Dual Nature of Radiation and Matter Test - 67

Statement I If the accelerating potential in a X-ray tube is increased, the wavelengths of the characteristic X-rays do not change
Statement II When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy

Statement I When ultraviolet light is incident on a photocell, its stopping potential is V₀ and the maximum kinetic energy of the photoelectrons is K_max. When the ultraviolet light is replaced by X-rays, both V₀ and K_max increase.
Statement II Photoelectrons are emitted with speeds ranging from zero to a maximum value because of the range of frequencies present in the incident light.

Statement I A metallic surface is irradiated by a monochromatic light of frequency v > v₀ (the threshold frequency). The maximum kinetic energy and the stopping potential are K_max and V₀ respectively. If the frequency incident on the surface is doubled, both the K_max and V₀ are also doubled.
Statement II The maximum kinetic energy and the stopping potential of photoelectrons emitted from a surface are linearly dependent on the frequency of incident light.

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electrons waves reflected from the planes of atoms in a crystal interfere constructively (see figure).
If a strong diffraction peak is observed when electrons are incident at an angle \'1\' from the normal to the crystal planes with distance \'d\' between them (see figure), de-Broglie wavelength λ_dB of electrons can be calculated by the relationship (n is an integer)

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electrons waves reflected from the planes of atoms in a crystal interfere constructively (see figure).
Electrons accelerated by potential V are diffracted from a crystal. If d = 1 Å and i = 30° , V should be about
(h = 6.6 × 10^–34 Js, m_e = 9.1 × 10^–31kg, e = 1.6 × 10^–19C)

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \'N\' be the number density of free electrons, each of mass \'m\'. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ‘ω_p\', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency ω, where a part of the energy is absorbed and a part of it is reflected. As ω approaches ω_p all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Taking the electronic charge as ‘e\' and the permittivity as \'ε₀\', use dimensional analysis to determine the correct expression for ω_p

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let \'N\' be the number density of free electrons, each of mass \'m\'. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency ‘ω_p\', which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency ω, where a part of the energy is absorbed and a part of it is reflected. As ω approaches ω_p all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.
Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons N = 4 × 10²⁷ m^–3. Take ε⁰ = 10^–11 and m = 10^–30, where these quantities are in proper SI units

A silver sphere of radius 1 cm and work function 4.7 eV is suspended from an insulating thread in free-space. It is under continuous illumination of 200 nm wavelength light. As photoelectrons are emitted, the sphere gets charged and acquires a potential. The maximum number of photoelectrons emitted from the sphere is A × 10^Z (where 1 < A < 10). The value of ‘Z’ is