Problem 19.33


Two light bulbs are 1.0 m apart.


Part A -

From what distance can these light bulbs be marginally resolved by a small telescope with a 3.0cm -diameter objective lens? Assume that the lens is limited only by diffraction and wavelenght = 600nm .

Express your answer using two significant figures.

The Rayleigh resolution used for telescopes is

1.22 (lambda)/D = limiting angle
= 2.44*10^-5 radians
= separation/X
= 1 meter/X

Solve for X

X = 40 km

To determine the distance from which the light bulbs can be marginally resolved by the small telescope, we can use the formula for the minimum resolvable angle, derived from the concept of diffraction:

θ = 1.22 * (λ / D)

Where:
θ is the minimum resolvable angle (in radians)
λ is the wavelength of light (600 nm = 600 x 10^(-9) m)
D is the diameter of the objective lens (3.0 cm = 3.0 x 10^(-2) m)

Plugging in the values:

θ = 1.22 * (600 x 10^(-9) m) / (3.0 x 10^(-2) m)

Now we can rearrange the formula to solve for the distance from which the light bulbs can be resolved:

D = (λ / θ)

D = (600 x 10^(-9) m) / (1.22 * (600 x 10^(-9) m) / (3.0 x 10^(-2) m))

D = (3.0 x 10^(-2) m) / 1.22

D ≈ 2.5 x 10^(-2) m

So, the distance from which the light bulbs can be marginally resolved by the small telescope is approximately 2.5 x 10^(-2) m or 0.025 m.

To find the distance from which two light bulbs can be marginally resolved by a small telescope, we can use the concept of Rayleigh's criterion. According to Rayleigh's criterion, two point sources can be just resolved when the first minimum of the diffraction pattern of one source falls on the maximum of the diffraction pattern of the other source.

The formula for calculating the minimum resolvable angle (θ) for two point sources is given by:
θ = 1.22 * (λ / D),
where θ is the minimum resolvable angle, λ is the wavelength of light, and D is the diameter of the objective lens.

In this case, the diameter of the objective lens (D) is given as 3.0 cm, which is equal to 0.03 m, and the wavelength (λ) is given as 600 nm, which is equal to 6.0 x 10^(-7) m.

Substituting these values into the formula for θ:
θ = 1.22 * (6.0 x 10^(-7) m / 0.03 m)
θ ≈ 2.44 x 10^(-5) radians

Now, we can use the formula for the angular resolution (θ) to find the distance (d) at which the two light bulbs can be marginally resolved.

The formula for calculating the angular resolution (θ) is given by:
θ = 1.22 * (λ / d),
where θ is the angular resolution and d is the distance between the two light bulbs.

Rearranging the formula to solve for d:
d = 1.22 * (λ / θ)

Substituting the given values for θ and λ:
d = 1.22 * (6.0 x 10^(-7) m / 2.44 x 10^(-5) radians)
d ≈ 3.01 x 10^(-2) m

Therefore, the two light bulbs can be marginally resolved by a small telescope from a distance of approximately 3.0 centimeters.