Question

A photon scatters off of a free electron. (a) What is the maximum possible change in wavelength? (b) Suppose a photon scatters off of a free proton. What is the maximum possible change in wavelength now? (e) Which more clearly demonstrates the particle nature of electromagnetic radiation--collision with an electron or collision with a proton?

Short answer

(a) The maximum possible change in wavelength of the electron is$4.85\times {10}^{-3}\hspace{0.33em}m$ .

(b) The maximum possible change in wavelength of the proton is$2.646\times {10}^{-15}\hspace{0.33em}m$ .

(c) The collision with an electron more clearly demonstrates the particle nature of electromagnetic radiation.

Step-by-step solution

Significance of the change in the wavelength

The change in the wavelength is directly proportional to the Planck’s constant and the angle subtended by a particle. Moreover, the wavelength is inversely proportional to the light’s speed and the mass of an electron.

(a) Determination of the maximum possible change in the wavelength initially

During the scattering, the maximum wavelength change occurs at an angle of .

The equation of the maximum possible change in the wavelength of the electron is expressed as:

${\lambda }^{\text{'}}-\lambda =\frac{h}{m_{e}C}(1-\cos \theta )$

Here, role="math" localid="1657610341456" ${\lambda }^{\text{'}}-\lambda$is the maximum change in the wavelength, $h$is the Planck’s constant, $m_e$is the mass of an electron and $\theta$is the angle subtended by the proton.

Substitute $6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}\text{for}h,9.1\times {10}^{-31}\mathrm{kg}\text{for}{m}_{\mathrm{e}},3\times {10}^8\mathrm{m}/\mathrm{s}\text{for}c\text{and}180°$

for $\theta$in the above equation.

role="math" localid="1657610242005" $$\begin{gathered} {\lambda }^{\text{'}}-\lambda =\frac{6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(9.1\times {10}^{-31}\mathrm{kg}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}\left(1-\cos 180°\right) \\ =\frac{6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(2.73\times {10}^{-22}\mathrm{kg}·\mathrm{m}/\mathrm{s}\right)}(1-(-1\left)\right) \\ =\frac{1.326\times {10}^{-33}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(2.73\times {10}^{-22}\mathrm{kg}·\mathrm{m}/\mathrm{s}\right)} \\ =4.85\times {10}^{-3}\mathrm{m} \end{gathered}$$

Thus, the maximum possible change in wavelength of the electron is$=4.85\times {10}^{-3}\mathrm{m}$ .

(b) Determination of the maximum possible change in the wavelength finally

During the scattering, the maximum wavelength change occurs at an angle of$180°$.

The equation of the maximum possible change in the wavelength of the proton is expressed as:

${\lambda }^{\text{'}}-\lambda =\frac{h}{m_{p}c}(1-\cos \theta )$

Here, ${\lambda }^{\text{'}}-\lambda$is the maximum change in the wavelength, $h$is the Planck’s constant, $m_p$is the mass of an electron and $\theta$is the angle subtended by the proton.

Substitute $6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}\text{for}h,1.67\times {10}^{-27}\mathrm{kg}\text{for}{m}_e,3\times {10}^8\mathrm{m}/\mathrm{s}\text{for}c$and ${180}^{°}$

in the above equation.

$$\begin{gathered} {\lambda }^{\text{'}}-\lambda =\frac{6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(1.67\times {10}^{-27}\mathrm{kg}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}\left(1-\cos 180°\right) \\ =\frac{6.63\times {10}^{-34}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(5.01\times {10}^{-19}\mathrm{kg}·\mathrm{m}/\mathrm{s}\right)}(1-(-1\left)\right) \\ =\frac{1.326\times {10}^{-33}\mathrm{kg}·{\mathrm{m}}^2/\mathrm{s}}{\left(5.01\times {10}^{-19}\mathrm{kg}·\mathrm{m}/\mathrm{s}\right)} \\ =2.646\times {10}^{-15}\mathrm{m} \end{gathered}$$

Thus, the maximum possible change in wavelength of the proton isrole="math" localid="1657610571650" $2.646\times {10}^{-15}\mathrm{m}$

(c) Determination of the particle nature of electromagnetic radiation

The collision with a particular electron more clearly demonstrates the particle nature of electromagnetic radiation as the wavelength change in noticeable. In the wavelength of the proton, there is an insignificant wavelength change. The photon in the x-ray band mainly interacts with other proton as a wave function and also the scattered protons also have a same type of wavelength.

Thus, the collision with an electron more clearly demonstrates the particle nature of electromagnetic radiation.