What is the area of the Airy disk on the screen based on the given information?
The area of the Airy disk on the screen can be calculated using the formula:A = πr², where r is the radius of the Airy disk. The radius of the Airy disk can be determined using the formula:r = 1.22 λ L/D, where λ is the wavelength of the light, L is the distance from the double-slit to the screen, and D is the diameter of the circular aperture.
Calculating the Radius of the Airy Disk
To calculate the radius of the Airy disk, we first need to find the radius of the first dark ring, r1, which forms the boundary for the bright Airy disk. The formula for r1 is:r1 = sqrt(1.22 λ L/d), where d is the distance between the slits. By substituting the given values, we find:r1 = sqrt(1.22 × 6.50 × 10⁻⁷ × 5.00 × 10⁻²/1.00 × 10⁻⁴)m= 4.76 × 10⁻³ m.
Next, we can calculate the radius of the Airy disk using the formula:r = 1.22 λ L/D. By substituting the given values, we find:r = 3.00 × 10⁻⁵ m.
Calculating the Area of the Airy Disk
Now that we have the radius of the Airy disk, we can calculate its area using the formula:A = πr². Substituting the radius value, we get:A = 3.14 × (3.00 × 10⁻⁵)²= 2.83 × 10⁻⁹ m².
Therefore, the area of the Airy disk on the screen is approximately 2.83 × 10⁻⁹ m² to three significant figures.