Question

Consider a photon and electron with equal linear momentum and wavelength of the photon is 400 nm, in this case, the maximum velocity of the electron is ______.

1. 720 m/s
2. 1500 m/s
3. 1813 m/s
4. 280 m/s

Answer 1

Correct Answer - Option 3 : 1813 m/s

CONCEPT:

Photon:

• According to Einstein's theory, the light propagates in the bundles of energy, each bundle is being called a photon.
• The energy of a photon is given as,

\(⇒ E=\frac{hc}{λ}\)

• ​The rest mass of the photon is zero. But its effective mass is given as,

\(⇒ m=\frac{h}{cλ}\)

Where h = Planck's constant, c = speed of light, and λ = wavelength

• Photon exerts pressure on the surface.

Linear momentum

• It is defined as the product of the mass of an object, m, and its velocity, v. 
• It is a vector quantity.​

​⇒ P = mv

CALCULATION:

Given λ_P = 400 nm = 4000 Å, m_e = 9.1 × 10^-31 kg and P_e = P_P

• The effective mass of the photon is given as,

\(⇒ m_P=\frac{h}{cλ_P}\)     -----(1)

• The momentum of a photon is given as,

⇒ P_P = m_Pc     -----(2)

By equation 1 and equation 2,

\(⇒ P_P=\frac{h}{λ_P}\)     -----(3)

• The momentum of the electron is given as,

⇒ Pe = me × v     -----(4)

Since the momentum of a photon is equal to the momentum of the electron, So,

⇒ P_e = P_P

\(⇒ P_e=\frac{h}{λ_P}\)     -----(5)

By equation 4 and equation 5,

\(⇒ m_e× v=\frac{h}{λ_P}\)

\(⇒ v=\frac{6.6\times10^{-34}}{9.1\times10^{-31}\times4000\times10^{-10}}\)

⇒ v = 1813 m/s

• Hence, option 3 is correct.