Question

The wavelength of an X-ray beam is 10 Å. The mass of a fictitious particle having the same energy as that of the X-ray photons is $\frac{x}{3}h$ kg. The value of x is ________ . (h=Planck's constant)

Hint:

When a question involves both the photon energy ($E=hc/\lambda$) and mass-energy equivalence ($E=mc^2$), it is often required to equate them to find the "equivalent mass" or "photon mass", which is $m=h/(c\lambda)$. Be wary of potential typos in units in exam questions.

Answer 1

Correct Answer: 10

The energy of a photon (E) is given by the Planck-Einstein relation: $E = \frac{hc}{\lambda}$, where h is Planck's constant, c is the speed of light, and $\lambda$ is the wavelength.
The energy of the fictitious particle is given by Einstein's mass-energy equivalence relation: $E = mc^2$, where m is the mass of the particle.
We are given that the energies are the same, so we can equate the two expressions:
$mc^2 = \frac{hc}{\lambda}$.
Solving for the mass m: $m = \frac{h}{c\lambda}$.
The problem states that the mass of the particle is $m = \frac{x}{3}h$. Note: The units 'kg' are physically incorrect here, as 'h' has units of J·s. We proceed by equating the algebraic expressions as intended by the question.
$\frac{x}{3}h = \frac{h}{c\lambda}$.
We can cancel h from both sides:
$\frac{x}{3} = \frac{1}{c\lambda}$.
$x = \frac{3}{c\lambda}$.
We are given $\lambda = 10$ Å $= 10 \times 10^{-10}$ m. The speed of light $c \approx 3 \times 10^8$ m/s.
$x = \frac{3}{(3 \times 10^8 \text{ m/s}) \times (10 \times 10^{-10} \text{ m})}$.
$x = \frac{3}{3 \times 10^8 \times 10^{-9}} = \frac{3}{3 \times 10^{-1}} = \frac{1}{10^{-1}} = 10$.
The value of x is 10.