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Ichigo449
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Homework Statement
Consider a given monochromatic component of sunlight. The electric field drives a given air molecule. Each oscillating charge of the air molecule radiates waves in all directions, some of which travel to the eye of a given observer. But, for a given molecule (call it No.1) there is another (No.2) that is one half-wavelength farther from the observer. If both molecules are driven with the same amplitude and phase constant, their waves should superpose to give zero at the position of the observer. For scattering near ##\frac{\pi}{2}/## radians, we can obviously satisfy these phase and amplitude conditions, provided the number of air molecules per unit volume is large enough so that there is nearly always a molecule "No.2" for every molecule "No.1". So that section of the sky corresponding to the scattering should be dark. This is obviously false for air but, surprisingly, is true for water. Why?
Hint: Consider a small region in space (region 1) and consider another region (region 2) of the same size, situated at the same distance from the sun and situated one half-wavelength (consider a single monochromatic component of sunlight) farther from the observer than region 1. Assume that these regions are small compared with the wavelength. Let there be n1 and n2 molecules in regions 1 and 2 respectively. Compute the total electric field due to these regions and average the amplitude over a single period. Now consider the effect of fluctuations in the number of molecules over a long enough period of time.
Homework Equations
Coloumb's Law, Conditions for Destructive Interference
The Attempt at a Solution
I can find the electric fields by Coloumb but am not sure how to include the effects of fluctuations.