Physics Class 12 NCERT Solutions: Chapter 11 Dual Nature of Radiation and Matter Part 7

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de Broglie wave equation

De Broglie Wave Equation

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Q: 17. (A) For what kinetic energy of a neutron will the associated de Broglie wavelength be?

(B) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter, having an average kinetic energy of at .

Answer:

(A) De Broglie wavelength of the neutron,

Mass of a neutron,

Planck’s constant,

Kinetic energy (K) and velocity (v) are related as:

De Broglie wavelength and velocity are related as:

Using equation in equation, we get:

Hence, the kinetic energy of the neutron is.

(B) Temperature of the neutron,

Boltzmann constant,

Average kinetic energy of the neutron:

The relation for the de Broglie wavelength is given as:

Where

Therefore, the de Broglie wavelength of the neutron is.

Q: 18. Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Answer:

The momentum of a photon having energy is given as:

Where,

Wavelength of the electromagnetic radiation

Speed of light

Planck’s constant

De Broglie wavelength of the photon is given as:

Where,

Mass of the photon

Velocity of the photon

Hence, it can be inferred from equations (i) and (ii) that the wavelength of the electromagnetic radiation is equal to the de Broglie wavelength of the photon.