Question

What wavelength must electromagnetic radiation have if a photon in the beam has the same momentum as an electron moving with a speed $1.1 \times 10^5$ m/s (Planck’s constant $= 6.6 \times 10^{-34}$ Js, rest mass of electron $= 9 \times 10^{-31}$ kg)?

• A. $2/3$ nm
• B. $20/3$ nm
• C. $4/3$ nm
• D. $40/3$ nm
• E. $3/20$ nm

Hint:

Photon momentum depends only on wavelength: $p = h/\lambda$.

Answer 1

The Correct Option is C

Concept:
Photon momentum: \[ p = \frac{h}{\lambda} \] Electron momentum: \[ p = mv \]

Step 1: Equate momenta.
\[ \frac{h}{\lambda} = mv \]

Step 2: Solve for $\lambda$.
\[ \lambda = \frac{h}{mv} \] \[ = \frac{6.6 \times 10^{-34}}{9 \times 10^{-31} \times 1.1 \times 10^5} \]

Step 3: Calculate.
\[ \lambda = \frac{6.6}{9.9} \times 10^{-8} = \frac{2}{3} \times 10^{-8} \text{ m} \] \[ = \frac{2}{3} \times 10^{-8} \text{ m} = \frac{4}{3} \text{ nm} \]