Problem 19.33
Two light bulbs are 1.5 m apart.
From what distance can these light bulbs be marginally resolved by a small telescope with a 4.0 cm-diameter objective lens? Assume that the lens is limited only by diffraction and lambda = 600 nm.
The angular resolution limit for two point sources of equal intensity is
d(theta) = 1.22(lambda)/D,
in radians, where D is the aperture diameter. This assume good atmospheric vewing conditions ("seeing").
If the viewing distance is X (meters),
1.5/X = 1.22*(600*10^-9 m)/(0.04 m)
Solve for X, in meters
X = 1.5*0.04/[1.22*600*10^-9)]
Solve for X
To determine the distance at which the light bulbs can be marginally resolved by a small telescope with a 4.0 cm-diameter objective lens, we can use the formula for the minimum resolvable angle (θ) based on diffraction:
θ = 1.22 * (lambda / D)
where:
- λ is the wavelength of light (given as 600 nm, which is 600 x 10^-9 m)
- D is the diameter of the objective lens (given as 4.0 cm, which is 4.0 x 10^-2 m)
First, we need to convert the diameter of the objective lens to meters:
D = 4.0 cm = 4.0 x 10^-2 m
Now we can substitute the values into the formula to calculate the minimum resolvable angle:
θ = 1.22 * (600 x 10^-9 m / 4.0 x 10^-2 m)
θ ≈ 0.0183 radians
The minimum resolvable angle represents the angular separation between the two light bulbs that can be just barely resolved by the telescope. To find the distance at which this angular separation occurs, we can use the formula:
Distance = (Diameter of object) / (Minimum resolvable angle)
In this case, the diameter of the object is the distance between the two light bulbs:
Distance = 1.5 m / 0.0183 radians
Distance ≈ 81.97 m
Therefore, the light bulbs can be marginally resolved by the small telescope from a distance of approximately 81.97 meters.
To determine the distance at which the light bulbs can be marginally resolved by a small telescope, we need to calculate the resolving power of the telescope. The resolving power is determined by the diameter of the objective lens and the wavelength of light being observed.
The formula for resolving power (RP) is given by:
RP = 1.22 * λ / D
Where:
RP = resolving power
λ = wavelength of light
D = diameter of the objective lens
In this case, the diameter of the objective lens (D) is given as 4.0 cm and the wavelength of light (λ) is 600 nm (600 * 10^-9 m).
Substituting these values into the formula, we can calculate the resolving power:
RP = 1.22 * 600 * 10^-9 / 4 * 10^-2
Simplifying the expression:
RP = 0.0183 / 0.04
RP = 0.4575
The resolving power (RP) of the telescope is approximately 0.4575.
Now, the minimum angular separation required for two objects to be resolved is given by the formula:
θ = λ / D
Where:
θ = minimum angular separation
λ = wavelength of light
D = diameter of the objective lens
Substituting the values:
θ = 600 * 10^-9 / 4 * 10^-2
θ = 0.00225
The minimum angular separation (θ) is approximately 0.00225 radians.
Now, to find the distance at which the light bulbs can be marginally resolved, we can use the formula:
D = 2 * r * tan(θ)
Where:
D = distance between the light bulbs
r = distance at which the light bulbs can be marginally resolved
θ = minimum angular separation
Substituting the values:
1.5 = 2 * r * tan(0.00225)
Dividing both sides by 2 * tan(0.00225):
r = 1.5 / (2 * tan(0.00225))
Calculating the value of r:
r ≈ 1.5 / (2 * 0.00225)
r ≈ 333.33 m
Therefore, the light bulbs can be marginally resolved from a distance of approximately 333.33 meters using the given telescope with a 4.0 cm-diameter objective lens.