The earth's magnetic field lines resemble that of a dipole at the centre of the earth. If the magnetic moment of this dipole is close to \(8 \times 10^{22}\, Am^2\), the value of earth 's magnetic field near the equator is close to....\(Gauss\) (radius of the earth \(= 6.4 \times 10^6\, m\))

Water falls from a height of 200 m into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool.
\(\left(\right.\)Take \(g=10 \mathrm{~m} / \mathrm{s}^2\), specific heat of water \(\left.=4200 \mathrm{~J} /(\mathrm{kg} \mathrm{K})\right)\)

Given below are two statements : Statement-I: When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be \(0^{\circ}\). Statement-II: The contact angle between a solid and a liquid is a property of the material of the solid and liquid as well : In the light of above statement, choose the correct answer from the options given below.

Three vectors \(\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}\) and \(\overrightarrow{\mathrm{OR}}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is _______.

A air bubble of radius \(1\,cm\) in water has an upward acceleration \(9.8\, cm\, s ^{-2}\). The density of water is \(1\, gm\, cm ^{-3}\) and water offers negligible drag force on the bubble. The mass of the bubble is\(.......gm\) \(\left( g =980 \,cm / s ^{2}\right)\)

The energy stored in the electric field produced by a metal sphere is \(4.5\, J\). lf the sphere contains \(4\,\mu C\) charge, its radius will be.......\(mm\) : [Take : \(\frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}\,N - {m^2}\,/{C^2}\, ]\)

The ratio of vapour densities of two gases at the same temperature is \(\frac{4}{25}\), then the ratio of r.m.s. velocities will be:

A block of mass \(0.1\, kg\) is connected to an elastic spring of spring constant \(640\, Nm^{-1}\) and oscillates in a damping medium of damping constant \(10^{-2}\, kg\,s^{-1}\) . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to ..... \(s\)

The half-life of \(Au ^{198}\) is \(2.7\, days.\) The activity of \(1.50\, mg\) of \(Au ^{198}\) if its atomic weight is \(198\, g mol ^{-1}\) is .......\(Ci\) \(\left( N _{ A }=6 \times 10^{23}\, / mol \right)\)

A mass of \(10\; kg\) is suspended by a rope of length \(4 \;\mathrm{m},\) from the ceiling. A force \(\mathrm{F}\) is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of \(45^{\circ}\) with the vertical. Then \(\mathrm{F}\) equals ........... \(N\) (Take \(\mathrm{g}=10 \;\mathrm{ms}^{-2}\) and the rope to be massless)

When a ball is dropped into a lake from a height \(4.9 \,m\) above the water level, it hits the water with a velocity \(v\) and then sinks to the bottom with the constant velocity \(v\). It reaches the bottom of the lake \(4.0,s\) after it is dropped. The approximate depth of the lake is ............ \(m\)