From what distance can these light bulbs be marginally resolved by a small telescope with a 6.0 cm-diameter objective lens? Assume that the lens is limited only by diffraction and wavelength = 600nm

These two bulbs are 2.0m apart. Sorry for not putting that in the question.

To determine the distance from which light bulbs can be marginally resolved by a small telescope with a 6.0 cm-diameter objective lens, we need to consider the phenomenon of diffraction limit.

The diffraction limit is the smallest resolvable detail that can be observed based on the properties of optics and the wavelength of light. In the case of a circular aperture like the objective lens of a telescope, the diffraction limit is given by a formula known as the Rayleigh criterion:

θ = 1.22 * λ / D

Where:
θ is the angular resolution (in radians),
λ is the wavelength of light,
D is the diameter of the objective lens.

First, we need to convert the diameter of the objective lens to meters. Since it is given as 6.0 cm, we divide it by 100: 6.0 cm / 100 = 0.06 m.

Next, we can plug the values into the formula.

θ = 1.22 * (600 nm) / (0.06 m)

Before we proceed, let's convert the wavelength of light to meters. Since 1 nm = 10^-9 m, 600 nm equals 600 * 10^-9 m = 6 * 10^-7 m.

θ = 1.22 * (6 * 10^-7 m) / (0.06 m)

Now we can perform the calculation.

θ = 7.32 * 10^-7 radians

The value we have calculated is the angular resolution. To find the distance from which the light bulbs can be marginally resolved, we can use the following approximation:

Distance = 1.22 * L / θ

Where L is the size of the light bulbs. Assuming each light bulb has a diameter of about 5 cm, we can convert it to meters: 5 cm / 100 = 0.05 m.

Distance = 1.22 * (0.05 m) / (7.32 * 10^-7 radians)

Simplifying the equation:

Distance ≈ 6.72 * 10^4 meters

Therefore, the light bulbs can be marginally resolved from a distance of approximately 67.2 kilometers.