5

The diagram below shows an arbitrary point, point A, that lies near two point sources of waves (labeled 5'1 and 5'2). l A i l l l d/2 d/2 -- In this problem, we consider how the path length difference to point A changes as point A is moved along the dark line shown, away from the sources. Point Z on the diagram below was chosen so that point Z and source 5'2 are equidistant from point A (notice how points Z, 81 and 52 form a right-triangle as a result with d as a hypotenuse). a. How do the angles a and 8 compare? Explain. b. Suppose that point A is moved away from the sources along the dark line. In the limit that point A is very far from the sources, what values do the angles a, ,8 and 7 approach? Explain your conclusion. c. Recall that the path length difference AD is the additional distance traveled by one of the waves to reach the same point (in this case A) as the other wave. Label AD in the above diagram. d. The diagram below illustrates the limit In which point A is moved very far from the sources. To point A (far away) In this limit, nd an expression for AD in terms of the angle 6 and the source separation d. (Hint: How do the angles (Sand 0 compare?) e. For what values of AD (in terms of A) will there be: i. maximum constructive interference? ii. complete destructive interference (Le. a node)? f. Use your answers from parts d and e to write equations that can be used to determine the angle(s) for which there will be: i. lines of maximum constructive interference ii. nodal lines 3. Determine the angles for which there will be nodal lines and lines of maximum constructive interference for the case of two sources in phase, a distance 1.5)l apart. h. Use your results from part g to draw accurate nodal lines and lines of maximum constructive interference on the diagram below. Label each line with the corresponding value of AD and 9. \\\\lll/ \\\\ l// \\\\ // \\ / \\ / \\~ / \\ / \"N ./ \\ \\' / m M 2 sources in phase d = 1.5/1 between sources (not shown)