Question

Obtain the
expression for area of the nth zone.

Answer 1

Answer on Question #77791, Physics / Optics

Question. Obtain the expression for area of the $n - th$ zone.

Solution.

Find the area of the $n - th$ Fresnel zone

$$\Delta S _ {n} = S _ {n} - S _ {n - 1}$$

From the figure

$$r _ {n} ^ {2} = a ^ {2} + (a - h _ {n}) ^ {2} = \left(b + n \frac {\lambda}{2}\right) ^ {2} - (b - h _ {n}) ^ {2}$$ $$\lambda \ll a \quad \text{and} \quad \lambda \ll b$$

We have

$$h _ {n} = \frac {b n \lambda}{2 (a + b)}$$ $$S _ {n} = 2 \pi a h _ {n} = \frac {\pi a b \lambda}{a + b} n$$ $$\Delta S _ {n} = S _ {n} - S _ {n - 1} = \frac {\pi a b \lambda}{a + b}$$

So, the area of the $n - th$ Fresnel zone

$$\Delta S _ {n} = \frac {\pi a b \lambda}{a + b}$$

and does not depend on $n$.

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