Fundamental Concept
Light and other Electromagnetic Waves have dual character, i.e. they show properties of both matter and waves. Young's Double Slit Experiment is a proof of the Wave Nature of Light while The Photoelectric Effect proves the partical Nature of Light.
The basic logic as I understand it is that, when particles meet at a point, they collide. When two waves meet at a point they interfere. It is like drawing two graphs on the same graph paper and then getting the resultant.
If two waves interfere at a point, then there Amplitudes are related as :
Anet = A1 + A2
A1 and A2 are the amplitudes of the two different waves and Anet is the resultant magnitude.
The Intensity of a wave is defined as the power transferred per unit area. It is directly proportional to the square of it's amplitude.
I ∝ A2So if two waves having Intensities I1 and I2 interfere, then the Intensities will add up as :
√Inet = √I1 + √I2Squaring this equation will give us the result :
Inet = I1 + I2 + 2√(I1 I2 )
Interference Of Waves
Light is a Transverse wave and the general equation of a Transverse wave is of the form :
Phase Difference :
The angle within the sin or cos function is called the Phase of the wave. Phase difference between two waves is the difference of the angles measured in radians.
Notice the difference between waves in Phase and waves Out of Phase.
In this diagram the two given waves have a phase difference of θ
When we consider two waves to be interfering at a point, kx becomes a constant so can be combined with φ.
Phase difference can also be represented as phasors ( or vectors whatever you want to call it ) :
y = A.cos( ωt - kx + φ )Here y and x represent position, k and φ are constants ( Can you guess their dimension? ), t represents time and ω is the angular frequency of the wave. A represents the Amplitude of the wave.
Phase Difference :
The angle within the sin or cos function is called the Phase of the wave. Phase difference between two waves is the difference of the angles measured in radians.
"Sine waves same phase". Licensed under Public Domain via Wikimedia Commons.
"Sine waves different phase". Licensed under Public Domain via Wikimedia Commons.
Notice the difference between waves in Phase and waves Out of Phase.
"Phase shift" by Peppergrower - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
In this diagram the two given waves have a phase difference of θ
When we consider two waves to be interfering at a point, kx becomes a constant so can be combined with φ.
Phase difference can also be represented as phasors ( or vectors whatever you want to call it ) :
 |
| "Amplitude Phasors" by Mridul Kothari |
This representation makes our job a lot easier. If I now need to determine the resultant Amplitude, I just need to vectorially add A1 and A2 . This would give me the length of the resultant vector as
Aresultant = √(A21In terms of Intensities the formula would be :+ A22 + 2A1A2cos(θ))
Iresultant = I1 + I2 + 2 √(I1 I2 ) . cos(θ)
Thus from the above equation we can conclude that Intensity is minimum when cos(θ) = -1 and Intensity is maximum when cos(θ) = 1. These two cases correspond to Destructive and Constructive Interference respectively.
Conditions for Sustained Interference :
Interference happens all around us all the time, but the following conditions need to be satisfied in order to obtain sustained interference which is noticeable.
- The two interfering waves must be Coherent. This means that the phase difference between the two sources must be a constant.
A constantly changing phase difference would mean interference pattern would also keep changing so it is going to be harder to observe it. - The two interfering sources must emit light of the same Wavelength and Time Period ( which implies that ω is also the same ).
If the Time Periods are different, a condition similar to the previous case will arise. Also if the Time Periods are equal, it implies that the frequencies and hence the Wavelengths are also equal ( Speed of Light remains constant ). - The Amplitudes should be approximately the same, as this would ensure that minimum intensity is zero.
the Amplitudes should be approximately the same because A1 = A2 implies that I1 = I2 . Thus when these waves Interfere, the minimum Intensity will be zero, so noticeable bright and dark fringes will be obtained. - The two sources must be narrow or extremely small.
A broad source is the same as a large number of smaller sources, each pair of which will be having its own pattern.
Path Difference :
The Path difference between two waves, is the difference in the length of paths traveled by the two waves. It is usually mentioned as a multiple of λ ( The Wavelength ).
The Path difference between two waves, is the difference in the length of paths traveled by the two waves. It is usually mentioned as a multiple of λ ( The Wavelength ).
