Electromagnetic radiation; light

The only two fundamental forces of nature that human beings can directly experience through their five senses are gravity and light. The other sensations such as smell, heat, sound and so on detect macroscopic properties of matter, and not its fundamental forces or constituents. Electricity is deeply linked with light. It is the accelerated movement of electric charge that is responsible for the generation of light: an accelerating charge emits radiation. That part of radiation that is visible to us is called light. Famous physicists such as Michael Faraday, James Clerk Maxwell, H. Coulomb, and so on are nineteenth century figures who were crucial in reaching our current understanding of electricity and magnetism. There is an anecdote that when Faraday had announced his discovery, he was asked as to what was its commercial value, to which he replied "Nothing". Today, industries based on his discoveries are valued at over US$300 billion! Light is a phenomenon that has fascinated human minds for millennia, and a satisfactory explanation of its properties and underlying principles has been possible only in the last century. Besides being essential for vision (the eye ``sees'' objects by receiving light that has bounced off the object in question), light has almost limitless applications in daily life. On a more general level, all life on earth is powered by the light energy that is received from the Sun, and it through the process of photosynthesis that all the energy of timber, fossil fuels and so on has become available on earth. The most important property of light that distinguishes it from a particle is that light is spread over a finite volume of space, be it in a lighted room at night, or the light coming from the Sun. A particle in contrast occupies a definite volume in space, and in Newtonian mechanics a particle is taken to be at a single point, that is, occupying zero volume in space. Light is a special case of the more general phenomenon of waves, and the discussion of light is made more coherent by first analyzing the general properties of waves. We will then go on to discuss the special case of electromagnetic waves - also called radiation - and show that light in fact is a form of electromagnetic wave. For a long time, physicists were of the view that - similar to all the other waves seen in nature -electromagnetic waves, or radiation in short, was also the result of the oscillations of an underlying medium, which was called the ether. It was only in the early twentieth century that precise observations finally ruled out the existence of an ether, and the theory of special relativity provided a more accurate explanation of the nature of radiation. We now know that radiation is a special case of transverse wave in that there is no medium undergoing transverse oscillations, as in the case of a string. Instead, radiation is the result of electric and magnetic fields simultaneously propagating in empty space. Similar to all wave phenomena, radiation is characterized by its wavelength $\lambda$ and its frequency $f$. Electromagnetic radiation propagates at the velocity of light $c$. We have from eq.(6.37), that

$$f \lambda=c$$ (6.1)

From the special theory of relativity, it is known that the velocity of light $c$ is a universal constant. Hence, in describing radiation, we can dispense with frequency of radiation $f$, and use only its wavelength $\lambda$ to fully describe its properties. The wavelength of radiation has a vast range, from waves with infinitely long wavelengths to waves of infinitely short wavelength. Starting from long wavelengths, we have radio waves which have a wavelength of about 500 m, followed by short wave radio waves, and so on down to around 1 m wavelength for TV waves. We then have microwave radiation with wavelength around 1 cm, followed by infrared radiation down to $10^{-6}$m. Visible light consists of radiation with wavelength from $10^{-6}$m to $10^{-7}$m. As the wavelength of light decreases, the human eye perceives it as light changing its colour from red to violet. Radiation of wavelength less than $10^{-7}$m is invisible and is called ultraviolet radiation. Radiation with wavelength from $10^{-9}$m to $10^{-12}$m are called X-rays, and radiation shorter than $10^{-12}$m all the way to vanishingly small wavelength are called gamma-rays. The electromagnetic spectrum is given in Figure 6.1.

