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In a plane EM wave, the electric field oscillates sinusoidally at a frequency of \( 5 \times 10^{10} \, \text{Hz} \) and an amplitude of \( 50 \, \text{V m}^{-1} \). The total average energy density of the electromagnetic field of the wave is:
[Use \( \varepsilon_0 = 8.58 \times 10^{-12} \, \text{C}^2 /\text{N m}^2 \)]

The parameter that remains the same for molecules of all gases at a given temperature is:

The given figure represents two isobaric processes for the same mass of an ideal gas, then:

The refractive index of a prism with apex angle \( A \) is \( \cot \frac{A}{2} \). The angle of minimum deviation is:

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is \( 60 \, \text{cm} \), the length of the closed pipe will be:

A coil is placed perpendicular to a magnetic field of \( 5000 \, \text{T} \). When the field is changed to \( 3000 \, \text{T} \) in \( 2 \, \text{s} \), an induced emf of \( 22 \, \text{V} \) is produced in the coil. If the diameter of the coil is \( 0.02 \, \text{m} \), then the number of turns is the coil is:

A rigid wire consists of a semicircular position of radius \( R \) and two straight sections. The wire is partially immerged in a perpendicular magnetic field \( B = B_0 \hat{k} \) as shown in figure. The magnetic force on the wire if it has current \( i \) is:

When a metal surface is illuminated by light of wavelength \( \lambda \), the stopping potential is \( 8 \,\text{V} \). When the same surface is illuminated by light of wavelength \( 3\lambda \), stopping potential is \( 2 \,\text{V} \). The threshold wavelength for this surface is:

Four identical particles of mass \( m \) are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \( \left( \frac{2\sqrt{2}+1}{32} \right)\frac{G m^2}{L^2} \), the length of the sides of the square is:

Identify the logic operation performed by the given circuit.

If the percentage errors in measuring the length and the diameter of a wire are \( 0.1\% \) each. The percentage error in measuring its resistance will be:

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

A coin is placed on a disc. The coefficient of friction between the coin and the disc is \( \mu \). If the distance of the coin from the center of the disc is \( r \), the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is:

A force is represented by \( F = a x^{2} + b t^{1/2} \) where \( x \) = distance and \( t \) = time. The dimensions of \( \frac{b^{2}}{a} \) are:

Two conductors have the same resistance at \( 0^\circ \text{C} \) but their temperature coefficients of resistance are \( \alpha_{1} \) and \( \alpha_{2} \). The respective temperature coefficients for their series and parallel combinations are:

Two charges \( q \) and \( 3q \) are separated by a distance \( r \) in air. At a distance \( x \) from charge \( q \), the resultant electric field is zero. The value of \( x \) is:

The relation between time \( 't' \) and distance \( 'x' \) is \( t = \alpha x^{2} + \beta x \), where \( \alpha \) and \( \beta \) are constants. The relation between acceleration \( (a) \) and velocity \( (v) \) is:

If the wavelength of the first member of Lyman series of hydrogen is \( \lambda \). The wavelength of the second member will be:

In the given arrangement of a doubly inclined plane two blocks of masses \( M \) and \( m \) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is \( 0.25 \). The value of \( m \), for which \( M = 10 \, \text{kg} \) will move down with an acceleration of \( 2 \, \text{m} \,/ \text{s}^{2} \), is: (take \( g = 10 \, \text{m} \, /\text{s}^{2} \) and \( \tan 37^\circ = \frac{3}{4} \) )

An artillery piece of mass \( M_1 \) fires a shell of mass \( M_2 \) horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

Equivalent resistance of the following network is ________ \( \Omega \).

Two waves of intensity ratio \( 1:9 \) cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is \( I_1 \) (b) Waves are coherent is \( I_2 \) and differ in phase by \( 60^\circ \). If \( \dfrac{I_1}{I_2} = \dfrac{10}{x} \) then \( x = \_\_\_\_\_\_\_\_ \).

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by \( 0.02\% \) is \( \_\_\_ \, \text{m} \). (Take density of sea water \( = 10^{3} \, \text{kg m}^{-3} \). Bulk modulus of rubber \( = 9 \times 10^{8} \, \text{N m}^{-2} \), and \( g = 10 \, \text{m s}^{-2} \) )

A particle performs simple harmonic motion with amplitude \( A \). Its speed is increased to three times at an instant when its displacement is \( \dfrac{2A}{3} \). The new amplitude of motion is \( \dfrac{nA}{3} \). The value of \( n \) is \( \_\_\_\_ \).

A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L$ ($L = l^2$). The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{-7}\text{ H}$, where $x = \text{________}$.

A body starts falling freely from height \( H \) hits an inclined plane in its path at height \( h \). As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of \( \frac{H}{h} \) for which the body will take the maximum time to reach the ground is \( \_\_\_\_\_\_\_\_ \).

The mass defect in a particular reaction is \( 0.4 \, \text{g} \). The amount of energy liberated is \( n \times 10^{7} \, \text{kWh} \), where \( n = \_\_\_\_\_\_\_\_ \). (speed of light \( = 3 \times 10^{8} \, \text{m/s} \))

A parallel plate capacitor with plate separation \( 5 \, \text{mm} \) is charged up by a battery. It is found that on introducing a dielectric sheet of thickness \( 2 \, \text{mm} \), while keeping the battery connections intact, the capacitor draws \( 25\% \) more charge from the battery than before. The dielectric constant of the sheet is \( \_\_\_\_\_\_\_\_ \).

A solid circular disc of mass \( 50 \, \text{kg} \) rolls along a horizontal floor so that its center of mass has a speed of \( 0.4 \, \text{m/s} \). The absolute value of work done on the disc to stop it is:

An electron moves through a uniform magnetic field $\vec{B} = B_0\hat{i} + 2B_0\hat{j}\text{ T}$. At a particular instant of time, the velocity of electron is $\vec{u} = 3\hat{i} + 5\hat{j}\text{ m/s}$. If the magnetic force acting on electron is $\vec{F} = 5e\hat{k}\text{ N}$, where $e$ is the charge of electron, then the value of $B_0$ is ________ $\text{T}$.

JEE Main 2024 31st Jan Morning Shift

Mathematics Physics Chemistry

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The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is \( 60 \, \text{cm} \), the length of the closed pipe will be: