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 485 nm are at a distance L from the hole. The separation between the sources is 2.70 cm, and they are just resolved by the camera. What is L?
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To find the distance L, we can use the concept of angular resolution in a pinhole camera.
The angular resolution is the minimum angle at which two objects can be distinguished as separate. When the separation between the two sources is just resolved, the central maximum of one source aligns with the first minimum of the other source.
In this case, the central maximum of one source aligns with the first minimum of the other source. If the separation between the sources is 2.70 cm, then the distance from the center of the central maximum to the first minimum is half of that, or 1.35 cm.
We can use the concept of small angle approximation to relate the angular separation to the distance L and the diameter of the aperture D.
Let θ be the angular separation between the two sources, D be the diameter of the aperture, and L be the distance from the aperture to the screen.
The small angle approximation states that for small angles, the tangent of the angle is approximately equal to the angle itself in radians. In this case, the angle θ is small because the sources are distant.
Using the small angle approximation, we can write:
θ = λ / D
where λ is the wavelength of light. In this case, λ = 485 nm = 4.85 x 10^-4 cm.
Now we can rearrange the equation to solve for L:
L = D / θ
Substituting the known values, we get:
L = (0.600 mm) / (4.85 x 10^-4 cm)
L = 1237.11 cm
Therefore, the distance L is approximately 1237.11 cm.