A path difference of λ means that the two waves are in Phase and there is Constructive Interference.
A path difference of λ/2 means that there is Destructive Interference between the two waves.
A path difference of 3λ/2 is equivalent to a path difference of λ/2 since the sinusoidal wave repeats itself after every λ.
We also know that waves repeat after a phase change of 2π . Therefore it can be said that a path difference of λ is equivalent to a phase difference of 2π .
Thus we have Constructive Interference at path differences of nλ and Destructive at path differences of nλ/2 ( here n is an Integer ).
If the path difference between two waves is δx and the phase difference between them is φ then these quantities are related as :
In Young's Double Slit Experiment (YDSE) light from one source is made to fall on two very narrow slits. These two narrow slits act as sources of Secondary wavefront. Since the Initial source was the same, the light from the secondary sources has the same Wavelength, Amplitude and both the waves are in the same Phase.
Therefore all conditions for sustained interference are met.
A path difference of 3λ/2 is equivalent to a path difference of λ/2 since the sinusoidal wave repeats itself after every λ.
We also know that waves repeat after a phase change of 2π . Therefore it can be said that a path difference of λ is equivalent to a phase difference of 2π .
Thus we have Constructive Interference at path differences of nλ and Destructive at path differences of nλ/2 ( here n is an Integer ).
If the path difference between two waves is δx and the phase difference between them is φ then these quantities are related as :
δx/λ = φ/2π
Young's Double Slit Experiment
"Double-slit schematic" by Peter Suppenhuhn, svg version by Trutz Behn - Own work. Licensed under Public Domain via Wikimedia Commons.
In Young's Double Slit Experiment (YDSE) light from one source is made to fall on two very narrow slits. These two narrow slits act as sources of Secondary wavefront. Since the Initial source was the same, the light from the secondary sources has the same Wavelength, Amplitude and both the waves are in the same Phase.
Therefore all conditions for sustained interference are met.
In the diagram above, d ( the distance between the source and screen ) is much larger than a ( the distance between the sources ). Therefore we can make two assumptions necessary for us to obtain the fringe width.
- α is approximately equal to α'.
- α is very small so sinα' = tanα .
The above assumptions can be made because a << d . And because we do not consider the fringes to be forming at a large distance ( x in given diagram ) from the center. Thus we only consider cases for which x << d.
Which basically means that α and α' are both very small angles ( This does not mean that whenever two angles tend to zero they are approximately equal ).
Once you've made those assumptions, using sin α' = tan α you will get the path difference to be :
Which basically means that α and α' are both very small angles ( This does not mean that whenever two angles tend to zero they are approximately equal ).
Once you've made those assumptions, using sin α' = tan α you will get the path difference to be :
Δs = ax / d
For Δs = 0 , both the rays would have travelled an equal path. So Δs = 0. And we obtain the First Maxima.
For Δs = ( + or - )λ/2 ( => x = λd/(2a) ) we will have Destructive Interference. so we shall obtain the First Minima.
For Δs = ( + or - )λ ( => x = λd/a ) we will have Constructive Interference. so we shall obtain the Second maxima.
This distance between the First and Second maxima ( or any two other maximas or any two other minimas ) is called the Fringe Width . Almost all questions from Young's Double Split Experiment require the equation for fringe width in some form. Remember this equation.
Fringe Width = λ * D/a
After the Second maxima will be the second Minima after which will be the third Maxima and Minima and so the pattern goes on.
According to our approximation the Intensity at each maxima should be 4I0 ( if you're wondering how that came about, go up and check the equation for intensity, don't be lazy :-P ). and each minima should have zero Intensity.
In reality however, the First Central Maxima is the Brightest and the First Minima is the Darkest. As you go away from the center on either side, the brightness of maxima keeps decreasing and that of minima keeps increasing, so if you go too far away there won't be a definite regular pattern.
The wonderful thing about Young's experiment is that you can use it to calculate the wavelength of light. You can get the fringe width, a and d from the experimental data, so you can obtain the wavelength of light!
There was only one main concept ( the addition of Amplitudes ). The rest of it has been pure application of that one theory. That's the beauty of Physics. You may forget formulas, but if you know your basics, you can easily derive the formula mentally. There's just no need to mug up!
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