The ratio of radii of Fresnel.s fourth and ninth zone is

Options

A. `1 : 4`
B. `4 : 9`
C. `9 : 4`
D. `2 : 3`

Answer

To find the ratio of the radii of Fresnel's fourth and ninth zones, we can use the relationship between the radii of the Fresnel zones. The formula states that the ratio of the radius of zone A to the radius of zone B is given by the square root of the ratio of their zone numbers.

### Step-by-Step Solution:

1. **Identify the Zone Numbers**:
- For the fourth zone, the zone number \( A = 4 \).
- For the ninth zone, the zone number \( B = 9 \).

2. **Use the Formula for Radius Ratio**:
\[ \frac{R_A}{R_B} = \sqrt{\frac{A}{B}} \]
Here, \( R_A \) is the radius of the fourth zone and \( R_B \) is the radius of the ninth zone.

3. **Substitute the Zone Numbers into the Formula**:
\[ \frac{R_{4}}{R_{9}} = \sqrt{\frac{4}{9}} \]

4. **Calculate the Square Root**:
\[ \frac{R_{4}}{R_{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \]

5. **Express the Ratio**:
The ratio of the radius of the fourth Fresnel zone to the radius of the ninth Fresnel zone is:
\[ R_{4} : R_{9} = 2 : 3 \]

### Final Answer:
The ratio of the radii of Fresnel's fourth and ninth zones is \( 2 : 3 \).

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