31.

A heating element of resistance r is fitted inside an adiabatic cylinder which carries a frictionless piston of mass m and cross-section A as shown in diagram. The cylinder contains one mole of an ideal diatomic gas. The piston current flows through the element such that the temperatures rises with time t as ΔT = αt + $\frac{1}{2}$ βt² ? (α and β are constants), while pressure remains constant. The atmospheric pressure above the piston is P₀. Then

      

• he rate of increase in internal energy is $\frac{5}{2} \mathrm{R} \left(\mathrm{\alpha } + \mathrm{\beta t}\right)$

• the current flowing in the element is $\sqrt{\frac{5}{2\mathrm{r}} \mathrm{R} \left(\mathrm{\alpha } + \mathrm{\beta t}\right)}$

• the piston moves upwards with constant acceleration

• the piston moves upwards with constant speed

Three capacitors 3 µF, 6 µF and 6 µF are connected in series to a source of 120 V. The potential difference, in volt, across the 3 µF capacitor will be

• 24

• 30

• 40

• 60

A galvanometer having internal resistance 10 Ω requires 0.01 A for a full scale deflection. To convert this galvanometer to a voltmeter of full-scale deflection at 120 V, we need to connect a resistance of

• 11990 Ω in series

• 11990 Ω in parallel

• 12010 Ω in series

• 12010 Ω in parallel

A parallel plate capacitor is charged and then disconnected from the charging battery. If the plates are now moved farther apart by pulling at them by means of insulating handles, then

• the energy stored in the capacitor decreases

• the capacitance of the capacitor increases

• the charge on the capacitor decreases

• the voltage across the capacitor increases

An electron in a circular orbit ofradius 0.05 nm performs 10¹⁶ revolutions per second. The magnetic moment due to this rotation of electron is (in Am²)

• 2.16 × 10^-23

• 3.21 × 10^-22

• 3.21 × 10^-24

• 1.26 × 10^-23

A very small circular loop of radius a is initially (at t = 0) coplanar and concentric with a much larger fixed circular loop of radius b. A constant current I flows in the larger loop. The smaller loop is rotated with a constant angular speed ω about the common diameter. The emf induced in the smaller loop as a function of time t is

• $\frac{{\mathrm{\pi a}}^2{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}} \mathrm{\omega } \cos \left(\mathrm{\omega t}\right)$

• $\frac{{\mathrm{\pi a}}^2{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}} \mathrm{\omega } \sin \left({\mathrm{\omega }}^2{\mathrm{t}}^2\right)$

• $\frac{{\mathrm{\pi a}}^2{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}} \mathrm{\omega } \sin \left(\mathrm{\omega t}\right)$

• $\frac{{\mathrm{\pi a}}^2{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{b}} \mathrm{\omega } {\sin }^2 \left(\mathrm{\omega t}\right)$

An infinite sheet carrying a uniform surface charge density σ lies on the xy-plane. The work done to carry a charge q from the point $\mathrm{A} = \mathrm{a}(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}})$ to the point $\mathrm{B}\hspace{0.17em}= \mathrm{a}(\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 6\hat{\mathrm{k}})$ (where a is a constant with the dimension of length and ε₀ is the permittivity of free space) is

• $\frac{3\mathrm{\sigma aq}}{2{\mathrm{\epsilon }}_0}$

• $\frac{2\mathrm{\sigma aq}}{{\mathrm{\epsilon }}_0}$

• $\frac{5\mathrm{\sigma aq}}{2{\mathrm{\epsilon }}_0}$

• $\frac{3\mathrm{\sigma aq}}{{\mathrm{\epsilon }}_0}$

The intensity of magnetization of a bar magnet is 5.0 x 10⁴ Am^-1.The magnetic length and the area of cross section of the magnet are 12 cm and 1cm² respectively. The magnitude of magnetic moment of this bar magnet is (in SI unit)

• 0.6

• 1.3

• 1.24

• 2.4

A proton of mass m and charge q is moving in a plane with kinetic energy E. If there exists a uniform magnetic field B, perpendicular to the plane of the motion, the proton will move in a circular path of radius

• $\frac{2\mathrm{Em}}{\mathrm{qB}}$

• $\frac{\sqrt{2\mathrm{Em}}}{\mathrm{qB}}$

• $\frac{\sqrt{\mathrm{Em}}}{2\mathrm{qB}}$

• $\sqrt{\frac{2\mathrm{Eq}}{\mathrm{mB}}}$

In which of the following phenomena, the heat waves travel along straight lines with the speed of light ?

• Thermal conduction

• Forced convection

• Natural convection

• Thermal radiation