Question

If n represents the order of a half period zone the area of this zone is approximately proportional to ${n^m}$ where m is equal to
A) Zero
B) Half
C) One
D) two

Answer 1

This is a characteristic of progressive waves. The entire area of the wave front can be divided into a '$n$' number of smaller regions of zones and these zones are referred to as half period zones. Formula for area of Half Period Zone: $\pi \left( {p\lambda + \dfrac{\lambda }{2}\left( {2n - 1} \right)\dfrac{1}{4}} \right)$ =$\lambda p\pi$ Hence it is independent of order of zone.

Complete step by step answer:
Step 1:
Characteristics of a progressive wave
(a) Each particle of the medium executes vibration about its mean position. The disturbance progresses onward from one particle to another.
(b) The particles of the medium vibrate with the same amplitude as their mean positions.
According to Fresnel's the entire wavefront can be divided into a large number of parts of zones which are known as Fresnel's half-period zones (HPZ's). The resultant effect at any point on the screen is due to the combined effect of all the secondary waves from the various zones.

Step 2:
The entire area of wave front can be divided into a 'n' number of smaller regions of zones and these zones are referred to as half period zones. Formula for area of Half Period Zone: $\pi \left( {p\lambda + \dfrac{\lambda }{2}\left( {2n - 1} \right)\dfrac{1}{4}} \right)$ =$\lambda p\pi$
The Area of the half period zone is independent of the order of zones. Therefore, m is equal to zero in ${n_m}$.

Hence option A is correct.

Note:
The path difference between two consecutive secondary waves reaching the screen (here point P) is $\dfrac{\lambda }{2}$.
Area of Second Half Period Time: We can find this by entering n=2 in the above formula and area of Third Half Period Time: We can find this by entering n=3 in the above formula.