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Step-by-Step Solution
Step 1: Identify the Relationship Between Photon Energy and Mass
A photon with wavelength \lambda has energy given by
E = \frac{hc}{\lambda} , where h is Planck's constant and c is the speed of light.
For a particle of mass m to have the same energy in terms of mass-energy equivalence, we use
E = m c^2 .
Step 2: Express the Mass in Terms of Known Constants
Equating the two expressions for energy:
\frac{hc}{\lambda} = m c^2
\quad \Longrightarrow \quad
m = \frac{hc}{\lambda c^2} = \frac{h}{c \lambda}.
Step 3: Relate to the Given Mass Form
The problem states that the mass of the fictitious particle is
\frac{x}{3} h \text{ kg} . Hence we set:
\frac{x}{3} h = \frac{h}{c \lambda}.
Dividing both sides by h :
\frac{x}{3} = \frac{1}{c \lambda}.
Therefore,
x = \frac{3}{c \lambda}.
Step 4: Substitute Given Values and Compute x
Given:
Speed of light, c = 3 \times 10^8 \, \text{m/s}
Wavelength, \lambda = 10 \, \text{Å} = 10 \times 10^{-10} \, \text{m} = 10^{-9} \, \text{m}
Substitute these into x = \frac{3}{c \lambda} :
x = \frac{3}{(3 \times 10^8) \times (10^{-9})} = \frac{3}{3 \times 10^{-1}} = \frac{3}{0.3} = 10.
Step 5: Conclude the Value of x
Hence, the value of x is
\boxed{10}.
