A pinhole camera has a small circular aperture of diameter D. Light from distant objects passes through the aperture into an otherwise dark box, falling on a screen at the other end of the box. The aperture in a pinhole camera has diameter D = 0.600 mm. Two point sources of light of wavelength 595 nm are at a distance L from the hole. The separation between the sources is 2.75 cm, and they are just resolved by the camera. What is L?
To solve this problem, we can use the concept of diffraction to determine the condition for resolving two point sources with a pinhole camera.
Diffraction is the bending of the waves as they pass through an aperture or around an obstacle. In the case of a pinhole camera, diffraction occurs when light passes through the small circular aperture.
According to the Rayleigh criterion, two point sources can be resolved if the first minimum of the diffraction pattern of one source falls on the maximum of the diffraction pattern of the other source.
The angle θ between the central maximum and the first minimum of the diffraction pattern can be determined using the formula:
sin(θ) ≈ 1.22 * (λ / D)
Where:
- θ is the angle between the central maximum and the first minimum in radians,
- λ is the wavelength of light in meters,
- D is the diameter of the aperture in meters.
In this problem, the wavelength of light, λ, is given as 595 nm, which is equal to 595 * 10^(-9) meters. The diameter of the aperture, D, is given as 0.600 mm, which is equal to 0.600 * 10^(-3) meters.
Using the given values, we can calculate the angle θ.
sin(θ) ≈ 1.22 * (595 * 10^(-9) / 0.600 * 10^(-3))
sin(θ) ≈ 1.22 * 9.92 * 10^(-4)
Now, we can solve for θ:
θ ≈ sin^(-1)(1.22 * 9.92 * 10^(-4))
θ ≈ sin^(-1)(1.218 * 10^(-3))
Once we have the value of θ, we can use trigonometry to determine the distance L between the pinhole and the point sources. In the problem, it is given that the separation between the sources is 2.75 cm, so we can form a right triangle with L as the hypotenuse, θ as one of the acute angles, and 2.75 cm as the opposite side.
Using the appropriate trigonometric function, we can solve for L:
L = 2.75 cm / sin(θ)
Substituting the value of θ, we can calculate L.
L = 2.75 cm / sin(sin^(-1)(1.218 * 10^(-3)))
L ≈ 2.75 cm / (1.218 * 10^(-3))
Evaluating this expression will give us the value of L.