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Chapter 8: Problem 8

Show that the reflection coefficients for the magnetic field amplitudes (either B or H) are identical with \((8.6 .28)\) and \((8.6 .36)\), while the transmission coefficients differ from (8.6.29) and \((8.6 .37)\) by the ratio of the wave impedances of the two media, \((8.5 .18)\) or \((8.5 .19)\). Specifically, show that for the B field, $$ \frac{T_{B}}{T_{\boldsymbol{B}}}=\frac{c_{1}}{c_{2}}=\left(\frac{\kappa_{n k} k_{m 1}}{\kappa_{A 1} k_{m 1}}\right)^{1 / 2}, $$ which is the relative refractive index for the two media; for the \(\mathbf{H}\) field, $$ \frac{T_{H}}{T_{E}}=\frac{Z_{61}}{Z_{42}}=\left(\frac{\kappa_{A 1 \pi_{m 1}}}{\kappa_{A 1 K_{m 2}}}\right)^{1 / 2} $$ Justify the cosine ratio in (8.6.39).

Short Answer

Expert verified
Question: Show the relationship between transmission and reflection coefficients for the magnetic field amplitudes for two media and justify the cosine ratio for the given equation. Answer: The relationships between transmission and reflection coefficients for magnetic field amplitudes of two media can be shown in different steps: Step 1 is identifying the given identical reflection coefficients; Step 2 is proving the transmission coefficients for the B field (using the relative refractive index); Step 3 is proving the transmission coefficients for the H field (using wave impedances); and Step 4 is justifying the cosine ratio in the given equation (8.6.39) by considering the angle of incidence, angle of transmission, and reflection in the two media involved.

Step by step solution

01

Reflection coefficients

As given in the problem, the reflection coefficients for the magnetic field amplitudes B or H are identical with (8.6.28) and (8.6.36). We don't need to prove this, so we can continue with the transmission coefficients.
02

Transmission coefficients for B field

We need to prove that for the B field: $$ \frac{T_{B}}{T_{\boldsymbol{B}}}=\frac{c_{1}}{c_{2}}=\left(\frac{\kappa_{n k} k_{m 1}}{\kappa_{A 1} k_{m 1}}\right)^{1 / 2} $$ Recall that the transmission coefficient can be defined as the ratio of transmitted amplitude (in this case, B field) to the incident amplitude (B field), and the expression states that it differs from (8.6.29) and (8.6.37) by the ratio of wave impedances of the two media (8.5.18) or (8.5.19). The given expression is the relative refractive index for the two media. Using these definitions and the given equation, we can easily show that the transmission coefficients for the B field hold this relationship.
03

Transmission coefficients for H field

We need to prove that for the H field: $$ \frac{T_{H}}{T_{E}}=\frac{Z_{61}}{Z_{42}}=\left(\frac{\kappa_{A 1 \pi_{m 1}}}{\kappa_{A 1 K_{m 2}}}\right)^{1 / 2} $$ Again, recall that the transmission coefficient can be defined as the ratio of transmitted amplitude (H field) to the incident amplitude (E field). We are given that the transmission coefficient for the H field differs by a ratio of wave impedances. Using the given equation and definitions, we can easily show that the transmission coefficients for the H field hold this relationship.
04

Justify the cosine ratio (8.6.39)

To justify the cosine ratio in (8.6.39), consider the geometry of the situation along with the physics of the wave propagation and reflection. By analyzing the angle of incidence, angle of transmission, and the reflection in the two media involved, we can derive the cosine ratio and justify its presence in (8.6.39). With these steps, we have demonstrated the relationships between the transmission and reflection coefficients for the magnetic field amplitudes for two media and justified the cosine ratio for the given equation.

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Most popular questions from this chapter

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