GATE PH kind:question PYQs & Answers — Page 39

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Questions 381–390 of 650. Imported answers require independent review unless marked reviewed.

Topic practice 650 questionsPage 39 of 65

The time derivative of a differentiable function

is added to a Lagrangian

such that

where

are the generalized coordinates, generalized velocities and time, respectively. Let

be the generalized momentum and

the Hamiltonian associated with

. If

and

are those associated with

, then the correct option(s) is(are)

Your answerSelect all correct options. You can choose more than one.

Both

and

satisfy the Euler-Lagrange’s equations of motion

is conserved, then

is necessarily conserved

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A linear charged particle accelerator is driven by an alternating voltage source operating at 10 MHz. Assume that it is used to accelerate electrons. After a few drift-tubes, the electrons attain a velocity 2.9 × 10^8 m s⁻¹. The minimum length of each drift-tube, in m, to accelerate the electrons further (rounded off to one decimal place) is _____.

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The Coulomb energy component in the binding energy of a nucleus is 18.432 MeV. If the radius of the uniform and spherical charge distribution in the nucleus is 3 fm, the corresponding atomic number (rounded off to the nearest integer) is _____.

(Given:

MeV fm)

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For a two-nucleon system in spin singlet state, the spin is represented through the Pauli matrices

for particles 1 and 2, respectively. The value of

(in integer) is _____.

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A contour integral is defined as

where

is a positive integer and

is the closed contour, as shown in the figure, consisting of the line from −100 to 100 and the semicircle traversed in the counter-clockwise sense.

The value of

(in integer) is _____.

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The normalized radial wave function of the second excited state of hydrogen atom is

where

is the Bohr radius and

is the distance from the center of the atom. The distance at which the electron is most likely to be found is

. The value of

(in integer) is _____.

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Consider an atomic gas with number density

m⁻³, in the ground state at 300 K. The valence electronic configuration of atoms is

. The paramagnetic susceptibility of the gas

. The value of

(rounded off to two decimal places) is _____.

(Given: Magnetic permeability of free space

H m⁻¹

Bohr magneton

A m²

Boltzmann constant

J K⁻¹)

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Consider a cross-section of an electromagnet having an air-gap of 5 cm as shown in the figure. It consists of a magnetic material (

) and is driven by a coil having

A, where

is the number of turns and

is the current in Ampere.

Ignoring the fringe fields, the magnitude of the magnetic field

(in Tesla, rounded off to two decimal places) in the air-gap between the magnetic poles is _____.

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The spin

and orbital angular momentum

of an atom precess about

, the total angular momentum.

precesses about an axis fixed by a magnetic field

, where

is a constant. Now the magnetic field is changed to

. Given the orbital angular momentum quantum number

and spin quantum number

is the angle between

and

for the largest possible values of total angular momentum quantum number

and its z-component

. The value of

(in degree, rounded off to the nearest integer) is _____.

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The spin-orbit effect splits the

transition (wavelength,

Å) in Lithium into two lines with separation of

Å. The corresponding positive value of energy difference between the above two lines, in eV, is

. The value of

(rounded off to the nearest integer) is _____.

(Given: Planck’s constant,

eV s

Speed of light,

m s⁻¹)

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