Wavelength of X-rays having the largest penetrating power is:
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This is a fundamental concept in electromagnetic radiation: Short Wavelength \(\leftrightarrow\) High Frequency \(\leftrightarrow\) High Energy \(\leftrightarrow\) High Penetrating Power. This applies to the entire EM spectrum, from radio waves to gamma rays.
Step 1: Relate penetrating power to the energy of a photon.
The penetrating power of electromagnetic radiation, like X-rays, is directly proportional to the energy of its photons. Higher energy photons are more penetrating.
Step 2: Recall the energy-wavelength relationship for a photon.
The energy (\(E\)) of a photon is inversely proportional to its wavelength (\(\lambda\)):
\[
E = hf = \frac{hc}{\lambda}
\]
where \(h\) is Planck's constant and \(c\) is the speed of light.
Step 3: Determine which wavelength corresponds to the highest energy.
To have the largest penetrating power, the X-ray must have the highest energy. According to the formula, the highest energy corresponds to the shortest (smallest) wavelength.
Step 4: Compare the given wavelengths.
The given options are \(1.2 \, \text{\AA}\), \(6 \, \text{\AA}\), \(9 \, \text{\AA}\), and \(12 \, \text{\AA}\). The smallest value among these is \(1.2 \, \text{\AA}\). This wavelength will have the highest energy and therefore the largest penetrating power.
