A plane electromagnetic wave is propagating along the direction \(\frac{\hat{i}+\hat{j}}{\sqrt{2}},\) with its polarization along the direction \(\hat{\mathrm{k}}\). The correct form of the magnetic field of the wave would be (here \(\mathrm{B}_{0}\) is an appropriate constant)

A potentiometer \(PQ\) is set up to compare two resistances as shown in the figure. The ameter \(A\) in the circuit reads \(1.0\, A\) when two way key \(K_3\) is open. The balance point is at a length \(l_1\, cm\) from \(P\) when two way key \(K_3\) is plugged in between \(2\) and \(1\) , while the balance point is at a length \(l_2\, cm\) from \(P\) when key \(K_3\) is plugged in between \(3\) and \(1\) . The ratio of two resistances \(\frac{{{R_1}}}{{{R_2}}}\) is found to be

A particle of mass \(m\) moves in a circular orbit in a central potential field \(U ( r )= U _{0} r ^{4}\). If Bohr's quantization conditions are applied, radii of possible orbitals \(r _{ n }\) vary with \(n ^{1 / \alpha}\), where \(\alpha\) is ....... .

The electric field in an electromagnetic wave is given by \({E}=\left(50\, {NC}^{-1}\right) \sin \omega({t}-{x} / {c})\) The energy contained in a cylinder of volume \({V}\) is \(5.5 \times 10^{-12} \, {J}\). The value of \({V}\) is \(......{cm}^{3}\) \(\left(\right.\) given \(\left.\in_{0}=8.8 \times 10^{-12} \,{C}^{2} {N}^{-1} {m}^{-2}\right)\)

In the figure given below. a block of mass \(M =490\,g\) placed on a frictionless table is connected with two springs having same spring constant \(\left( K =2 N m ^{-1}\right)\). If the block is horizontally displaced through ' \(X\) 'm then the number of complete oscillations it will make in \(14 \pi\) seconds will be \(.........\)

The critical angle for a denser-rarer interface is \(45^{\circ}\). The speed of light in rarer medium is \(3 \times 10^8\) ms. The speed of light in the denser medium is:

Two trains ' \(A\) ' and ' \(B\) ' of length ' \(l\) ' and ' \(4 l\) ' are travelling into a tunnel of length ' \(L\) ' in parallel tracks from opposite directions with velocities \(108\,km / h\) and \(72\,km / h\), respectively. If train '\(A\)' takes \(35\,s\) less time than train ' \(B\) ' to cross the tunnel then, length ' \(L\) ' of tunnel is \(...........\,m\) \(\text { (Given } L=60 l \text { ) }\)

A piece of wood of mass \(0.03\, kg\) is dropped from the top of a \(100\, m\) height building. At the same time, a bullet of mass \(0.02\, kg\) is fired vertically upward, with a velocity \(100\, ms^{- 1}\), from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ \(m\). \((g = 10\, ms^{-2})\)

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

An electric bulb rated as \(100 \mathrm{~W}-220 \mathrm{~V}\) is connected to an ac source of rms voltage 220 V. The peak value of current through the bulb is :