In an interference pattern, the slit widths are in the ratio 1:9. Then find out the ratio of minimum and maximum intensity.
Understand the Problem
The question is asking to determine the ratio of minimum and maximum intensity in an interference pattern where the slit widths are given in the ratio of 1:9. This involves applying principles from wave interference and intensity calculations.
Answer
The ratio of minimum to maximum intensity is \( \frac{32}{41} \).
Answer for screen readers
The ratio of minimum to maximum intensity is ( \frac{32}{41} ).
Steps to Solve
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Identify the slit widths Let the slit widths be ( w_1 = 1 ) and ( w_2 = 9 ).
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Calculate maximum intensity The maximum intensity ( I_{\text{max}} ) in an interference pattern is given by the formula: $$ I_{\text{max}} = (I_1 + I_2) = k(w_1^2 + w_2^2) $$ where ( k ) is a proportionality constant. Thus, $$ I_{\text{max}} = k(1^2 + 9^2) = k(1 + 81) = 82k $$
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Calculate minimum intensity The minimum intensity ( I_{\text{min}} ) using the formula: $$ I_{\text{min}} = (I_1 - I_2)^2 $$ So, $$ I_{\text{min}} = k(w_1 - w_2)^2 = k(1 - 9)^2 = k(-8)^2 = 64k $$
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Calculate the ratio of minimum and maximum intensity The ratio ( R ) of minimum to maximum intensity can be calculated as: $$ R = \frac{I_{\text{min}}}{I_{\text{max}}} = \frac{64k}{82k} = \frac{64}{82} = \frac{32}{41} $$
The ratio of minimum to maximum intensity is ( \frac{32}{41} ).
More Information
In this interference problem, we demonstrated how changing slit widths affects the resulting intensities. The calculated ratio ( \frac{32}{41} ) shows a notable difference between the minimum and maximum intensities in the interference pattern.
Tips
- Confusing the formulas for maximum and minimum intensity can lead to incorrect calculations.
- Neglecting to square the differences in calculating minimum intensity is a frequent error.
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