A boy is rolling a \(0.5\, kg\) ball on the frictionless floor with the speed of \(20\, ms ^{-1}\). The ball gets deflected by an obstacle on the way. After deflection it moves with \(5 \%\) of its initial kinetic energy. What is the speed of the ball now ? (in \(ms ^{-1}\))

A slab of dielectric constant \(K\) has the same crosssectional area as the plates of a parallel plate capacitor and thickness \(\frac{3}{4}\,d\), where \(d\) is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be.(Given \(C _{0}=\) capacitance of capacitor with air as medium between plates.)

A series \(LCR\) circuit is connected to an ac source of \(220\,V , 50 Hz\). The circuit contain a resistance \(R=100 \Omega\) and an inductor of inductive reactance \(X_L=79.6 \Omega\). The capacitance of the capacitor needed to maximize the average rate at which energy is supplied will be \(..........\mu F\).

A particle is moving with speed \(v= b\sqrt x\) along positive \(x-\) axis. Calculate the speed of the particle at time \(t = \tau\) (assume that the particle is at origin at \(t = 0\) ).

In a series \(LCR\) circuit, the inductance, capacitance and resistance are \(L =100\, mH , C =100 \mu F\) and \(R =\) \(10 \Omega\) respectively. They are connected to an \(AC\) source of voltage \(220\, V\) and frequency of \(50\, Hz\). The approximate value of current in the circuit will be.....\(A.\)

Charges \(-q\) and \(+ q\) located at \(A\) and \(B\), respectively, constitute an electric dipole. Distance \(AB = 2a\), \(O\) is the mid point of the dipole and \(OP\) is perpendicular to \(AB\). A charge \(Q\) is placed at \(P\) where \(OP = y\) and \(y >> 2a\). The charge \(Q\) experiences an electrostatic force \(F\). If \(Q\) is now moved along the equatorial line to \(P'\) such that \(OP' = \frac{y}{3}\), the force on \(Q\) will be close to: \(\left( {\frac{y}{3} >  > 2a} \right)\)

A proton with a kinetic energy of \(2.0\,eV\) moves into a region of uniform magnetic field of magnitude \(\frac{\pi}{2} \times 10^{-3}\,T\). The angle between the direction of magnetic field and velocity of proton is \(60^{\circ}\). The pitch of the helical path taken by the proton is \(..........cm\) (Take, mass of proton \(=1.6 \times 10^{-27}\,kg\) and Charge on proton \(=1.6 \times 10^{-19}\,kg)\)

A galaxy is moving away from the earth at a speed of \(286\, kms ^{-1}\). The shift in the wavelength of a red line at \(630\, nm\) is \(x \times 10^{-10}\, m\). The value of \(x ,\) to the nearest integer, is........ [Take the value of speed of light \(c,\) as \(3 \times 10^{8}\) \(\left. ms ^{-1}\right]\)

A moving coil galvanometer has \(100\) turns and each turn has an area of \(2.0 \mathrm{~cm}^2\). The magnetic field produced by the magnet is \(0.01 \mathrm{~T}\) and the deflection in the coil is \(0.05\) radian when a current of \(10 \mathrm{~mA}\) is passed through it. The torsional constant of the suspension wire is \(\mathrm{x} \times 10^{-5} \mathrm{~N}-\mathrm{m} / \mathrm{rad}\). The value of \(\mathrm{x}\) is _______.

Two beams of light having intensities \(I\) and \(4\,I\) interfere to produce a fringe pattern on a screen. The phase difference between the two beams are \(\pi / 2\) and \(\pi / 3\) at points \(A\) and \(B\) respectively. The difference between the resultant intensities at the two points is \(x I\). The value of \(x\) will be.

An electric field is given by \((6 \hat{i}+5 \hat{j}+3 \hat{k}) \ N / C\). The electric flux through a surface area \(30 \hat{\mathrm{i}}\; m^2\) lying in \(YZ-\)plane (in SI unit) is _______.