The electric field in an electromagnetic wave is given by \(E =56.5 \sin \omega( t - x / c ) \;NC ^{-1}\). Find the intensity of the wave if it is propagating along \(x-\)axis in the free space. (Given \(\left.\varepsilon_{0}=8.85 \times 10^{-12} \;C ^{2} N ^{-1} m ^{-2}\right)\)

A parallel plate capacitor is formed by two plates each of area \(30 \pi\, cm ^{2}\) separated by \(1\, mm\). A material of dielectric strength \(3.6 \times 10^{7} \,Vm ^{-1}\) is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is \(7 \times 10^{-6}\, C\), the value of dielectric constant of the material is \(\left\{ Use : \frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right\}\)

Two blocks (\(P\) and \(Q\)) with respectively masses \(2\) kg and \(1.5\) kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube (\(S\)), as shown in the figure below. Block \(P\) is positioned on the top surface which has no friction and block \(Q\) is in contact with side-surface, having coefficient friction \(\mu\). The cube (\(S\)) moves towards the right with acceleration of \(\dfrac{g}{2}\), where \(g\) is gravitational acceleration. During this movement the block \(P\) and \(Q\) remain stationary. The value of \(\mu\) is _______. (take \(g = 10\) m/s\(^2\))

A metallic sphere cools from \(50^{\circ} C\) to \(40^{\circ} C\) in \(300 \,s.\) If atmospheric temperature around is \(20^{\circ} C ,\) then the sphere's temperature after the next \(5\) minutes will be close to\(.....C\)

A block '\(A\)' takes \(2\,s\) to slide down a frictionless incline of \(30^{\circ}\) and length ' \(l\) ', kept inside a lift going up with uniform velocity ' \(v\) '. If the incline is changed to \(45^{\circ}\), the time taken by the block, to slide down the incline, will be approximately \(........\,s\)

The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is _______.

A block of mass \(5\,kg\) is placed at rest on a table of rough surface. Now, if a force of \(30\,N\) is applied in the direction parallel to surface of the table, the block slides through a distance of \(50\,m\) in an interval of time \(10\,s\). Coefficient of kinetic friction is (given, \(g =10\,ms ^{-2}\))

Let \(\left\{a_k\right\}\) and \(\left\{b_k\right\}, k \in N\), be two G.P.s with common ratio \(r_1\) and \(r_2\) respectively such that \(a_1=b_1=4\) and \(r_1 < r_2\). Let \(c_k=a_k+k, \in N\). If \(c_2=5\) and \(c_3=13 / 4\) then \(\sum \limits_{k=1}^{\infty} c_k - \left(12 a _6+8 b _4\right)\) is equal to

A cubical block of side \(30\,cm\)  is moving with  velocity \(2\,\,ms^{-1}\) on a smooth horizontal surface. The surface has a bump at a point \(O\) as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is

Water flows in a horizontal tube (see figure). The pressure of water changes by \(700\; \mathrm{Nm}^{-2}\) between \(\mathrm{A}\) and \(\mathrm{B}\) where the area of cross section are \(40\; \mathrm{cm}^{2}\) and \(20\; \mathrm{cm}^{2},\) respectively. Find the rate of flow of water through the tube. ........ \(\mathrm{cm}^{3} / \mathrm{s}\) (density of water \(=1000\; \mathrm{kgm}^{-3}\) )

Let \(\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, - \,1),\) where the function \(f\) satisfies \(f(x + y) = f(x) f(y)\) for all natural numbers \(x, y\) and \(f(1) = 2.\) Then the natural number \(‘ a '\) is

\(ABC\) is a triangular park with \(AB = AC = 100\) \(metres\). A vertical tower is situated at the mid-point of \(BC\). If the angles of elevation of the top of the tower at \(A\) and \(B\) are \({\cot ^{ - 1}}\left( {3\sqrt 2 } \right)\) and \(\cos e{c^{ - 1}}\left( {2\sqrt 2 } \right)\) respectively, then the height of the tower (in metres) is

The efficiency of an ideal heat engine working between the freezing point and boiling point of water, is ........ \(\%\)