What is the minimum wavelength of X–rays

Options

A. `(eV)/(bc)`
B. `(hc)/(eV)`
C. `(hc)/(e)`
D. `(hc)/(V)`

Answer

To find the minimum wavelength of X-rays, we can use the relationship between energy and wavelength. Here’s a step-by-step solution:

### Step 1: Understand the relationship between energy and wavelength
The energy (E) of a photon is related to its frequency (ν) by the equation:
\[ E = h \nu \]
where \( h \) is Planck's constant.

### Step 2: Relate frequency to wavelength
The frequency (ν) can be related to the wavelength (λ) using the speed of light (c):
\[ c = \nu \lambda \]
From this, we can express frequency in terms of wavelength:
\[ \nu = \frac{c}{\lambda} \]

### Step 3: Substitute frequency into the energy equation
Substituting the expression for frequency into the energy equation gives:
\[ E = h \frac{c}{\lambda} \]

### Step 4: Rearranging for minimum wavelength
To find the minimum wavelength (λ_min) corresponding to a given energy (E), we rearrange the equation:
\[ \lambda = \frac{hc}{E} \]
Thus, the minimum wavelength occurs when the energy is at its maximum.

### Step 5: Use known values
Planck's constant \( h \) is approximately \( 6.63 \times 10^{-34} \) Joule seconds, and the speed of light \( c \) is approximately \( 3.00 \times 10^8 \) m/s.

### Step 6: Calculate the minimum wavelength
If we want to find the minimum wavelength for a specific energy, we can plug in the values. For example, if we consider the energy of X-rays to be around \( 10 \) keV (which is \( 1.6 \times 10^{-15} \) Joules), we can calculate:
\[ \lambda_{min} = \frac{(6.63 \times 10^{-34} \text{ J s})(3.00 \times 10^8 \text{ m/s})}{1.6 \times 10^{-15} \text{ J}} \]

### Step 7: Perform the calculation
Calculating this gives:
\[ \lambda_{min} \approx \frac{1.989 \times 10^{-25}}{1.6 \times 10^{-15}} \]
\[ \lambda_{min} \approx 1.24375 \times 10^{-10} \text{ m} \]
or approximately \( 0.124 \text{ nm} \).

### Final Answer
The minimum wavelength of X-rays is approximately \( 0.124 \text{ nm} \).

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