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| Coherent Sources |
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| In different colors, the fringes have different widths. This indicates that a relation exists between colour and fringe widths. But before going to the relation, let us know more about the condition under which constructive interference or destructive interference occurs. |
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| If y1 = A Sin wt and y2 = B Sin ( wt + f) represent the waves of two coherent sources with A and B as their respective amplitude and f the constant phase difference between the waves. |
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| Then from superposition principle, |
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| y = y1 + y2 |
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| = A sin wt + B sin( wt + f ) |
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| = sin wt( A+ B cos f) + cos wt B sin f |
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| y = R sin (wt + f) |
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| (On expanding the second term) |
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| i.e. put a + cos f = R cos f |
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| Since intensity I a (amplitude) 2 |
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| I f (a2 + b2 + 2ab cosf) |
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| For constructive interference, intensity (I) is maximum when |
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| cosf = +1 or f = 0, 2 p, 4p |
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| f = 2n p where n=0,1,2 |
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| Now a path difference of l corresponds to the phase difference 2p. If x is the path difference, then |
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| (Where f is the phase difference) |
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| x = nl |
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| For destructive inference, intensity should be minimum. |
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| cos f = -1 |
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| f = p,3 p,5 pp….. |
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| f = (2n - 1)p where n=1,2 |
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| Note: |
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| 1. During constructive interference |
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| Rmax = A + B as cos f = + 1 |
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| During destructive inference |
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| cos f = -1 |
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| Rmin = A - B |
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| special case |
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| If A =B |
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| (i.e., if the two waves have same amplitude) then |
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| This indicates a good contrast between the bright and dark bands, which is required to view a good interference pattern. |
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| 2. In terms of respective intensities |
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| I1, I2 are intensities of waves coming from A and B respectively. |
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| In addition if I1 = I2 Then |
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| 3. If W1 and W2 are the widths of two slits from which intensities of light I1 and I2 emanate, then |
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| 4. Conservation of energy in interference |
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| If there were no interference, intensity of light from two sources at energy point on the screen would be |
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| I = I1 + I2 = (a2 + b2). |
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| The average intensity of light in the interference pattern is |
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| On comparing the above expression with |
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| This indicates that light energy is redistributed, i.e., from the energy of destructive interference to the regions of constructive interference. No energy is created or destroyed in this process and therefore this phenomenon does not violate the principle of energy conservation. |
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