A prism of angle \(75^{\circ}\) and refractive index \(\sqrt{3}\) is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ___________ .
\(\left(\sin 15^{\circ}=0.25\right.\) and \(\left.\sin 25^{\circ}=0.43\right)\)

One end of a straight uniform \(1\; \mathrm{m}\) long bar is pivoted on horizontal table. It is released from rest when it makes an angle \(30^{\circ}\) from the horizontal (see figure). Its angular speed when it hits the table is given as \(\sqrt{\mathrm{n}}\; \mathrm{s}^{-1},\) where \(\mathrm{n}\) is an integer. The value of \(n\) is

The parallel combination of two air filled parallel plate capacitors of capacitance \(C\) and \(nC\) is connected to a battery of voltage, \(V\). When the capacitor are fully charged, the battery is removed and after that a dielectric material of dielectric constant \(K\) is placed between the two plates of the first capacitor. The new potential difference of the combined system is

An ideal gas is enclosed in a cylinder at pressure of \(2\,atm\) and temperature, \(300\,K.\) The mean time between two successive collisions is \(6 \times 10^{-8}\, s.\) If the pressure is doubled and temperature is increased to \(500\,K,\) the mean time between two successive collisions will be close to

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 \(...\,\Omega\)

The equivalent resistance of the given circuit between the terminals \({A}\) and \({B}\) is  \(....\Omega\)

A charge \(q\)  is spread uniformly over an insulated loop of radius \(r\) . If it is rotated with an angular velocity \(\omega \) with respect to normal axis then the magnetic moment of the loop is

As shown in the figure, two infinitely long, identical wires are bent by \(90^o\) and placed in such a way that the segments \(LP\) and \(QM\) are along the \(x-\) axis, while segments \(PS\) and \(QN\) are parallel to the \(y-\) axis. If \(OP = OQ = 4\, cm\), and the magnitude of the magnetic field at \(O\) is \(10^{-4}\, T\), and the two wires carry equal current (see figure), the magnitude of the current in each wire and the direction of the magnetic field at \(O\) will be \((\mu_ 0 = 4\pi \times10^{-7}\, NA^{-2})\)

A transmitting station releases waves of wavelength \(960\, m\). A capacitor of \(2.56\, \mu F\) is used in the resonant circuit. The self inductance of coil necessary for resonance is ............. \(-\times 10^{-8} H\)

A parallel plate capacitor is having separation between plates \(0.885\) mm. It has a capacitance of \(1\,\mu\)F when the space between the plates is filled with an insulating material of resistivity \(1\times 10^{13}\,\Omega\)m and resistance \(17.7\times 10^{14}\,\Omega\). Relative permittivity of the insulating material is \(\alpha\times 10^7\). The value of \(\alpha\) is ________. (Take permittivity of free space \(=8.85\times 10^{-12}\) F/m)