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Step-by-Step Solution
Step 1: Understand the Problem
The question states that when yellow light propagates through equal thicknesses of air and vacuum, the difference in the number of waves is exactly one. We are given:
Refractive index of air: \mu_{\text{air}} = 1.0003
Wavelength of yellow light in vacuum: \lambda_{\text{vac}} = 6000 \, \text{\AA}
We need to find the thickness of the air column, t , that satisfies this condition.
Step 2: Relate the Number of Waves in Vacuum and in Air
The number of waves that fit into a given thickness t is given by the thickness divided by the wavelength in that medium. Therefore:
Number of waves in vacuum: N_{\text{vac}} = \dfrac{t}{\lambda_{\text{vac}}}
Number of waves in air: N_{\text{air}} = \dfrac{t}{\lambda_{\text{air}}} , where \lambda_{\text{air}} = \dfrac{\lambda_{\text{vac}}}{\mu_{\text{air}}}
The problem states that the difference in the number of waves is 1:
N_{\text{vac}} - N_{\text{air}} = 1
Step 3: Express the Difference in Terms of Known Quantities
Substitute the expressions for N_{\text{vac}} and N_{\text{air}} :
\dfrac{t}{\lambda_{\text{vac}}} \;-\; \dfrac{t}{\lambda_{\text{vac}} / \mu_{\text{air}}} \;=\; 1
Simplify the second term:
\dfrac{t}{\lambda_{\text{vac}}} - \dfrac{t\,\mu_{\text{air}}}{\lambda_{\text{vac}}} = 1 \\
t \left(\dfrac{1}{\lambda_{\text{vac}}} - \dfrac{\mu_{\text{air}}}{\lambda_{\text{vac}}}\right) = 1 \\
t \left(\dfrac{1 - \mu_{\text{air}}}{\lambda_{\text{vac}}}\right) = 1
Hence,
t = \dfrac{\lambda_{\text{vac}}}{\mu_{\text{air}} - 1}.
Step 4: Substitute the Numerical Values
Here, \lambda_{\text{vac}} = 6000 \,\text{\AA} = 6000 \times 10^{-10}\,\text{m} and \mu_{\text{air}} = 1.0003 . So,
t = \dfrac{6000 \times 10^{-10}\,\text{m}}{1.0003 - 1}
First, compute the denominator:
1.0003 - 1 = 0.0003
Thus,
t = \dfrac{6000 \times 10^{-10}}{0.0003}\,\text{m}
Performing the calculation:
t = \dfrac{6000 \times 10^{-10}}{3 \times 10^{-4}}\,\text{m}
= \dfrac{6000}{3} \times \dfrac{10^{-10}}{10^{-4}}\,\text{m}
= 2000 \times 10^{-6}\,\text{m}
= 2 \times 10^{-3}\,\text{m}
= 2\,\text{mm}.
Step 5: Conclude the Result
The required thickness of the air column is 2\,\text{mm} .
