Taking the electronic charge as 'e' and the permittivity as $$'{\varepsilon _0}'$$. Use dimensional analysis to determine the correct expression for $${\omega _p}$$.

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons N $$ \approx $$ 4 $$ \times $$ 10²⁷ m^-3. Taking $${{\varepsilon _0}}$$ = 10^{- 11} and m $$ \approx $$ 10^{- 30}, where these quantities are in proper SI units.

A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is

A block is moving on an inclined plane making an angle $$45^\circ $$ with the horizontal and the coefficient of friction is $$\mu $$. The force required to just push it up the inclined plane is 3 times the force required to just prevent it from sliding down. If we define N = 10 $$\mu $$, then N is

Four solid spheres each of diameter $$\sqrt 5 $$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N $$ \times $$ 10⁻⁴ kg-m², then N is

5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be T₁, the work done in the process is

Steel wire of lenght ‘L’ at 40^oC is suspended from the ceiling and then a mass ‘m’ is hung from its free end. The wire is cooled down from 40^oC to 30^oC to regain its original length ‘L’. The coefficient of linear thermal expansion of the steel is 10⁻⁵ /^oC, Young’s modulus of steel is 10¹¹ N/m² and radius of the wire is 1 mm. Assume that L >> diameter of the wire. Then the value of ‘m’ in kg is nearly

A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a tall building which reflects the sound waves. The speed of sound in air is 320 m/s. The frequency of the siren heard by the car driver is

Consider an electric field $$\overrightarrow E = {E_0}\widehat x$$ where $${E_0}$$ is a constant. The flux through the shaded area (as shown in the figure) due to this field is

A $$2$$ $$\mu F$$ capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch $$S$$ is turned to position $$2$$ is

A spherical metal shell A of radius $${R_A}$$ and a solid metal sphere $$B$$ of radius $${R_B}\left( { < {R_A}} \right)$$ are kept far apart and each is given charge $$' + Q'.$$ Now they are connected by a thin metal wire. Then

The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 $$\mathop A\limits^o $$. The wavelength of the second spectral line in the Balmer series of singly-ionized helium atom is

A meter bridge is set up as shown, to determine an unknown resistance X using a standard 10 $$\Omega$$ resistor. The galvanometer shows null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm, respectively, for the ends A and B. The determined value of X is

A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius R ( < L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached : (case A). The disc is not free to rotate about its centre and (case B) the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is(are) true?

A composite block is made of slabs A, B, C, D and E of different thermal conductivities (given in terms of a constant K) and sizes (given in terms of length, L) as shown in the figure. All slabs are of same width. Heat Q flows only from left to right through the blocks. Then, in steady-state

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

The phase space diagram for a ball thrown vertically up from ground is

The phase space diagram for simple harmonic motion is a circle centred at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and E₁ and E₂ are the total mechanical energies respectively. Then

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.2 m/s². The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is _________.

Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side a. The surface tension of the soap film is $$\gamma$$. The system of charges and planar film are in equilibrium, and $$a = k{\left[ {{{{q^2}} \over \gamma }} \right]^{1/N}}$$, where k is a constant. Then N is __________.

The activity of a freshly prepared radioactive sample is 10¹⁰ disintegrations per second, whose mean life is 10⁹ s. The mass of an atom of this radioisotope is 10^$$-$$25 kg. The mass (in mg) of the radioactive sample is _________.

A long circular tube of length 10 m and radius 0.3 m carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 $$\Omega$$ and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $$I = {I_0}\cos (300t)$$, where I₀ is constant. If the magnetic moment of the loop is $$N{\mu _0}{I_0}\sin (300t)$$, then N is ___________.