Q: The Inductors of two LR circuits are placed next to each other , as shown in the figure . The values of the self-Inductance of the inductors , resistances , mutual-inductance and applied voltage are specified in the given circuit . After both the switches are closed simultaneously , the total work done by the batteries against the induced EMF in the inductors by the time the currents reach their steady state values is …..

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Ans: (55)

Sol: Mutual inductance is producing flux in same direction as self inductance

$\displaystyle U = \frac{1}{2}L_1 I_1^2 + \frac{1}{2}L_2 I_2^2 + M I_1 I_2 $

Q: A cubical solid aluminium (bulk modulus = -VdP/dV = 70 GPa) block has an edge length 1 m on the surface of earth . It is kept on the floor of a 5 km deep ocean . Taking the average density of water and the acceleration due to gravity to be 10^{3} kg/m^3 and 10 m/s^2 respectively , the change in edge length of the block in mm is …..

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Ans: (0.24)

Sol: $\displaystyle \frac{\Delta V}{V} = -\frac{\Delta p}{B}$ ; Where B = Bulk modulus

V = l^{3}

$\frac{\Delta V}{V} = 3 \frac{\Delta l}{l}$

$\displaystyle 3\frac{\Delta l}{l} = |-\frac{\Delta p}{B}|=\frac{\rho g h}{B} $

$\displaystyle \Delta l = \frac{\rho g h l}{3 B} $

Q: Two capacitors with capacitance values C_{1} = 2000 ± 10 pF and C_{2} = 3000 ± 15 pF are connected in series . The voltage applied across this combination is C = 5.00 ±0.02 V . The percentage error in the calculation of the energy stored in this combination is ………….

Q: A beaker of radius r is filled with water (Refractive index = 4/3) up to height H as shown in the figure on the left . The beaker is kept on a horizontal table rotating with angular speed ω . This makes water surface curved so that the difference in height of water level at the center and at the circumference of the beaker is h (h<< H , h<< r) as shown in the figure on the right . Take this surface to be approximately spherical with a radius of curvature R . Which of the following is/are correct ? (g is the acceleration due to gravity )

(a) $\displaystyle R = \frac{h^2 + r^2}{2h}$

(b) $\displaystyle R = \frac{3 r^2}{2h}$

(c) Apparent depth of the bottom of beaker is close to $\displaystyle v = [\frac{3H}{2}(1 + \frac{\omega^2 H}{2g})^{-1}] $

(d) Apparent depth of the bottom of beaker is close to $\displaystyle v = [\frac{3H}{4}(1 + \frac{\omega^2 H}{4g})^{-1}] $

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Ans: (a,d)

Sol:

$\displaystyle (R-h)^2 + r^2 = R^2 $

$\displaystyle R^2 + h^2 – 2 R h + r^2 = R^2 $

$\displaystyle h^2 + r^2 = 2Rh $

$\displaystyle R = \frac{h^2 + r^2}{2h} $

Since h << r

$\displaystyle R = \frac{r^2}{2h} $

$\displaystyle h = \frac{\omega^2 r^2}{2 g} $

$\displaystyle R = \frac{r^2 2 g}{2\omega^2 r^2} $

Q: A spherical bubble inside water has radius R . Take the pressure inside the bubble and the water pressure to be P_{o} . The bubble now gets compressed radially in an adiabatic manner so that its radius becomes (R-a) . For a << R the magnitude of the work done in the process is given by (4πP_{o}R^{2} a )X , Where X is a constant and γ = C_{p}/C_{v} = 41/30 . The value of X is ……