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Continuity of a wave at a boundary. For light (or other electromagnetic radiation) incident from medium 1 to medium 2, we found that, provided the magnetic permeability of the medium is unity (or does not change at the discontinuity) and provided the "geometry" is constant (parallel-plate transmission line of constant cross-sectional shape or slab of material in free space), then the reflection and transmission coefficients for the electric field \(E_{x}\) and magnetic field \(B_{v}\) are given by $$ \begin{array}{ll} R_{E}=\frac{k_{1}-k_{2}}{k_{1}+k_{2}}, & T_{E}=1+R_{E}=\frac{2 k_{1}}{k_{1}+k_{2}} \\ R_{B}=\frac{k_{2}-k_{1}}{k_{2}+k_{1}}, \quad T_{B}=1+R_{B}=\frac{2 k_{2}}{k_{2}+k_{1}} \end{array} $$ where \(k=n \omega / c\) and \(n\) is the index of refraction. Show that the reflection and transmission coefficients for \(E_{z}\) imply that \(E_{x}\) and \(\partial E_{z} / \partial z\) are both continuous at the discontinuity, i.e., that they have the same instantaneous values on either side of the discontinuity. (By the field on the left side (medium 1) we mean, of course, the superpasition of the incident and reflected waves.) Similarly, show that the reflection and transmission coefficients for the magnetic fleld \(B_{y}\) imply that \(B_{v}\) is continuous at the boundary but that \(\partial B_{y} / \partial z\) is not continuous. Show that \(\partial B_{v} / \partial z\) increases by a factor \(\left(k_{2} / k_{1}\right)^{2}=\left(n_{2} / n_{1}\right)^{2}\) in crossing from medium 1 to medium 2. It is important to notice that we mean the total field, not just the part traveling in a particular direction.

Step by step solution

01

Define Variables and Parameters

We begin by defining the given parameters. We have incident electromagnetic radiation moving from medium 1 to medium 2 with reflection and transmission coefficients provided for electric field \(E_x\) and magnetic field \(B_v\). The wave numbers \(k_1\) and \(k_2\) are given by \(k_i = n_i \omega / c\), where \(n_i\) is the refractive index of the medium, \(\omega\) is angular frequency, and \(c\) is the speed of light.

02

Analyzing Electric Field Continuity

We know that \(E_x\) should be continuous at the boundary. The total electric field in medium 1 is the sum of the incident and reflected waves, given by \(E_{x_{total,1}} = E_{x_i} + E_{x_r}\). In medium 2, the transmitted wave exists, \(E_{x_{total,2}} = E_{x_t}\). At the boundary, the boundary condition implies \(E_{x_{total,1}} = E_{x_{total,2}}\), showing the continuity of \(E_x\).

03

Analyzing Magnetic Field Continuity

The magnetic field \(B_v\) must also be continuous at the boundary. The total magnetic field in medium 1 is \(B_{v_{total,1}} = B_{v_i} - B_{v_r}\), since the direction of the reflected wave changes the sign. In medium 2, we have \(B_{v_{total,2}} = B_{v_t}\). Continuity at the boundary requires \(B_{v_{total,1}} = B_{v_{total,2}}\), which verifies the continuity of \(B_v\).

04

Discontinuity of \(\partial E_z / \partial z\)

For the electric field component \(E_z\), the partial derivative with respect to \(z\), \(\partial E_z / \partial z\), does not need to be continuous across the boundary. This can be shown through examining the reflected wave which implies a change in phase or amplitude due to the mechanical structure or transition between different media. Thus, \(\partial E_z / \partial z\) may differ across media.

05

Calculate \(\partial B_v / \partial z\) Increase

Considering the magnetic field, \(\partial B_v / \partial z\) is not continuous. The coefficient relationship \(R_B\) and \(T_B\) implies that the actual field changes such that \(\partial B_v / \partial z\) will increase by a factor \(\left(k_2 / k_1\right)^2 = \left(n_2 / n_1\right)^2\) as it moves from medium 1 to medium 2. This is found by substituting \(k\) values into the expression derived from boundary conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boundary Conditions

When electromagnetic waves encounter a boundary between two different media, certain conditions must be satisfied to ensure a smooth transition. These are known as boundary conditions. They dictate how electric and magnetic fields behave at interfaces. In essence:

• Both the tangential component of the electric field and the perpendicular component of the magnetic field should be continuous across the boundary.
• This continuity ensures that the fields don't suddenly "jump" when crossing from one medium to another, which would violate physical principles like conservation of energy.
• Instead, these quantities adjust in a manner that reflects the properties of the involved media, like their refractive indices.

Boundary conditions help us understand phenomena like reflection and refraction, defining how much of the wave is reflected back into the original medium and how much is transmitted into the second medium.

Reflection and Transmission Coefficients

Reflection and transmission coefficients come into play when light moves from one medium to another. They quantify how much of an electromagnetic wave is reflected back into the first medium or transmitted into the second.

• The reflection coefficient (\( R \)) indicates the proportion of the wave that is reflected, while the transmission coefficient (\( T \)) describes the fraction that gets through to the next medium.
• For instance, if the reflection coefficient is 0.3, 30% of the wave's energy is reflected back, while the remaining 70% transmits, assuming no absorption losses.

These coefficients depend on the refractive indices of the involved media. They help in the precise calculation of wave behavior at boundaries and are crucial for designing systems like lenses and coatings that manipulate wave transmission and reflection effectively.

Continuity of Electric and Magnetic Fields

Maintaining continuity of electric and magnetic fields is crucial when waves cross between different media. Generally, the total electric field, comprising both incident and reflected components, must equal the transmitted field at the boundary.

• This means the electric field value just inside medium 1 plus any reflected wave must equal the transmitted wave field value just inside medium 2.
• Magnetic fields, on the other hand, also undergo similar continuity conditions, essential for maintaining magnetic field consistency across boundaries.

The continuous behavior of these fields ensures a coherent transition of wave energy across media. It assures that wave energy remains conserved, fulfilling fundamental physics laws. This principle is fundamental to wave optics, influencing phenomena like interference and refraction.

Refractive Index

The refractive index, denoted as \( n \), of a medium is critical in wave propagation. It defines how much the speed of light changes as it moves from one medium to another.

• The refractive index is a ratio comparing the speed of light in a vacuum to that in the medium, hence \( n = c / v \), where \( v \) is the speed of light in the medium.
• Materials with a higher refractive index slow down light more than those with a lower index.

A change in refractive index at a boundary leads to the bending or refraction of light waves. This property is crucial in various applications, from eyeglasses to fiber optics, affecting everything about how light travels through mediums, from direction to speed.