If \(\epsilon_0\) denotes the permittivity of free space and \(\Phi_{\mathrm{E}}\) is the flux of the electric field through the area bounded by the closed surface, then dimension of \(\left(\epsilon_0 \frac{\mathrm{~d} \phi_{\mathrm{E}}}{\mathrm{dt}}\right)\) are that of :

Dimension of \(\frac{1}{\mu_0 \varepsilon_0}\) should be equal to

An \(AC\) source is connected to an inductance of \(100\,mH\) a capacitance of \(100\,\mu F\) and a resistance of \(120\,\Omega\) as shown in figure. The time in which the resistance having a thermal capacity \(2\,J^{\circ } C\) will get heated by \(16^{\circ} C\) is ..........\(S\)

Mass numbers of two nuclei are in the ratio of \(4: 3\). Their nuclear densities will be in the ratio of

An inductor of inductance \(10\) mH having resistance of \(100\,\Omega\) is connected to battery of E.M.F. \(1.0\) V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is \(2\) mA and \(4\) mA is _______.

A body is projected from the ground at an angle of \(45^{\circ}\) with the horizontal. Its velocity after \(2s\) is \(20 \,ms ^{-1}\). The maximum height reached by the body during its motion is \(m\). (use \(g =10\, ms ^{-2}\) )

Read the following statements: \(A.\) When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of the liquid becomes twice. \(B.\) Two bodies \(P\) and \(Q\) having equal surface areas are maintained at temperature \(10^{\circ}\,C\) and \(20^{\circ}\,C\). The thermal radiation emitted in a given time by \(P\) and \(Q\) are in the ratio \(1: 1.15\) \(C.\) A carnot Engine working between \(100\,K\) and \(400\,K\) has an efficiency of \(75 \%\) \(D.\) When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice. Choose the correct answer from the options given below :

Two simple pendulums of length \(1\, m\) and \(4\, m\) respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to

When 300 J of heat given to an ideal gas with \(C _{ p }=\frac{7}{2} R\) its temperature raises from \(20{ }^{\circ} C\) to \(50^{\circ} C\) keeping its volume constant. The mass of the gas is (approximately) _________ g. \(( R =8.314 J / mol . K )\).

A radioactive sample is undergoing \(\alpha\) decay. At any time \(t_{1}\), its activity is \(A\) and another time \(t _{2}\), the activity is \(\frac{ A }{5}\). What is the average life time for the sample?

Let \(S\) be the sum of all solutions (in radians) of the equation \(\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0\) in \([0,4 \pi]\) Then \(\frac{8 \mathrm{~S}}{\pi}\) is equal to ...... .

A source of sound \(A\) emitting waves of frequency \(1800\,Hz\) is falling towards ground with a terminal speed \(v.\) The observer \(B\) on the ground directly beneath the source receives waves of frequency \(2150\,Hz.\) The source \(A\) receives waves, reflected from ground of frequency nearly ..... \(Hz\) (Speed of sound \(= 343\,m/s\) )

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is