} = K\left( {1 + {m \over N}} \right)$$. For a very large N(>10³), the capacitance C is $$\alpha \left( {{{K{\varepsilon _0}A} \over {d\ln 2}}} \right)$$

The value of $$\alpha $$ will be ..................

[$$ \in $$₀ is the permittivity of free space.]

A planar structure of length L and width W is made of two different optical media of refractive indices n₁ = 1.5 and n₂ = 1.44 as shown in figure. If L >> W, a ray entering from end AB will emerge from end CD. CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle $$\theta $$ is varied, the maximum time taken by a ray to exit the plane CD is t $$ \times $$ 10^-9 s, where, t is ................

[Speed of light, c = 3 $$ \times $$ 10⁸ m/s]

A block of weight 100 N is suspended by copper and steel wires of same cross-sectional area 0.5 cm² and length $$\sqrt 3 $$ m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30$$^\circ $$ and 60$$^\circ $$, respectively. If elongation in copper wire is ($$\Delta {l_c}$$) and elongation in steel wire is ($$\Delta {l_s}$$), then the ratio $${{\Delta {l_c}} \over {\Delta {l_s}}}$$ is .............. .

[Young's modulus for copper and steel are 1 $$ \times $$ 10¹¹ N/m² and 2 $$ \times $$ 10¹¹ N/m² respectively.]

A liquid at 30$$^\circ $$C is poured very slowly into a Calorimeter that is at temperature of 110$$^\circ $$C. The boiling temperature of the liquid is 80$$^\circ $$C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be 50$$^\circ $$C. The ratio of the latent heat of the liquid to its specific heat will be ...........$$^\circ $$C.

[Neglect the heat exchange with surrounding]