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Chapter 21: Reflection and Transmission of

Electromagnetic Waves

Chapter Learning Objectives: After completing this chapter the student will be able to:

Analyze the behavior of an electric or magnetic field at a boundary between two different materials.

Calculate the transmission and reflection coefficients for an electromagnetic wave at a boundary between two materials.

Calculate the strength of the transmitted and reflected waves when a wave hits a boundary between two different materials.

You can watch the video associated

with this chapter at the following link:

Historical Perspective: Although Snell’s Law is named after

Willebrond Snellius (1580-1626), it was first discovered by Ibn

Sahl (c. 965-1040) in 984. Ibn Sahl was a Persian

mathematician in the court of Baghdad who extended the work

of Egyptian scientist Claudius Ptolemy (c. 100-170) to

understand the behavior of lenses and parabolic mirrors.

Photo credit: https://alchetron.com/Ibn-al-Haytham [Public domain].

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Boundary Conditions on Electric and Magnetic Fields

We now have a good understanding of how an electromagnetic wave propagates in free space

and also how it propagates in a dielectric material. But what happens at a boundary between free

space and a dielectric? Or what about at a boundary between two different dielectric materials?

Does the wave reflect, or does it pass right through the boundary, or does it do a little bit of

each? These questions are very important, and we will thoroughly understand the answers

before the end of this chapter.

Let’s first consider a simple electric field and a simple magnetic field at the boundary between

two regions, each of which has its own value of r and its own value of r, as shown in Figure

21.1. In this figure, both of the fields are perpendicular (normal) to the boundary. We will

consider what happens if the fields are parallel to the boundary in a little bit.

Figure 21.1. Perpendicular (Normal) Electric and Magnetic Fields at a Boundary

We will first analyze the perpendicular electric field by beginning with Gauss’s Law:

(Copy of Equation 8.7)

Since the field is entirely perpendicular to the surface, the integral reduces to a multiplication.

Note that D2n is positive because it is pointing out of the surface, while D1n is negative because it

is pointing into the surface.

(Equation 21.1)

Dividing by s, we find:

(Equation 21.2)

21.1

y

zx

r1, r1 r2, r2

D1n and B1n D2n and B2n

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This is equation is always true, even in the somewhat unusual circumstance that there is a surface

charge density at the boundary. This is not typically the case, so we usually assume that s=0,

giving:

(Equation 21.3)

We can also rewrite this equation in terms of the electric field E. First, we will recall that D and

E are related by the following equation:

(Copy of Equation 8.11)

Substituting this into Equation 21.3, we find:

(Equation 21.4)

We can now perform a similar analysis of the perpendicular magnetic flux density. We will

begin with Gauss’s Law for Magnetic Fields, which states that there are no magnetic monopoles:

(Copy of Equation 11.7)

Applying this equation to Figure 21.1, we obtain:

(Equation 21.5)

This can be simplified as follows:

(Equation 21.6)

Applying the relationship between B and H:

(Copy of Equation 14.7)

We find:

(Equation 21.7)

Next, we will consider tangential electric and magnetic fields, as shown in Figure 21.2.

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Figure 21.2. Tangential Electric and Magnetic Fields at a Boundary

Here, we will begin with the vector form of Kirchhoff’s Voltage Law:

(Copy of Equation 7.26)

Applying this to the dotted-line path, we find:

(Equation 21.8)

E2t is positive because it is in the same direction as the counter-clockwise path, while E1t is

negative because it is in the opposite direction as the path. This equation can be simplified to:

(Equation 21.9)

In turn, this can be written in terms of D by applying Equation 8.11:

(Equation 21.10)

Finally, we will consider the tangential magnetic field. We will begin with Ampere’s Law,

omitting the displacement current term because these fields are static.

y

zx

r1, r1 r2, r2

E1

ta

nd

H1

t

E2

ta

nd

H2

t

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(Copy of Equation 14.3)

This can be applied to Figure 21.2 as follows:

(Equation 21.11)

Again, H2t is positive because it is pointing in the same direction as the path, while H1t is

negative because it is pointing in the opposite direction as the path. The current density Js is

pointing along the boundary, either into the page or out of the page. Although such a current

density is possible, it would be highly unusual, so we will assume it is not present.

(Equation 21.12)

Finally, this equation can be rewritten in terms of magnetic flux density as follows:

(Equation 21.13)

The most important results from this analysis are summarized in Table 21.1.

Table 21.1: Field Continuity at Boundary Conditions

ElectricField

MagneticField

Perpendicular Tangential

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Example 21.1: An electric field of 20V/m is incident on a boundary from air into a

dielectric material with r=10. It is at an incident angle of 35° away from perpendicular to

the boundary. What is the angle of the electric field relative to the boundary once it has entered

the dielectric.

We can derive an equation that will allow us to quickly and easily solve problems like the one in

Example 21.1. Such a problem is shown in Figure 21.3. Here, we can break the total fields E1

and E2 down into normal and tangential components and apply the results derived above to these

individual components. As shown in Figure 21.3, this analysis will show that the electric field

changes direction at the boundary.