Figure 6.1: The Electromagnetic Spectrum

One can easily see from the list of radiation that the various forms of radiation have unique properties that have been put to society's use in varied manners, from X-rays in medical science to TV, radios, mobile phones, microwaves, communications and so on. The most accurate description of radiation is provided by the quantum theory of radiation, which is briefly discussed in Section 11.3. In the quantum description of radiation, electromagnetic waves are composed out of quantum packets of energy and momentum called photons. At very small wavelengths of radiation, the description based on photons is the only accurate one, and no classical approximation is adequate. Also, the concept of photons is required for understanding very low intensity radiation since this phenomenon involves only a few photons. However, photons of wavelength greater than X-rays can be approximately treated as a classical transverse wave, which is what is called radiation. As the wavelength varies, different descriptions of radiation become more convenient, of which descriptions (a) and (b) are only approximately correct. (a) Geometrical Optics. The wavelength of radiation is much smaller than the size of the equipment or probe being used to study light, and the energy of the photon is smaller than the sensitivity of the apparatus. This is the case for visible light. (b)Classical electromagnetic theory. The wavelength of radiation is comparable to the size of the equipment, as is the case with radio waves. (c) Photons. The need for the quantum theory of photons becomes necessary when the energy of the photon is larger than the sensitivity of the apparatus. This is the case with human vision, with the human eye being able to detect incoming light consisting of as few as 4-5 photons.

Geometrical Optics :Fermat's Principle of Least Time

So far, we have considered light to be a special case of a (transverse) wave, namely an electromagnetic wave. Recall in Section 6.3, we discussed a regime of radiation for which its wavelength is much smaller than the size of the equipment or probe being used. In the case of light this regime can be approximated by what has been earlier called Geometrical Optics. Note the important fact that since we are taking the wavelength $\lambda$ to be very small the relation

$$f \lambda=c$$ (6.2)

implies that we simultaneously need to make the frequency of light very large. Fortunately, this is the case with visible light, and hence the possibility of making a consistent approximation of photons. In geometrical optics light behaves like a ray that is specified by its direction of propagation. The fundamental postulate of geometrical optics is that light energy travels in rays. The ray travels in a straight line from point to point; that is, in going from point $A$ to point $B$, light rays will follow a straight line. As shown in Figure 6.2, light energy which is converging (diverging) is represented by a set of converging (diverging) set of rays.

Figure 6.2: Light rays

The question naturally arises, how do light rays propagate in the presence of material bodies. In particular, two well known cases are the reflection of light off polished surfaces (mirrors) as well as the refraction of light in passing from one medium to another. Although the speed of light in vacuum is known to be a universal constant, the speed of light varies in different media. The refractive index $n>1$ of a medium, which allows for the propagation of light, is defined by how much light slows down in that medium compared to vacuum. The refractive index in general depends on the wavelength of light, since different wavelengths behave in a different manner; we will ignore this subtlety in our discussion on refraction. If the speed of light in the medium is $c_m$, the definition of refractive index $n$ is given by

$$c_m = \frac{c}{n}<c \mbox{\rm { ; n: refractive index}}$$ (6.3)

Fermat's Principle provides an answer to the question of propagation of light in the presence of different media. The first guess one might have is that light should take the path that minimizes the distance covered in going from point to point, as this would be a straight forward generalization of light rays moving in a straight line in empty space. Take the example of refraction, in which a light ray goes from say point $A$ in air to point $B$ in glass, as in Figure 6.3.

Figure 6.3: refraction

Clearly, the straight line between $A$ and $B$ is the shortest distance. But light taking this path would contradict the observed fact that light ``bends'' in going from air to glass. This phenomenon can easily be seen by inserting one's hand into water, which looks bent. The hint for Fermat's principle can be seen from eq.(6.3); since a light ray is traveling slower in a more dense medium, it could choose to travel in a manner which minimizes the time it takes from going from point $A$ to point $B$, with the points being in any arbitrary media. We hence have Fermat's Principle which states: Light rays follow the path of least time in going from point to point. For light rays traversing a single medium, a path with minimum distance covered is equivalent to path with minimum time taken of travel. Hence Fermat's principle of minimizing time of travel reproduces the result that light, in vacuum, travels in a straight line. The power of Fermat's principle lies in the fact that it can also account for the behaviour of light rays in arbitrary and multiple media.

We restate Fermat's principle in a more mathematical form. Let $T$ be time taken for the light ray to go from point $A$ to point $B$ by taking some path. Then Fermat's principle states that

$$\delta T=0 \mbox{\rm { : Points A and B fixed}}$$ (6.4)

The symbol $\delta$ acting on $T$ and resulting in zero means that a small variation in the path of light near its path of minimum time yields approximately no change in the time taken to go from initial to final point. Light ``chooses'' a path for which the time taken by it is a minimum; or equivalently, from the definition of what a minimum means, light ``chooses'' a minimal path such that there are many other nearby paths which take almost the same time.

Reflection

Consider the reflection of light off a mirror. As shown in Figure xxx, the light ray starts at point $A$, is incident at point $O$ of the mirror at an angle $\theta_i$ with respect to the perpendicular to the mirrored surface, and then is reflected through point $B$, making a reflected angle $\theta_r$. Given the angle of incidence $\theta_i$, we apply Fermat's principle of least time to determine the angle of reflection $\theta_r$. The path which goes directly from point $A$ to point $B$ is a trivial solution; we are interested only in paths that reflect off the mirror. Since the entire path of the ray is in a single medium, the path with a minimum time is equivalent to a path with minimum distance. The distance traversed by the light ray in going from $A$ to $B$ is given by $AOB$. This distance can be determined by reflecting point $B$ about the mirror, with reflected point denoted by $B'$ as shown in Figure6.4.

Figure 6.4: Reflection

The distance $AOB'$ is equal to the distance $AOB$. As shown in Figure 6.4, consider a light ray incident at point $O$, and suppose the reflected ray goes through points $B_1$, or through point $B_2$, and so on. We need to determine what is the correct direction of the reflected ray. Consider the reflections, namely $B_1'$ and $B_2'$. Clearly, the distance between $A$ and $B_1'$, $B_2'$ and so on is a minimum if the line connecting them is a straight line. This in turn implies from elementary geometry shown in Figure 6.4 that

$$\theta_r=\theta_i$$ (6.5)

Hence, Fermat's principle yields that the angle of incidence is equal to the angle of reflection, a well known result of geometrical optics.

Refraction

Refraction is the phenomenon of light propagating through several media with different indices of refraction. Consider light rays propagating in air with refractive index $n_1$, and then refracting (propagating) through a piece of glass with refractive index $n_2$, and follows a path shown in Figure 6.3. Recall the index of refraction has been defined in eq.(6.3). From its definition, we see that $n$ is dimensionless. Table 6.1 gives the indices of refraction for some typical transparent materials at a particular frequency of light.

Table 6.1: Indices of Refraction for Light with $\lambda =589\times 10^{-9}$m

Medium	$n=c/c_m$
Vacuum	$1.0000$
Air	$1.0003$
Water	$1.33$
Plexiglas	$1.51$
Crown glass	$1.52$
Quartz	$1.54$
Diamond	$2.42$

The relation of the angles made by light in the two media goes under the name of Snell's law. We now derive what the value of angle $\theta_2$ should be, given the inputs of $\theta_1, n_1$ and $n_2$. This derivation is more difficult than the case of reflection because, due to the varying velocity of light in the two different media, the path of minimum distance is no longer the same as the path of minimum time.

Instead of plunging into the details of the derivation, let us illustrate the problem with an analogy. Since the velocity of light is faster in air than in glass, an analogy of the propagation of light in air is a man running. The analogy of light propagating in glass can be taken to be that of a man swimming. Suppose a lifeguard standing on the ground is trying to reach a buoy floating in a swimming pool in the shortest possible time. If he goes in a straight line, he will spend a certain fraction of time running on the ground, and the remaining fraction in swimming. Since he swims slower than he runs, he should try run for a longer time on the ground before diving into the pool. On the other hand, if he runs too far on the ground, he will end up swimming more than he should. The exact point where he should dive into the pool for minimizing the time he takes to reach the buoy is the analogy of how light rays should bend as they leave one medium and enter the other.

Figure 6.5: Lifeguard Choosing Optimum Path, with Dashed Paths being Non-Optimal

The result of minimizing the time taken by light yields Snell's law of refraction, namely

$$n_1\sin \theta_1=n_2\sin \theta_2$$ (6.6)

Recall in deriving the law of reflection, we found the shortest path by a geometrical method, and this was sufficient. In the derivation of Snell's law we need to have a deeper understanding of Fermat's principle, in particular, what does it mean to say that the time taken for a light ray is a minimum? To implement Fermat's principle, in analogy with the discussion on how to find the minima of a function, we need to compare two paths close to each other. We will try and fix the value of $\theta_2$ by the requirement that the time taken by these paths is equal. If we can carry out this procedure successfully, we can then conclude that we have found the path which takes light the minimum time to go from $A$ to $B$. Consider, as shown is Figure 6.6, Path 1 along AOB and Path 2 along APB. We compare the time taken by light, along Path 1 and Path 2, in traveling from point $A$ to point $B$. Path 1 and Path 2 hit the boundary of the two media at points $O$ and $P$ respectively. A shown in Figure 6.3, Path 1 has $\theta_1$ as its angle of incidence and $\theta_2$ as its angle of refraction. To determine the difference in the distance traveled by light for the two paths, draw $XP$ perpendicular to $AO$ and $OY$ perpendicular to $BP$. See Figure 6.6. Path 1 travels an extra distance of $OX\equiv d_1$ in medium 1 compared to Path 2, and in medium 2 Path 2 travels an extra distance of $PY \equiv d_2$ compared to Path 1.

Figure 6.6: Path of Minimum Time

Recall velocity of light in air is $$\frac{c}{n_1}$$, and in glass is $$\frac{c}{n_2}$$. The extra time spent by light in traversing Paths 1 and 2 is, consequently, given by $$n_1\frac{d_1}{c}$$ and $$n_2\frac{d_2}{c}$$ respectively. We can choose either Path 1 or Path 2 to be the path of minimum time, as they are taken to be very close to each other. Note

$$T_1 = \mbox{\rm {Extra Time Taken for Path 1}}= n_1\frac{d_1}{c}$$ (6.7)

$$T_2 = \mbox{\rm {Extra Time Taken for Path 2}}=n_2\frac{d_2}{c}$$ (6.8)

For Path 1 to be the path of minimum time, Fermat's principle requires that the difference in time taken by light for Paths 1 and 2 should be zero, that is

$$0 = \delta T=T_1-T_2$$ (6.9)

$$\Rightarrow n_1 d_1 = n_2 d_2$$ (6.10)

From Figure 6.6 we have that

$$d_1 = OP\sin \theta_1$$ (6.11)

$$d_2 = OP\sin \theta_2$$ (6.12)

Hence, from equations above we have the condition for minimum time given by

$$n_1\sin \theta_1 =n_2 \sin \theta_2 \mbox{\rm { : Snell's Law}}$$ (6.13)

Question: In our derivation of Snell's law, where did we use the condition that Path 1 and Path 2 are close to each other?

Total Internal Reflection

If light travels from a more dense to a less dense medium, as in the case of light traveling from water into air, we have the phenomenon of total internal reflection in which light is refracted back into the more dense medium as shown in Figure 6.7.

Figure 6.7: tir

If the incident angle is equal to a critical angle, called $\theta_C$, the refracted angle becomes $90^o$, and is given by Snell's law by the relation

$$\sin \theta_C=\frac{n_2}{n_1}; \mbox{ }n_1>n_2$$ (6.14)

For all incident angles greater than $\theta_C$, light undergoes total internal reflection and is reflected back into the more dense medium. One important application of this phenomenon is the entire field of fibre optics. Fibre optic cables are made out of dense optically transparent fibres, and light propagates inside these light pipes due to total internal reflection with almost no loss of energy. Fibre optics have many uses from transmitting high volumes of telecommunication signals, as well as in medical laproscopic procedures.

Electromagnetic Field

What are electromagnetic waves made out of? We have mentioned a few times that, unlike ordinary sound or water waves, electromagnetic waves are not the result of the oscillations of any underlying medium. So what is the entity that is undergoing wave motion? To answer this question, we need to introduce a major new concept, which is the concept of a field. To understand the concept of a field, we need to first re-examine the conception in physics as to what is a material entity. From the time of Newton until the mid-nineteenth century, all material entities were thought to be similar to a solid body. That is,a solid body has a definite position in space, and with the body ``carrying'' its energy and momentum. Recall that in Section 5.1.4, in our discussion on the propagation of energy for a taut string, we constructed the continuous string out of infinitesimal point particles. All solid bodies in turn can similarly be thought of as being composed out of small point-like constituents, and hence all physical entities can be thought of as being composed out of a collection of point particles. This view has had great success, including the formulation of fluid mechanics and of aerodynamics. In the nineteenth century, a major conceptual revolution was wrought by physicists such as Michael Faraday and James Clerk Maxwell. In their study of electricity and magnetism, they discovered that the effects of, for example an electric charge could be best understood by postulating a physical entity that exists in its own right, and carries the effects of the charge. This entity was called a field. So, for example, if we introduce an electric charge into a space, the electric charge generates an electric field which then acts on say other charges. If the charge introduced is a positive charge, the field will act on other neighboring charges and create an attractive force on negative charges and a repulsive force on positive charges and so on. The fundamental and unique feature of a field, in contrast with a particle, is that the field is spread all over space, whereas the particle occupies a definite position in space. A particle is described by specifying its position in space, and in three dimensions, this means specifying three numbers, namely the particles $x,y$ and $z$ coordinates. In contrast, the electric field, for example, is described by assigning numerical value for the field at every point in space. In other words, a field is a function of space (and of time), and has much more information than say a particle. The field is a material entity that is as physical and as ``substantial'' as a particle. Just like a particle, a field has energy, momentum, angular momentum, entropy and so on. The behaviour of a particle is typically as shown in Figures 3.2 and fig1b. A wave is an example of a field, and as shown in Figure 5.2, is spread over space. In addition to the electric field, we also have the magnetic field, denoted by ${\bf B}$. The magnetic field, for example, describe how two magnets affect each other. The great discovery of Faraday and Maxwell was the understanding that electric and magnetic fields are really the same entity, and appear to be different only in certain special circumstances. A charge moving with a constant velocity, for example, generates a combination of both electric and magnetic fields. Furthermore, by transforming from one inertial frame to another, the values of the electric and magnetic fields change, showing that these are frame dependent, and not physical, quantities. The electric and magnetic fields ${\bf E}$ and ${\bf B}$ are generated by arbitrary collections of moving charges. ${\bf E}(t,x)$ and ${\bf B}$(t,x) are vectors that depend on time, and are spread all over space; what this means is that at every instant $t$, at every point $x$ in space there are three real numbers that specify the electric field, and similarly for the magnetic field. In other words, we need an infinite collection of vectors, one for each instant $t$ and each point $x$ to specify the electric and magnetic fields.

Electromagnetic Waves

An accelerating charge generates electromagnetic waves. Electromagnetic fields are a special case of transverse waves. As is the case for any wave propagation, energy has to be constantly pumped into the system, in this case, the work done to accelerate a charged particle for sustaining electromagnetic waves.

Intuitively speaking, if one attaches an electric charge to the end of a stick, and shakes the stick back and forth with a frequency $f$, the charge will ``radiate off'' electromagnetic waves of frequency $f$. The energy in the waves comes from the energy being expended in shaking the charge. The only reason that the the radiation that we create by shaking a piece of stick is not easily observable is because the amplitude of the radiation is very low, making detection difficult. A modern antennae is not much different from our intuitive picture, since in an antennae charge is made to flow back and forth - with some frequency - along the length of an antennae, and results in the emission of electromagnetic radiation. Radio waves are generated in this manner, and constitute the signals that are up picked by a radio receiver. Electromagnetic radiation is the result of the simultaneous oscillations of electric and magnetic fields. A changing electric field creates a magnetic field, and in turn, a changing magnetic field creates an electric field. It is this positive feedback mechanism that sustains the propagation of electromagnetic wave over billions of light years of distance. Since the electric field ${\bf E}$and magnetic field ${\bf B}$ are three-dimensional vectors, the propagation of radiation is only possible in three-dimensions. For simplicity, we study the electric and magnetic fields far from the charges that have generated the radiation. Suppose the electromagnetic wave is propagating in vacuum along the $x$-direction. Since the electromagnetic radiation is a transverse wave, the oscillations of the electric and magnetic field are in the directions perpendicular to the direction of propagation, and hence lie in the $yz$-plane. Let us denote the unit vector in the $xyz$-directions by $${\bf e}_x,{\bf e}_y$$ and ${\bf e}_z$ respectively. The simplest example of radiation is given by the electric field ${\bf E}$ always lying along the $y$-axis, and the magnetic field ${\bf B}$ always lying along the $z$-axis. In other words, the ${\bf E}$ and ${\bf B}$ fields for radiation are similar to the transverse wave given in eq.(5.22), with the added property that all electromagnetic waves propagate in vacuum at the speed of light. From eq.(6.1), we have

$$f=\frac{c}{\lambda}$$ (6.15)

Hence, from eqs.(5.22) and (6.23), the electric and magnetic fields that constitute electromagnetic wave propagation are given by

$${\bf E}(t,x) = E_0\sin\big( \frac{2 \pi}{\lambda}(x-c t)\big){\bf e}_y$$ (6.16)

$${\bf B}(t,x) = \frac{E_0}{c}\sin\big( \frac{2 \pi}{\lambda}(x-c t)\big){\bf e}_z$$ (6.17)

The propagation of the radiation given by eqs. (6.24) and (6.25) is shown in the Figure 6.8, with the direction of propagation being in the $x$-direction, and the sinusodial oscillations of the ${\bf E}$ and ${\bf B}$ field drawn in the $yz$-planes.

Figure 6.8: Electromagnetic waves

The energy, per unit volume, of the electric and magnetic fields at point $(x,y,z)$, denoted by $u$, is given by

$$u=\frac{1}{2}\epsilon_0[{\bf E}^2+c^2{\bf B}^2]$$ (6.18)

To obtain the total energy contained in the electromagnetic field, we have to sum the contribution to energy from all the volume of space. To develop a physical sense of the energy of a field, one can imagine breaking up space into volume elements, say into little cubes of side $a$ and with elemental volume $\Delta V (= a^3)$. In every volume of $a^3$, one can think of the field in that volume as being a tiny harmonic oscillator that can each carry a tiny amount of energy equal to

$$u\Delta V$$ (6.19)

The total energy of the field $U$ is then obtained by summing over the energy of all the oscillators in the volume elements, and we obtain

$$U=\sum_{\mbox{\rm { All volume elements}}}u\Delta V$$ (6.20)

Energy propagates along the direction of motion of the wave, which in the case of the electromagnetic wave being considered is in the $x$-direction. The intensity of light is a measure of how much radiant energy is received at a point, and as can be seen by the formula above, is proportional to ${\bf E}^2$. To understand the dimensions of the ${\bf E}$ and ${\bf B}$, recall $\epsilon_0$ is the permittivity constant defined in eq.(3.98), and is a measure of the strength of electromagnetic fields in vacuum. In a medium other than vacuum, the permittivity constant of the medium would appear in the energy equation eq.(6.26). From eqs.(3.97) and (3.98) $[\epsilon_0]=M^{-1}L^{-3}T^2C^2$. Since $U$ is the energy per unit volume, $[U]=ML^{-1}T^{-2}$, and hence from eq.(6.26) we have

$$[{\bf E}] = \frac{ML}{T^2}\frac{1}{C}$$ (6.21)

$$= \frac{\mathrm{Force}}{\mathrm{Charge}}$$ (6.22)

From a simple exercise in dimensional analysis, we have obtained an important result that the electric field ${\bf E}$ is proportional to a force. The SI unit for electric field

$$[{\bf E}]= \mbox{\rm {newton per coulomb}}$$ (6.23)

From eq.(6.26) it also follows that

$$[{\bf B}] = \frac{[{\bf E}]}{[c]}$$ (6.24)

$$= \frac{M}{T}\frac{1}{C}$$ (6.25)

and hence the SI unit of ${\bf B}$ is given by

$$[{\bf B}]= \mbox{\rm {newton per (coulomb-meter/sec)}}$$ (6.26)

Recall that an accelerated charge generate electromagnetic radiation; we see from equation (6.37) that electric and magnetic in turn react back on charged particles, and exert a force on them. We will study these effects in some detail in the chapter on electric and magnetic fields. The momentum of the electromagnetic field is along its direction of propagation, and the field momentum per unit volume is given by

$${\bf P}=\epsilon_0\frac{\vert{\bf E}\vert\vert{\bf B}\vert}{c}{\bf e}_x$$ (6.27)

Light, as has been mentioned in our earlier discussion on the electromagnetic spectrum, is a special case of electromagnetic waves when its wavelength is in the range of $10^{-6}$m to $10^{-7}$m, and which is visible to the human eye.

Interference and Diffraction

The phenomena of interference and diffraction of light are essentially similar, and both are special cases of the superposition of waves. When we studied the superposition of waves, we considered the simple example of a waves propagating along a fixed direction, which was taken to be along the x-axis. We now need to consider a more general case of wave propagation in two dimensions.

Interference

Consider the famous experiment of Thomas Young in 1801 which conclusively showed that light is a wave. Consider a bright source of light that goes through an opaque screen which has two slits. These slits let through light, and act as two sources of light. We want to ascertain the intensity of light on a screen as some fixed distance from the two slits.

Figure 6.9: Interference Experiment

As shown in Figure 6.9,the distance between the two slits $S_1$ and$S_2$ is denoted by $d$, and the distance of the point $P$ from $S_1$ and $S_2$ is $r_1$ and $r_2$ respectively. We assume $d«r_1,r_2$. Since we want to determine the intensity of light being received at point $P$ at a distance $y$ along the screen, we need to know the electric field ${\bf E}$ at point $P$ due to to light arriving at $P$ from $S_1$ and $S_2$. Since the intensity of light is proportional to ${\bf E}^2$, we will, for simplicity, ignore the vector nature of the electric field and only examine the magnitude of ${\bf E}$ due to light emanating from the two slits. Let the electric fields at point $P$ due to $S_1$ and $S_2$ be given, respectively, by

$$E_1 = E_0 \sin(2\pi \frac{r_1}{\lambda}-2\pi f t)$$ (6.28)

$$\mathrm{and}$$

$$E_2 = E_0\sin(2\pi \frac{r_2}{\lambda}-2\pi f t)$$ (6.29)

Although the electric field is in phase at the slits (since light is from a common source), the electric field at $P$ from $S_1$ and $S_2$ has a phase difference $\phi$ since light from $S_1$ and $S_2$ have traveled different distances to arrive at $P$. The superposition of waves yields the total electric field at $P$ to be

$$E = E_1+E_2$$ (6.30)

$$= 2E_0\cos(\frac{\phi}{2})\sin(2\pi \frac{r_1+r_2}{\lambda}-2\pi f t)$$ (6.31)

$$\mathrm{where}$$

$$\phi = 2\pi \frac{r_1-r_2}{\lambda}$$ (6.32)

and eq.(6.42) follows from eq.(5.61).

Figure 6.10: Interference

The intensity of light received at the screen is given in Figure 6.10. We analyze phase $\phi$ to determine the pattern of intensity shown in Figure 6.10. Note that the phase $\phi$ is a result of the path difference in the distance from $S_1$ and $S_2$ to $P$. From the Figure 6.9 we see that

$$r_1-r_2 = \delta$$ (6.33)

$$= d\sin A$$ (6.34)

Hence, from eq.(6.43)

$$\Rightarrow \phi=2\pi \frac{d}{\lambda}\sin A$$ (6.35)

The intensity at $P$ is given by

$$I=4E_0^2\cos^2(\frac{\phi}{2})$$ (6.36)

The maxima of intensity is at

$$\frac{\phi}{2}=m\pi, m=0,1,2,3...$$ (6.37)

Hence, from eqs.(6.46) and (6.48), the maxima's of intensity are given by

$$2m\pi = 2\pi \frac{d}{\lambda}\sin A$$ (6.38)

$$\Rightarrow d\sin A = m\lambda, m=0,1,2,3...\mbox{\rm { : maxima's}}$$ (6.39)

If light waves from $S_1$ and $S_2$ arrive at $P$ with a path difference of $d\sin A$ that is a multiple of wavelength $\lambda$,then they are in phase leading to constructive interference. Note the important point that the intensity of light is the same at each of the maxima's. Similarly, the minima's in intensity are the result light arriving at $P$ out of phase and resulting in destructive interference. Similar to above derivation, it can be shown that

$$d\sin A=(m+\frac{1}{2})\lambda, m=0,1,2,3...\mbox{\rm { : minima's}}$$ (6.40)

As one moves the point $P$ on the screen, the angle $A$ changes; only for certain values of $A$ will the equation $d\sin A=m\lambda$ be satisfied for some $m$, and at these points there will be maximum intensity. At other points, the equation for the minima's will be satisfied, leading to minimum intensity. Hence there will be alternating dark and bright bands of light arriving at the screen, and is shown in Figure 6.10. Young's experiment found the maxima's and minima's for the intensity of light at exactly the points that have been derived above, and was a major landmark experiment in proving that light is a wave phenomenon. Two hundred years later, a similar result for electrons verified the wave description of particles in quantum theory.

*Diffraction

Consider a source of light passing through a circular aperture, and being received on a screen. The pattern of intensity received at the screen will show a sharp maxima at the point of the screen closest to the aperture, and then have a series of maxima's (separated by minima's) of lower and lower intensity. See Figure 6.11. This phenomenon is called diffraction, and is unlike interference patterns where all the maxima's have the same intensity.

Figure 6.11: Diffraction Pattern

To understand the physics of diffraction, a N-slit experiment in which light from a source passes through equally spaced N-slits and is then received on a screen.

Doppler Effect for Light

Radiation of light from a source moving at a velocity $w_s$ is more complicated than the Doppler effect discussed for sound waves in Section 5.6. The reason being that in theory of special relativity, the frame in which the source emits the light is different from the frame in which the emitted light is absorbed, since the two frames are moving with a relative velocity given by $w_s$. Due to there being two different frames of reference, the time variable needs to be transformed from one frame to the other according to laws of special relativity. The result we have derived in eq.(5.91)for sound waves gets modified. The speed of light $c$ replaces the velocity of sound $v$. and there is a factor to reflect the difference in the flow of time in the two frames; this yields, similar to eq.(5.91), the following.

$$f'_{\mathrm{Light}} = \sqrt{1-\frac{w_s^2}{c^2}}f'_s$$ (6.41)

$$= \sqrt{1-\frac{w_s^2}{c^2}}\frac{c}{c-w_s}f$$ (6.42)

$$= \sqrt{\frac{c+w_s}{c-w_s}}f$$ (6.43)

Due to laws of special relativity, the velocity of the light source $w_s$ can never exceed velocity of light in vacuum, namely $c$. However, in a medium light travels at a speed less than $c$, and a source can then exceed light's velocity leading to a phenomenon similar to shock waves for sound, and is called Crenenkov radiation.