Question

1 (a) An EM plane wave, linearly polarised such that the electric field vector lies in the plane of the interface, is incident from air on water with an angle of incidence $\theta_i = 45^\circ$. The amplitude of the incident wave is 100 V/m; the water temperature is $20^\circ$. Calculate the amplitude of the H field in the water. You may use the fact that the refractive index of water is approximately 1.33. (b) Consider a vacuum with a magnetic field given by the equation $\mathbf{H}(x, y, z, t) = \begin{pmatrix} 1\\1\\0 \end{pmatrix} \exp(iu(x, y, z, t))$, where $\mathbf{u}(x, y, z, t) = \begin{pmatrix} 0\\0\\11 \end{pmatrix} \cdot \begin{pmatrix} x\\y\\z \end{pmatrix} - 10t$. i. At the origin, in which direction does the magnetic field point as a func- tion of time? [3] ii. Calculate and simplify an expression for the corresponding electric field, $\mathbf{E}(x, y, z, t)$. [5] (c) The components of electric and magnetic fields are not complex (but real), and uniform plane waves are unphysical. And yet we considered complex- valued uniform plane waves. i. Why it is sometimes advantageous to use complex fields? [1] ii. What is the physical interpretation of a complex electric field? [1] iii. What does it mean when the components of the wave vector are complex? [1] iv. Why do we use uniform plane waves? [1] v. How can we use complex-valued uniform plane waves to describe general waves? [1] 3/6 Paper continued over...

          1 (a) An EM plane wave, linearly polarised such that the electric field vector
lies in the plane of the interface, is incident from air on water with an angle of
incidence $\theta_i = 45^\circ$. The amplitude of the incident wave is 100 V/m; the water
temperature is $20^\circ$. Calculate the amplitude of the H field in the water.
You may use the fact that the refractive index of water is approximately 1.33.
(b) Consider a vacuum with a magnetic field given by the equation
$\mathbf{H}(x, y, z, t) = \begin{pmatrix} 1\\1\\0 \end{pmatrix} \exp(iu(x, y, z, t))$,
where
$\mathbf{u}(x, y, z, t) = \begin{pmatrix} 0\\0\\11 \end{pmatrix} \cdot \begin{pmatrix} x\\y\\z \end{pmatrix} - 10t$.
i. At the origin, in which direction does the magnetic field point as a func-
tion of time?
[3]
ii. Calculate and simplify an expression for the corresponding electric field,
$\mathbf{E}(x, y, z, t)$.
[5]
(c) The components of electric and magnetic fields are not complex (but real),
and uniform plane waves are unphysical. And yet we considered complex-
valued uniform plane waves.
i. Why it is sometimes advantageous to use complex fields?
[1]
ii. What is the physical interpretation of a complex electric field?
[1]
iii. What does it mean when the components of the wave vector are complex? [1]
iv. Why do we use uniform plane waves?
[1]
v. How can we use complex-valued uniform plane waves to describe general
waves?
[1]
3/6
Paper continued over...

$1 (a) An EM plane wave, linearly polarised such that the electric field vector lies in the plane of the interface, is incident from air on water with an angle of incidence = 45^∘. The amplitude of the incident wave is 100 V/m; the water temperature is 20^∘. Calculate the amplitude of the H field in the water. You may use the fact that the refractive index of water is approximately 1.33. (b) Consider a vacuum with a magnetic field given by the equation 𝐇(x, y, z, t) = < p m a t r i x > (iu(x, y, z, t)), where 𝐮(x, y, z, t) = < p m a t r i x > · < p m a t r i x > - 10t. i. At the origin, in which direction does the magnetic field point as a func- tion of time? [3] ii. Calculate and simplify an expression for the corresponding electric field, 𝐄(x, y, z, t). [5] (c) The components of electric and magnetic fields are not complex (but real), and uniform plane waves are unphysical. And yet we considered complex- valued uniform plane waves. i. Why it is sometimes advantageous to use complex fields? [1] ii. What is the physical interpretation of a complex electric field? [1] iii. What does it mean when the components of the wave vector are complex? [1] iv. Why do we use uniform plane waves? [1] v. How can we use complex-valued uniform plane waves to describe general waves? [1] 3/6 Paper continued over...$

Added by Juan Carlos M.

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