Superposition of WavesReturn to Table of ContentsPhotons and Light


Electromagnetic Theory.

We will see that light is a form of electromagnetic radiation, so it is instructive to review some of the properties of electricity and magnetism. The original studies of electricity and magnetism date back to at least the early Greek times. By the start of the nineteenth century, it was known that some objects could possess an electrical charge, and that these charges could exert a force on each other even through a vacuum. This force could be described mathematically as

 formula,                                                                 (4.1)

where q is the electrical charge on the object in question and E is the electric field produced by all the other charges in the universe. The charge was discovered to take on a discrete set of values, one of the first examples of quantization. In its turn, the electric field can be described by a scalar potential field V, which is related to the electric field by

 formula.                                                                (4.2)

In addition, it was also noted that a moving charge may experience another force which is proportional to its velocity v. This led to the definition of another field; namely the magnetic field B, such that

                                                          formula.                                                               (4.3)

As with the electric field, the magnetic field is generated by all the other currents in the universe. The magnetic field can be described in terms of a vector potential field A, which is related to the magnetic field by

                                                          formula.                                                                  (4.4)

If the electric and magnetic forces occur concurrently, then the force on the charge is given by the Lorentz force law

 formula.                                                           (4.5)

Maxwell's Equations

The fact that the electric field was described in terms of stationary charges, while the magnetic field was described in terms of moving charges led people to suspect that some relationship existed between the two fields. This was confirmed when it was found that an electric current could be generated by changing the magnetic field. In the mid-1800's, the theories of electricity and magnetism were finally united by James Clerk Maxwell in four equations now known as Maxwell's equations.

 formula,                                                         (4.6)

 formula,                                                                      (4.7)

 formula,                                                    (4.8)

                                                  formula.                            (4.9)

Each one of these can be understood separately.

The first of Maxwell's equations, equation (4.6), is known as Gauss's Law. It relates the flux of electric field intensity to the total charge enclosed by the surface. The flux is defined as

                                                     formula,                                                            (4.10)

where dS is the vector outwardly normal to the surface and the integral is over the entire surface enclosing the region in question. In words, Gauss's law tells us that the total flux through a closed surface, i.e. the change in the number of field lines passing through a closed surface, is proportional to the total charge contained within the volume defined by the surface. Thus if there is no charge inside the surface, the net flux is zero. If there is a positive net charge, the enclosed region acts as a source; if the net charge is negative, the enclosed region acts as a sink.

The constant epsilon is called the electric permittivity of the medium. If the medium is a vacuum, then epsilonepsilon0, where epsilon0 is known as the permittivity of free space and has a value of formula. The electric permittivity was originally used to act as a medium dependent proportionality constant that connects a parallel plate capacitor's capacitance with its geometric characteristics. Conceptually, we can view the permittivity as encompassing the electrical behavior of the medium: in a sense, it is a measure of the degree to which the material is permeated by the electric field in which it is immersed. We can relate the electric permittivity to the dielectric constant by the following formula

                                                      formula.                                                                      (4.11)

The second equation is also a form of Gauss's law, this time applied to the magnetic field. The fact that the enclosed charge is zero tells us that, at least according to classical electromagnetic theory, there is no such thing as a magnetic monopole. In other words, whereas the electrical charge could be viewed as either a positive or negative charge individually, we can never find magnetic charges which do not include both a positive and negative pole. Since the total enclosed charge is the algebraic sum of the charges, this lack of magnetic monopoles automatically insures that the sum is zero.

The third equation is known as Faraday's law. In a manner similar to the electric flux, the magnetic flux is defined as

                                                 formula,                                                                (4.12)

where the surface is now an open surface bounded by a conducting loop. Faraday found that if the induced emf that was developed in the loop depended on the rate at which the magnetic flux changed,

                                                 formula.                                                                   (4.13)

However, the emf exists only as a result of the presence of an electric field, which is related to the emf by

                                                   formula.                                                               (4.14)

Combining (4.13) and (4.14), any direct reference to the induced emf is removed and we get Faraday's law. Physically, this shows us that if the magnetic flux changes, in other words if either the surface area or the magnetic field changes with time, then an electrical field is produced as result. This electrical field creates an emf which acts in such a way as to resist the changes in the magnetic flux. Thus, a time varying magnetic field creates an electric field. Since there are no charges which act as a source or a sink, the field lines close on themselves, forming loops.

The last of Maxwell's equations is known as Ampere's Law. In its original form as expressed by Ampere, it related the number of magnetic field lines which passed through a surface formed by a closed loop to the total amount of current which was enclosed by the loop

 formula,                                                         (4.15)

where j is known as the current density. The open surface is bounded by the loop, and the quantity mu is called the permeability of the medium. In a vacuum, mumu0, where mu0 is called the permeability of free space and has a value of formula. We can relate the permeability of free space with the permeability via the equation

                                                       formula,                                                                   (4.16)

where KB is called the relative permeability. In a manner similar to the dielectric constant, the relative permeability can be viewed as a measurement of how well the magnetic field permeates a material.

While Ampere's law in its original formulation explained many important effects, such as the operation of a solenoid, it was found to also create larger problems. In particular, use of Ampere's law in the form of equation (4.15) led to violation of conservation of energy for the electric and magnetic fields. In order to correct this, Maxwell hypothesized the existence of an additional current, the displacement current, which is defined as

                                                     formula.                                                          (4.17)

When this is combined with Ampere's law in a region with no physical currents, we get

formula.

In other words, just as a time varying magnetic flux lead to the creation of a circular electric field, so to does a time varying electric flux lead to the creation of a linear magnetic field. If a physical current also exists, we again regain the last of Maxwell's equations.

Differential Form of Maxwell's Equation

We can recast Maxwell's equations into a differential form. This form will be necessary later when we begin discussing the wave nature of light. In order to do this conversion, we first need two important results from vector calculus, Gauss's divergence theorem and Stokes theorem. Gauss's divergence theorem tells us that the net flux of a vector field through a closed surface is equal to the integral of the divergence of that field over the volume contained in the surface

 formula.                                                 (4.18)

Similarly, Stokes theorem states that the flux through a closed loop is equal the integral of the curl of the field over the area enclosed by the loop

 formula.                                                 (4.19)

Let's start with Gauss's divergence theorem and apply it to the first two of Maxwell's equations. Then we get

formula

and

formula.

These relations must be equal for any volume, so the first two Maxwell's equations become

 formula                                                              (4.20)

and

 formula.                                                               (4.21)

Applying Stokes theorem to the last two of Maxwell's equations yields

formula

and

formula.

These relations must hold for any surface bounded by a closed loop, so the last two Maxwell's equations become

 formula                                                             (4.22)

and

 formula.                                                 (4.23)

Electromagnetic Wave Equation

How are Maxwell's equations used to show wave motion? Consider the electric and magnetic fields in a charge free vacuum region. Then Maxwell's equations become
formula.

To derive the wave equation for the electric field, start with the third of Maxwell's equations and take the curl of both sides

 formula.                                               (4.24)

The left hand side can be simplified by using the vector relationship

 formula                                        (4.25)

to get

 formula,                                    (4.26)

where the last step used the fact that formula. To evaluate the right hand side of (4.24), we start with the fact that the spatial derivatives (del) and the time derivative can be interchanged. We then use the last of Maxwell's equations to find

 formula.                                          (4.27)

Combining (4.26) and (4.27), we get

 formula,                                                      (4.28)

which we recognize as the three dimensional wave equation for each component of the electric field. Comparing (4.28) with the standard result for a wave, we see that

 formula.                             (4.29)

Using the fact that the experimentally determined speed of light is also 3.00 x 108 m/s, we are lead to the inescapable conclusion that light is just one form of electromagnetic wave propagation. When the electromagnetic disturbance is moving in a vacuum, we denote its speed by a special symbol, c.

In a manner similar to those leading to (4.28), we can start with the last of Maxwell's equations to find the wave equation for the magnetic field. Thus,

 formula.                                       (4.30)

Light as Transverse Waves

We can also determine whether light waves are longitudinal or transverse waves. Remember that longitudinal waves oscillate in the same direction as the direction of propagation, while transverse waves oscillate in a direction perpendicular to the direction of propagation. For simplicity, let the direction of propagation be in the x direction. Then E = E(x,t). Now look at a Gaussian box oriented along the coordinate axes. The flux is through the faces in the y-z planes, so Gauss's law becomes
formula.

From this, we see that the electromagnetic wave has no electric field component in the direction of propagation. Thus, the electric field is exclusively transverse. A similar argument can be used on Gauss's law for magnetic fields to show that it is also transverse to the direction of propagation. In particular, Faraday's law tells us that

 formula.                                          (4.31)

In other words, the time dependent magnetic field can only have a component in the z direction when the electric field is exclusively in the y direction. From these, we see that, in free space, the plane electromagnetic wave is transverse.

Orthogonality of the Electric and Magnetic Fields

We saw that we can write any general wave as a superpositioning of harmonic waves. Therefore, for simplicity, let's assume that

 formula.                                              (4.32)

Then, from (4.31), we have that

 formula.                                      (4.33)

Comparing (4.32) and (4.33), we see that

      formula.                                                                   (4.34)

Notice that even though this was derived using a plane wave, (4.34) is true for waves moving in the x direction in general.

Energy in an Electromagnetic Wave

As with any wave, the electromagnetic wave transports energy. Recall that the energy density stored in an electric field could be written as

 formula.                                                                 (4.35)

Similarly, the energy density stored in the magnetic field is

 formula.                                                                (4.36)

Since formula, we have that

 formula.                                                         (4.37)

Thus, the total energy density is shared between the constituent electric and magnetic fields

 formula.                                                                  (4.38)

To represent the flow of electromagnetic energy, let S symbolize the transport of energy per unit area. During a very small interval of time capital deltat, only the energy contained in the volume V = cAcapital deltat will cross the area A. Thus,

                                                        formula.                                       (4.39)

We now assume that, for an isotropic media, the energy flows in the direction of propagation of the wave. Then the corresponding vector S is

                                                         formula.                                                            (4.40)

This vector is known as the Poynting vector. If E and B are both harmonic waves, then we see that formula also cycles harmonically. At optical frequencies S is an extremely rapidly varying function of time, so its instantaneous value is practically impossible to measure. Instead, we want to somehow average S over a cycle. The time averaged value of the magnitude of the Poynting vector is a measure of the quantity known as the irradiance, I. For S defined in (4.40), we have that

                                                       formula,                                                    (4.41)

so

                                                        formula.                                                        (4.42)

Since the electric field is considerably more effective at exerting forces and doing work on charges than the magnetic field, the electric field E is referred to as the optical field. 


Superposition of WavesReturn to Table of ContentsPhotons and Light

Last updated: June 28, 1997

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