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Question Stem

In a circuit, a metal filament lamp is connected in series with a capacitor of capacitance C $$\mu F$$ across a 200 V, 50 Hz supply. The power consumed by the lamp is 500 W while the voltage drop across it is 100 V. Assume that there is no inductive load in the circuit. Take rms values of the voltages. The magnitude of the phase-angle (in degrees) between the current and the supply voltage is $$\psi$$. Assume, $$\pi \sqrt{3} \approx 5$$.

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A special metal 𝑆 conducts electricity without any resistance. A closed wire loop, made of 𝑆, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius 𝑎, with its center at the origin. A magnetic dipole of moment 𝑚 is brought along the axis of this loop from infinity to a point at distance $$r (\gg a)$$ from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is $$\frac{\mu_0}{2 \pi} \frac{m}{r^3}$$, where $$\mu_0$$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, 𝑚1 and 𝑚2, separated by a distance 𝑟 on the common axis, with their north poles facing each other, is $$\frac{k m_1 m_2}{r^4}$$, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question 32

When the dipole 𝑚 is placed at a distance 𝑟 from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

Question 33

The work done in bringing the dipole from infinity to a distance 𝑟 from the center of the loop by the given process is proportional to

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, $$C_V = 2R$$. Here, 𝑅 is the gas constant. Initially, each side has a volume $$V_0$$ and temperature $$T_0$$. The left side has an electric heater, which is turned on at very low power to transfer heat 𝑄 to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to $$V_0/2$$. Consequently, the gas temperatures on the left and the right sides become $$T_L$$ and $$T_R$$, respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Instructions

For the following questions answer them individually

Question 36

In order to measure the internal resistance $$r_1$$ of a cell of emf 𝐸, a meter bridge of wire resistance $$R_0 = 50 Ω$$, a resistance $$R_0/2$$, another cell of emf E/2 (internal resistance 𝑟) and a galvanometer G are used in a circuit, as shown in the figure. If the null point is found at l = 72 cm, then the value of $$r_1 = ___ Ω$$.

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Question 37

The distance between two stars of masses $$3 M_S$$ and $$6 M_S$$ is 9𝑅. Here 𝑅 is the mean distance between the centers of the Earth and the Sun, and $$M_S$$ is the mass of the Sun. The two stars orbit around their common center of mass in circular orbits with period 𝑛𝑇, where 𝑇 is the period of Earth’s revolution around the Sun.

The value of n is ___.

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Question 38

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals 𝑃, 𝑄 and 𝑅 are $$E_P, E_Q$$ and $$E_R$$, respectively, and they are related by $$E_P = 2 E_Q = 2 E_R$$. In this experiment, the same source of monochromatic light is used for metals 𝑃 and 𝑄 while a different source of monochromatic light is used for the metal 𝑅. The work functions for metals 𝑃, 𝑄 and 𝑅 are 4.0 eV, 4.5 eV and 5.5 eV, respectively. The energy of the incident photon used for metal 𝑅, in eV, is ___.

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