The hale telescope of mount Polamor has a diameter of 200 inches. What is its limiting angle of resolution for 600 nm light?

Options

A. `7.2 xx 10^(-8) rad`
B. `7.2 xx 10^(-4) rad`
C. `1.44 xx 10^(-7) rad`
D. `14.4 xx 10^(-10) rad`

Answer

To find the limiting angle of resolution for the Hale telescope, we will use the Rayleigh criterion formula. Here is a step-by-step solution:

### Step 1: Convert the diameter from inches to meters
The diameter of the telescope is given as 200 inches. We need to convert this measurement into meters.

1. **Conversion factor**: 1 inch = 2.54 cm
2. **Convert inches to centimeters**:
\[ 200 \text{ inches} \times 2.54 \text{ cm/inch} = 508 \text{ cm} \]
3. **Convert centimeters to meters**:
\[ 508 \text{ cm} \div 100 = 5.08 \text{ m} \]

### Step 2: Identify the wavelength of light
The wavelength of light is given as 600 nm. We need to convert this into meters.

1. **Conversion**:
\[ 600 \text{ nm} = 600 \times 10^{-9} \text{ m} = 6.00 \times 10^{-7} \text{ m} \]

### Step 3: Apply the Rayleigh criterion formula
The limiting angle of resolution (Δθ) can be calculated using the Rayleigh criterion formula:
\[ \Delta \theta = \frac{1.22 \lambda}{D} \]
Where:
- \( \Delta \theta \) = limiting angle of resolution in radians
- \( \lambda \) = wavelength of light in meters
- \( D \) = diameter of the telescope in meters

### Step 4: Substitute the values into the formula
Now, we will substitute the values of \( \lambda \) and \( D \) into the formula.

1. **Substituting values**:
\[ \Delta \theta = \frac{1.22 \times (600 \times 10^{-9} \text{ m})}{5.08 \text{ m}} \]

### Step 5: Calculate the angle of resolution
Now we will perform the calculation.

1. **Calculating**:
\[ \Delta \theta = \frac{1.22 \times 600 \times 10^{-9}}{5.08} \]
\[ = \frac{7.32 \times 10^{-7}}{5.08} \approx 1.44 \times 10^{-7} \text{ radians} \]

### Conclusion
The limiting angle of resolution for the Hale telescope for 600 nm light is approximately \( 1.44 \times 10^{-7} \) radians.

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