In a coil of resistance \(8 \Omega\), the magnetic flux due to an external magnetic field varies with time as \(\phi=\frac{2}{3}\left(9- t ^{2}\right)\). The value of total heat produced in the coil, till the flux becomes zero, will be \(....J\)

One main scale division of a vernier caliper is equal to \(m\) units. If \(n^{\text {th }}\) division of main scale coincides with \((n+1)^{\mathrm{th}}\) division of vernier scale, the least count of the vernier caliper is _______.

In an experiment with a closed organ pipe, it is filled with water by \(\left(\frac{1}{5}\right)\) th of its volume. The frequency of the fundamental note will change by

The value of numerical aperature of the objective lens of a microscope is \(1.25\). If light of wavelength \(5000\,\mathop A\limits^o \) is used, the minimum separation between two points, to be seen as distinct, will be....\(\mu m\)

In the given figure, the face \(AC\) of the equilateral prism is immersed in a liquid of refractive index ' \(n\) '. For incident angle \(60^{\circ}\) at the side \(AC\), the refracted light beam just grazes along face \(AC\). The refractive index of the liquid \(n =\frac{\sqrt{ x }}{4}\). The value of \(x...\) is (Given refractive index of glass \(=1.5\) )

A square aluminium (shear modulus is \(25 \times 10^{9}\,Nm ^{-2}\) ) slab of side \(60\,cm\) and thickness \(15\,cm\) is subjected to a shearing force (on its narrow face) of \(18.0 \times 10^{4}\,N\). The lower edge is riveted to the floor. The displacement of the upper edge is \(.......\mu\,m\).

Two capacitors of capacitances \(C\) and \(2\, C\) are charged to potential differences \(V\) and \(2\, V\), respectively. These are then connected in parallel in such a manner that the positive terminal of one is connected to the negative terminal of the other. The final energy of this configuration is\(.....CV^2\)

In the given figure switches \(S_{1}\) and \(S_{2}\) are in open condition. The resistance across \(a b\) when the switches \(S_{1}\) and \(S_{2}\) are closed is \(...\,\Omega\)

A wire of length \(L\) and radius \(r\) is clamped rigidly at one end. When the other end of the wire is pulled by a force \(F\), its length increases by \(5\,cm\). Another wire of the same material of length \(4 L\) and radius \(4\,r\) is pulled by a force \(4\,F\) under same conditions. The increase in length of this wire is \(....cm\).

In case of vertical circular motion of a particle by a thread of length \(r\) if the tension in the thread is zero at an angle \(30^{\circ}\) shown in figure, the velocity at the bottom point (A) of the circular path is ( \(g =\) gravitational acceleration)

An electron of mass \(m_{e}\) and a proton of mass \(m_{p}\) are accelerated through the same potential difference. The ratio of the de-Broglie wavelength associated with the electron to that with the proton is

A particle moves from the point \(\left( {2.0\hat i + 4.0\hat j} \right)\,m\), at \(t = 0\) with an initial velocity \(\left( {5.0\hat i + 4.0\hat j} \right)\,m{s^{ - 1  }}\). It is acted upon by a constant force which produces a constant acceleration \(\left( {4.0\hat i + 4.0\hat j} \right)\,m{s^{ - 2}}\). What is the distance of the particle from the origin at time \(2\,s\)

Determine the charge on the capacitor in the following circuit ................ \(\mu  C\)