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Problem 1. Maxwell's equations (10 points) (a) Show that from the Maxwell's equations in charge-free, homogeneous, isotropic medium, one can derive a waveform equation for both E and H fields, respectively. Hint: use the vector identity (A × B) × C = (A \cdot C)B – (B \cdot C)A. (b) Show that the plane wave solutions E = E$_0$exp [i(k \cdot r – ?t)] and H = H$_0$exp [i(k \cdot r – ?t)] where E$_0$ and H$_0$ are space and time independent vectors, are indeed the solution for the wave equation derived from the Maxwell's equations, and the speed of the electromagnetic waves satisfy the following relation: $v = \frac{\omega}{k} = \frac{1}{\sqrt{\epsilon\mu}}$ (c) Show that k \cdot E = 0, k \cdot H = 0, H = \frac{k \times E}{\omega\mu}, and E = \frac{H \times k}{\omega\epsilon} that E, H, and k are orthogonal to each other, showing the transverse nature of the waves. (d) Find the time-averaged Poynting vector within a period of oscillation for this wave. What does this say about the direction of energy flow?

          Problem 1. Maxwell's equations (10 points)
(a) Show that from the Maxwell's equations in charge-free, homogeneous, isotropic medium, one
can derive a waveform equation for both E and H fields, respectively.
Hint: use the vector identity (A × B) × C = (A \cdot C)B – (B \cdot C)A.
(b) Show that the plane wave solutions E = E$_0$exp [i(k \cdot r – ?t)] and H = H$_0$exp [i(k \cdot r –
?t)] where E$_0$ and H$_0$ are space and time independent vectors, are indeed the solution for the
wave equation derived from the Maxwell's equations, and the speed of the electromagnetic
waves satisfy the following relation:
$v = \frac{\omega}{k} = \frac{1}{\sqrt{\epsilon\mu}}$
(c) Show that k \cdot E = 0, k \cdot H = 0, H = \frac{k \times E}{\omega\mu}, and E = \frac{H \times k}{\omega\epsilon} that E, H, and k are orthogonal to each
other, showing the transverse nature of the waves.
(d) Find the time-averaged Poynting vector within a period of oscillation for this wave. What does
this say about the direction of energy flow?

Problem 1. Maxwell's equations (10 points)
(a) Show that from the Maxwell's equations in charge-free, homogeneous, isotropic medium, one
can derive a waveform equation for both E and H fields, respectively.
Hint: use the vector identity (A × B) × C = (A ·C)B – (B ·C)A.
(b) Show that the plane wave solutions E = E0exp [i(k ·r – ?t)] and H = H0exp [i(k ·r –
?t)] where E0 and H0 are space and time independent vectors, are indeed the solution for the
wave equation derived from the Maxwell's equations, and the speed of the electromagnetic
waves satisfy the following relation:
v = (ω)/(k) = (1)/(√(ϵμ))
(c) Show that k ·E = 0, k ·H = 0, H = (k ×E)/(ωμ), and E = (H ×k)/(ωϵ) that E, H, and k are orthogonal to each
other, showing the transverse nature of the waves.
(d) Find the time-averaged Poynting vector within a period of oscillation for this wave. What does
this say about the direction of energy flow?

Added by Colleen H.

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