.. CP An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than that of light. (a) Express the energy of a photon in terms of the kinetic energy Kof one of the electrons. (b) Which is greater, the energy of a photon or the kinetic energy of an electron?

Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. The higher the photon's frequency, the higher its energy.

From equation 36.2, when a beam of Wavelength A is diffracted from a single slit of width (1, the condition of a dark fringe is:

Where am is the Angle of line from centre of slit to mth dark fringe on screen.

From equation 39.], the de Broglie wavelength of a particle of momentum p is given by:

From equation 582, the energy of a photon of wavelength A is given by:

\

Calculations

Since both beams are diffracted through the same slit (the same width (1) and the first dark fringe of both beams occurs at

the same anole 01-

Calculations

Since both beams are diffracted through the same slit (the same width (1) and the first dark fringe of both beams occurs at

the same angle 91-

From equation (1), both beams must have the same Wavelength:

Substituting for A6 into equation (2), we get the momentum of the electrons;

Since the speed of the electrons is much SIOWer than that of light, so We use the non relativistic equations.

The nonrelativistic momentum is given by:

And the nonrelativistic kinetic energy is given by:

Combining these tWO equations, we get:

\

Now, we substitute for p and me, so we get:

Solving for A6, we get:

This is also the wavelength of the photons of light, so we substitute for this wavelength into equation (3), so We get the

energy of the photon:

therefore the energy of the photon in terms of kinetic energy is

(b) Let us compare the energy of the photons and the kinetic energy of the electrons as follows:

Substituting for K , from eqaution (4), we get:

;

Therefore, the energy of the photons is much larger than the kinetic energy of the electrons-