Physics Class 12 NCERT Solutions: Chapter 11 Dual Nature of Radiation and Matter Part 8

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The de Broglie Wavelength

The De Broglie Wavelength

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Q: 19. What is the de Broglie wavelength of a nitrogen molecule in air at? Assume that the molecule is moving with the root-mean square speed of molecules at this temperature. (Atomic mass of nitrogen)

Answer:

Temperature of the nitrogen molecule,

Atomic mass of nitrogen

Hence, mass of the nitrogen molecule,

But

Planck’s constant,

Boltzmann constant,

We have the expression that relates mean kinetic energy of the nitrogen molecule with the root mean square speed as:

Hence, the de Broglie wavelength of the nitrogen molecule is given as:

Therefore, the de Broglie wavelength of the nitrogen molecule is.

Q: 20. (A) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be.

(B) Use the same formula you employ in (a) to obtain electron speed for an collector potential of. Do you see what is wrong? In what way is the formula to be modified?

Answer:

(A)Potential difference across the evacuated tube,

Specific charge of an electron,

The speed of each emitted electron is given by the relation for kinetic energy as:

Therefore, the speed of each emitted electron is.

(B)Potential of the anode,

The speed of each electron is given as:

This result is wrong because nothing can move faster than light. In the above formula, the expression for energy can only be used in the non-relativistic limit, i.e., for .

For very high speed problems, relativistic equations must be considered for solving them. In the relativistic limit, the total energy is given as:

Where,

Relativistic mass

Mass of the particle at rest

Kinetic energy is given as: