The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
\(E_y=20 \sin \left(3 \times 10^6 x-4.5 \times 10^{14} t\right) V / m\)
(where \(x, t\) and other values have S.I. units). The dielectric constant of the medium is __________ .
(speed of light in free space is \(3 \times 10^8 m / s\) )

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is \(I_{1}\). The same rod is bent into a ring and its moment of inertia about a diameter is \(I _{2}\). If \(\frac{ I _{1}}{ I _{2}}\) is \(\frac{ x \pi^{2}}{3}\), then the value of \(x\) will be ...............

The equivalent resistance between \(A\) and \(B\) as shown in figure is \(........\,k\Omega\)

Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be

When heat \(Q\) is supplied to a diatomic gas of rigid molecules at constant volume its temperature increases by \(\Delta T.\) The heat required to produce the same change in temperature, at constant pressure is

In a meter bridge experiment the balance point in obtained if the gaps are closed by \(2\,\Omega\) and \(3\,\Omega\). A shunt of \(x\,\Omega\) is added to \(3\,\Omega\) resistor to shift the balancing point by \(22.5\,cm\). The value of \(X\) is \(................\)

A particle experiences a variable force \(\overrightarrow{ F }=\left(4 x \hat{ i }+3 y ^{2} \hat{ j }\right)\) in a horizontal \(x - y\) plane. Assume distance in meters and force is newton. If the particle moves from point \((1,2)\) to point \((2,3)\) in the \(x-y\) plane, the Kinetic Energy changes by............\(j\)

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)