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θ = 1 . 22 λ D , size 12{θ=1 "." "22" { {λ} over {D} } } {}

where λ size 12{λ} {} is the wavelength of light (or other electromagnetic radiation) and D size 12{D} {} is the diameter of the aperture, lens, mirror, etc., with which the two objects are observed. In this expression, θ size 12{θ} {} has units of radians.

Part a of the figure shows a graph of intensity versus theta. The curve has a central maximum at theta equals zero and its first minima occur at plus one point two two lambda over D and minus one point two two lambda over D. Farther from the central peak, several small peaks occur, but they are much much smaller than the central maximum. Part b of the figure shows a drawing in which two light bulbs, labeled object one and object two, appear in the foreground positioned next to each other. Two rays of light, one from each light bulb, pass through a pinhole aperture and continue on to strike a screen that is farther back in the drawing. On the screen is an x y plot of the two resulting intensity patterns. Because the rays cross in the pinhole, the ray from the left light bulb makes the right-hand intensity pattern, and vice versa. The angle between the rays coming from the light bulbs is labeled theta min. Each ray hits the screen at the central maximum of the intensity pattern that corresponds to the object from which the ray came. The central maximum of object one is at the same position as the first minimum of object two, and vice versa.
(a) Graph of intensity of the diffraction pattern for a circular aperture. Note that, similar to a single slit, the central maximum is wider and brighter than those to the sides. (b) Two point objects produce overlapping diffraction patterns. Shown here is the Rayleigh criterion for being just resolvable. The central maximum of one pattern lies on the first minimum of the other.

Connections: limits to knowledge

All attempts to observe the size and shape of objects are limited by the wavelength of the probe. Even the small wavelength of light prohibits exact precision. When extremely small wavelength probes as with an electron microscope are used, the system is disturbed, still limiting our knowledge, much as making an electrical measurement alters a circuit. Heisenberg’s uncertainty principle asserts that this limit is fundamental and inescapable, as we shall see in quantum mechanics.

Calculating diffraction limits of the hubble space telescope

The primary mirror of the orbiting Hubble Space Telescope has a diameter of 2.40 m. Being in orbit, this telescope avoids the degrading effects of atmospheric distortion on its resolution. (a) What is the angle between two just-resolvable point light sources (perhaps two stars)? Assume an average light wavelength of 550 nm. (b) If these two stars are at the 2 million light year distance of the Andromeda galaxy, how close together can they be and still be resolved? (A light year, or ly, is the distance light travels in 1 year.)

Strategy

The Rayleigh criterion stated in the equation θ = 1 . 22 λ D size 12{θ=1 "." "22" { {λ} over {D} } } {} gives the smallest possible angle θ size 12{θ} {} between point sources, or the best obtainable resolution. Once this angle is found, the distance between stars can be calculated, since we are given how far away they are.

Solution for (a)

The Rayleigh criterion for the minimum resolvable angle is

θ = 1 . 22 λ D . size 12{θ=1 "." "22" { {λ} over {D} } } {}

Entering known values gives

θ = 1 . 22 550 × 10 9 m 2 . 40 m =  2.80 × 10 7 rad.

Solution for (b)

The distance s size 12{s} {} between two objects a distance r size 12{r} {} away and separated by an angle θ size 12{θ} {} is s = size 12{s=rθ} {} .

Substituting known values gives

s = ( 2.0 × 10 6 ly ) ( 2.80 × 10 −7 rad ) = 0 . 56 ly.

Discussion

The angle found in part (a) is extraordinarily small (less than 1/50,000 of a degree), because the primary mirror is so large compared with the wavelength of light. As noticed, diffraction effects are most noticeable when light interacts with objects having sizes on the order of the wavelength of light. However, the effect is still there, and there is a diffraction limit to what is observable. The actual resolution of the Hubble Telescope is not quite as good as that found here. As with all instruments, there are other effects, such as non-uniformities in mirrors or aberrations in lenses that further limit resolution. However, [link] gives an indication of the extent of the detail observable with the Hubble because of its size and quality and especially because it is above the Earth’s atmosphere.

Questions & Answers

how can we find absolute uncertainty
ayesha Reply
it what?
Luke
in physics
ayesha
the basic formula is uncertainty in momentum multiplied buy uncertainty In position is greater than or equal to 4×pi/2. same formula for energy and time
Luke
I have this one question can you please look it up it's 9702/22/O/N/17 Question 1 B 3
ayesha
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Ojeh Reply
what is the dimension of strain
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Is there a formula for time of free fall given that the body has initial velocity? In other words, formula for time that takes a downward-shot projectile to hit the ground. Thanks!
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Sahim
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Kulsum
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watt is the unit of power
Rahul
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Rahul
watt can also be expressed as Nm/s
Rahul
what s i unit of mass
Maxamed
SI unit of mass is Kg(kilogram).
Robel
what is formula of distance
Maxamed
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i can't understand
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Imoreh
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50 N... (50 *1.732)N
Sahim
Plz cheak the ans and give reply..
Sahim
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Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
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Irungu
Most welcome
Sahim
Hey.. I've a question.
Sahim Reply
Is earth inertia frame?
Sahim
only the center
Shii
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Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
someone please give answer to this.
prakash
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
so solve it
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please
Ebenezer
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prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
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Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
I want learn physics
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Vinodhini
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Vinodhini
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Vinodhini Reply
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revolutionary
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Vinodhini Reply
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Vinodhini
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by reading it
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understanding difficult
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check out "LMES" youtube channel
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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