Question 10 of 10intermediate💡 UnderstandLong Answer3 marks

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

Correct Answer

According to the de Broglie relation, the wavelength λ\lambda associated with a particle of momentum pp is given by λ=h/p\lambda = h/p. This relation is satisfied by photons as well.

For a photon, the momentum pp is given by p=hv/cp = hv/c, where hh is Planck's constant, vv is the frequency, and cc is the speed of light. Substituting this into the de Broglie relation, we get:

λ=hp=hhv/c=cv\lambda = \frac{h}{p} = \frac{h}{hv/c} = \frac{c}{v}

Since the wavelength of electromagnetic radiation is defined as c/vc/v, the de Broglie wavelength of the photon is equal to the wavelength of the electromagnetic radiation.

Exercise: EXERCISES | Q: 11.11 | (Page 285)
For More Understanding

Explanation

The textbook context explicitly derives this equality in Eq. (11.7). It starts with the de Broglie hypothesis (Eq. 11.5) and applies the photon momentum formula (Eq. 11.6). By substituting the photon momentum into the de Broglie equation, the term hh cancels out, leaving c/vc/v, which is the standard definition of the wavelength for electromagnetic waves. This demonstrates the consistency of the wave-particle duality concept for radiation.

Solution Steps

  1. Step 1: State the de Broglie relation for a photon: λ=h/p\lambda = h/p.

  2. Step 2: State the momentum for a photon from context: p=hv/cp = hv/c.

  3. Step 3: Substitute the momentum into the de Broglie relation: λ=h/(hv/c)\lambda = h / (hv/c).

  4. Step 4: Simplify the equation to show λ=c/v\lambda = c/v.

  5. Step 5: Conclude that since c/vc/v is the wavelength of electromagnetic radiation, the two wavelengths are equal.