In Young's double slit experiment, the ratio of maximum and minimum intensities in the fringe system is 9:1. The ratio of amplitudes of coherent sources is
Show Hint
- 9:1
- 3:1
- 2:1
- 1:1
The Correct Option is C
Approach Solution - 1
In Young's double slit experiment, the intensity at any point on the screen is determined by the superposition of two wave amplitudes from the coherent sources. The relationship between the maximum intensity \(I_{\text{max}}\) and the minimum intensity \(I_{\text{min}}\) in terms of amplitude \(A_1\) and \(A_2\) is given by:
\(I_{\text{max}} = (A_1 + A_2)^2\)
\(I_{\text{min}} = (A_1 - A_2)^2\)
Given that the ratio of maximum and minimum intensities is 9:1:
\(\frac{I_{\text{max}}}{I_{\text{min}}} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 9\)
Taking the square root on both sides, we get:
\(\frac{A_1 + A_2}{A_1 - A_2} = 3\)
Cross-multiplying gives:
\(A_1 + A_2 = 3(A_1 - A_2)\)
Expanding and rearranging terms:
\(A_1 + A_2 = 3A_1 - 3A_2\)
\(A_2 + 3A_2 = 3A_1 - A_1\)
\(4A_2 = 2A_1\)
\(\frac{A_1}{A_2} = \frac{4}{2} = 2:1\)
Thus, the ratio of the amplitudes of coherent sources is 2:1.
Approach Solution -2
In Young's double slit experiment, the intensity of the interference fringes depends on the amplitudes of the coherent sources.
The maximum intensity \( I_{\text{max}} \) and the minimum intensity \( I_{\text{min}} \) are related to the amplitudes \( A_1 \) and \( A_2 \) of the two waves as follows: \[ I_{\text{max}} = (A_1 + A_2)^2 \] \[ I_{\text{min}} = (A_1 - A_2)^2 \] The ratio of maximum and minimum intensities is given by: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} \] Given that the ratio of maximum and minimum intensities is 9:1, we have: \[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 9 \] Taking the square root of both sides: \[ \frac{A_1 + A_2}{A_1 - A_2} = 3 \] Solving for the ratio of the amplitudes: \[ A_1 + A_2 = 3(A_1 - A_2) \] Expanding and simplifying: \[ A_1 + A_2 = 3A_1 - 3A_2 \] \[ A_1 + A_2 - 3A_1 + 3A_2 = 0 \] \[ -2A_1 + 4A_2 = 0 \] \[ A_1 = 2A_2 \]
Thus, the ratio of the amplitudes is \( A_1 : A_2 = 2:1 \). Therefore, the correct answer is: \[ \text{(C) } 2:1 \]
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