Q65P - University Physics with Modern Physics

Question

.. DATA At the metal fabrication company where you work, you are asked to measure the diameter D of a very small circular hole in a thin, vertical metal plate. To do so, you pass coherent monochromatic light with wavelength   through the hole and observe the diffraction pattern on a screen that is a distance x from the hole. You measure the radius r of the first dark ring in the diffraction pattern (see Fig. 36.26). You make the measurements for four values of x. Your results are given in

the table.

 

(a) Use each set of measurements to calculate D. Because the measurements contain some error, calculate the average of the four values of D and take that to be your reported result. (b) For , what are the radii of the second and third dark rings in the diffraction pattern?

Step-by-Step Solution

Answer

the average value of D is

 the radii of second and third ring are 11cm and 15 cm  respectively 

Step 1 : Determine the value of D

IDENTIFY and SET UP:

This problem uses the ideas about circular diffraction In part (a) the angular radius 01 of the First dark ring in the diffraction

pattern follOWS from the geometrical relationship, where 1' is the radius of the ?rst dark ring (see Fig. 3625).

Then We use Eq- (36.17) to solve for the diameter D of the circular hole and thus calculate the average of its four values (our target variable)- In part (b) the angular rad? of the second dark ring and the third dark ring   are given by the Mo

equations (Eq- 36.18), respectively. Then we use the relationship

to the radii of the second (m = 2) and third (77?. = 3) dark rings in the diffraction pattern (the target variables)-

EXECUTE:

Solving the equation : for 01 gives

 

But we know , then we have

 

 

Substituting the known value of A and the values of m and 7' from the given table, we ?nd

 

 

 

 

 

Hence, the average of the four values of D is 

Therefore the average value of D is

Step 2: Determine the radii of the second and third ring

(b) Solving the equations or 02 and 03, and respectively, and substituting the knOWn

values of A and D, We find

 

 

Hence, the radii of the second and third dark rings in the diffraction pattern are

 

 

EVALUATE:

Studying the diffraction pattern formed by using a very small hole and monochromatic light with a known wavelength

constitutes a practical way to determine the diameter of the hole-

Therefore the radii of second and third ring are 11cm and 15 cm   respectively