Figure 21.3. Electric Field Changes Direction at a Boundary

We can write equations that can be used to break each of the electric fields down into their

components:

(Equation 21.14)

(Equation 21.15)

y

zx

r1, r1 r2, r2

q1

E1

E2

q2

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Combining these equations with Equation 21.4, we find:

(Equation 21.16)

Applying them to Equation 21.9, we see that:

(Equation 21.17)

Dividing Equation 21.7 by Equation 21.6 yields:

(Equation 21.18)

Simplifying this equation gives:

(Equation 21.19)

And finally rearranging this equation gives:

(Equation 21.20)

This equation is known as Snell’s Law, and it is very well-known among those who work in

optics. It is a significant accomplishment to have been able to derive this equation directly from

Maxwell’s Equations (with an assist from Kirchhoff’s Voltage Law).

Example 21.2: Repeat the calculation from Example 21.1 using Snell’s Law to confirm

that the same result is obtained.

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Example 21.3: An electric field is incident on a boundary with an angle of 15° away from

perpendicular to the boundary. The transmitted wave has an angle of 30° away from

perpendicular to the boundary. The wave is moving from air into a dielectric material. What is

the dielectric constant of the second material?

Reflection and Transmission Coefficients

Now we will apply the knowledge just gained to calculate how much of an incoming

electromagnetic wave is reflected at a boundary and how much is transmitted through the

boundary. Consider Figure 21.4, which shows the experiment we will be considering.

Figure 21.4. Incident, Reflected, and Transmitted Electromagnetic Waves at a Boundary

We will need to be very particular with notation in the rest of this chapter, because there will be

many waves interacting with each other. Let’s begin by writing equations for the electric fields

associated with each of these three waves:

21.2

y

zx

Ei (z,t)

Hi (z,t)

ki

Incident Wave

Er (z,t)

kr

Hr (z,t)

Reflected Wave

Et (z,t)

Ht (z,t)

kt

Transmitted Wave

r1, r1 r2, r2

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(Equation 21.21)

(Equation 21.22)

(Equation 21.23)

Notice that the incident and transmitted waves, which are moving to the right, are assigned an

amplitude with the letter A, while the reflect wave, moving to the left, has an amplitude

represented by the letter B. Also, the transmitted wave is the only one of the three that is in the

second material, so it is the only one with k2 rather than k1.

We can use the characteristic impedance of the two materials and our knowledge of the

directions of E, H, and k to write equations for the three magnetic fields:

(Equation 21.24)

(Equation 21.25)

(Equation 21.26)

To simplify our analysis, we will assume that the boundary occurs at z=0. This is not necessary,

but it makes the math easier for us by eliminating the kz term in the exponents.

(Equation 21.27)

The tangential component of the electric field in region 1 is the sum of the incoming wave and

the reflected wave:

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(Equation 21.28)

The tangential component of the electric field in region 2 is simply the transmitted wave:

(Equation 21.29)

Substituting Equations 21.28 and 21.29 into Equation 21.9, we find:

(Equation 21.30)

Substituting Equations 21.21-21.23 into this equation and applying the simplification of

Equation 21.27, we see that:

(Equation 21.31)

This can be simplified by canceling all three of the exponents:

(Equation 21.32)

We will now set this result aside for a moment and turn our attention to the magnetic fields. The

tangential magnetic component in the first region is the sum of the incident wave and the

reflected wave. (Don’t worry about the sign difference right now, that will come in soon.)

(Equation 21.33)

The magnetic field in the second region is simply the transmitted component:

(Equation 21.34)

Substituting Equations 21.33 and Equation 21.34 into Equation 21.12, then substituting

Equations 21.24-21.26 into that result with the simplification of Equation 21.27, we find:

(Equation 21.35)

(That’s where the negative signs come in.) We can now simplify this by canceling the

exponents:

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(Equation 21.36)

Now, we will define the reflection coefficient to be the amplitude of the reflected wave divided

by the amplitude of the incident wave. Using this definition, Equation 21.32, and Equation

21.36, we can derive the following equation for :

(Equation 21.37)

Example 21.4: Derive Equation 21.37 from Equations 21.32 and 21.36.

The transmission coefficient T is defined to be the amplitude of the transmitted wave divided by

the amplitude of the incident wave. It can also be derived from Equation 21.32 and Equation

21.36 as follows:

(Equation 21.38)

Recalling the equation used to calculate characteristic impedance:

(Copy of Equation 19.42)

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We can rewrite our equations for and T in terms of relative dielectric constants:

(Equation 21.39)

(Equation 21.40)

The transmission and reflection coefficients are always related by the following equation:

(Equation 21.41)

Example 21.5: An electromagnetic wave is moving from region 1, which is free space, into

region 2, which has a relative dielectric constant of 4. Calculate the reflection and transmission

constants.

Example 21.6: Calculate the reflection and transmission constants for the case of Example

21.5 if the wave is going from the dielectric into free space.

Example 21.7: Calculate the amplitude of the electromagnetic waves that are reflected and

transmitted when a wave with amplitude of 10V/m is incident from air into a region with r=10.

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Summary

At a boundary between materials with different values of r and/or r, we must consider the

normal and tangential components of the electric and magnetic fields separately. Assuming

that there is no surface charge density or two-dimensional current density at the boundary,

we can use the following equations to analyze the fields at the boundary:

These equations can be used to derive Snell’s Law, which is one of the most important

results in optics:

When an electromagnetic wave is incident (incoming) at a boundary between two regions

with different values of r, we can calculate the reflection and transmission coefficients either

from the characteristic impedances of the two regions or from the relative dielectric

constants:

Here, Ai is the amplitude of the incident/incoming wave, Br is the amplitude of the reflected

wave, and At is the amplitude of the transmitted wave.

21.3

ElectricField

MagneticField

Perpendicular Tangential