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 :

A particle is moving in a circle of radius \(50 \mathrm{~cm}\) in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at \(t=0\) is \(4 \mathrm{~m} / \mathrm{s}\), the time taken to complete the first revolution will be \(\frac{1}{\alpha}\left[1-\mathrm{e}^{-2 \pi}\right] \mathrm{s}\), where \(\alpha=\) _______.

The time period of simple harmonic motion of mass \(\mathrm{M}\) in the given figure is \(\pi \sqrt{\frac{\alpha M}{5 K}}\), where the value of \(\alpha\) is _______.

Unpolarized light of intensity \(I\) passes through an ideal polarizer \(A\) Another indentical polarizer \(B\) is placed behind \(A\) The intensity of light beyond \(B\) is found to be \(\frac{I}{2}\) Now another identical polarizer \(C\) is placed between \(A\) and \(B\). The intensity beyond \(B\) is now found to be \(\frac{I}{8}\) The angle between polarizer \(A\) and \(C\) is:......\(^o\)

In Young's double slit experiment, the distance between slits and the screen is \(1 .0\,\,m\) and monochromatic light of \(600\,\,nm\)  is being used. A person standing near the slits is looking at the fringe pattern. When the separation between the slits is varied, the interference pattern disappears for a particular distance \(d_0\) between the slits . If the angular resolution of the eye is \(\frac {1}{60}^o,\) the value of \(d_0\)  is close to......\(mm\)

A driver in a car, approaching a vertical wall notices that the frequency of his car horn, has changed from \(440\, Hz\) to \(480 \,Hz ,\) when it gets reflected from the wall. If the speed of sound in air is \(345 \,m / s ,\) then the speed of the car is \(.......km/hr\)

Following plots show Magnetization \((M)\) \(vs\) Magnetising field \((H)\) and Magnetic susceptibility \((\chi)\) vs temperature \((T)\) graph : Which of the following combination will be represented by a diamagnetic material?

Four identical solid spheres each of mass \('m'\) and radius \('a'\) are placed with their centres on the four corners of a square of side \('b'\). The moment of inertia of the system about one side of square where the axis of rotation is parallel to the plane of the square is

Two satellites \(A\) and \(B\) of masses \(200\, kg\) and \(400\, kg\) are revolving round the earth at height of \(600\, km\) and \(1600\, km\) respectively. If \(T _{ A }\) and \(T _{ B }\) are the time periods of \(A\) and \(B\) respectively then the value of \(T _{ B }- T _{ A }\) [Given : radius of earth \(=6400\, km\), mass of earth \(=6 \times 10^{24}\, kg\) ]

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)

The output of the given logic circuit is

A capacitor of capacitance \(50 \; pF\) is charged by \(100 \; V\) source. It is then connected to another uncharged identical capacitor. Electrostatic energy loss in the process is \(\dots \; nJ\).