To find the frequency of the electromagnetic (e.m) wave that is best suited to observe a particle of radius \(3 \times 10^{-4} \, \text{cm}\), we can follow these steps:
### Step 1: Convert the particle radius to meters
The radius of the particle is given in centimeters. We need to convert it to meters for consistency in SI units.
\[
3 \times 10^{-4} \, \text{cm} = 3 \times 10^{-4} \times 10^{-2} \, \text{m} = 3 \times 10^{-6} \, \text{m}
\]
### Step 2: Understand the relationship between wavelength and particle size
To observe a particle correctly, the wavelength (\(\lambda\)) of the e.m wave must be less than the size of the particle. Therefore, we can write:
\[
\lambda < 3 \times 10^{-6} \, \text{m}
\]
### Step 3: Relate wavelength to frequency
The wavelength of an electromagnetic wave is related to its frequency (\(f\)) and the speed of light (\(c\)) by the equation:
\[
\lambda = \frac{c}{f}
\]
Where \(c\) (the speed of light) is approximately \(3 \times 10^8 \, \text{m/s}\).
### Step 4: Substitute the wavelength into the equation
We can substitute the expression for wavelength into the inequality:
\[
\frac{c}{f} < 3 \times 10^{-6} \, \text{m}
\]
### Step 5: Rearrange the inequality to find frequency
Rearranging the inequality gives us:
\[
f > \frac{c}{3 \times 10^{-6}}
\]
Substituting \(c = 3 \times 10^8 \, \text{m/s}\):
\[
f > \frac{3 \times 10^8}{3 \times 10^{-6}} = 10^{14} \, \text{Hz}
\]
### Step 6: Determine the order of frequency
The frequency must be greater than \(10^{14} \, \text{Hz}\). For the best observation, we want the frequency to be as high as possible.
### Step 7: Identify the suitable frequency from options
If the options provided include \(10^{15} \, \text{Hz}\), it is suitable as it is greater than \(10^{14} \, \text{Hz}\).
### Final Answer
The frequency of the e.m wave which is best suited to observe a particle of radius \(3 \times 10^{-4} \, \text{cm}\) is of the order of \(10^{15} \, \text{Hz}\).
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