Problems#
In a homogeneous medium with permittivity \(\epsilon\) and conductivity \(\sigma\) and without external sources (\(\mathbf{\mathcal{J}}_{ext}=\mathbf{0}\), \(\varrho_{ext}=0\)), derive from (11) and (16) that for \(t.0\)
where \(\varrho_c(0)\) is the charge at time \(t=0\) and \(\tau=\sigma/\epsilon\) is the relaxation time. Even for a moderate conductor such as sea water, \(\tau\) is only \(2\times 10^{-10} s\). This shows that the charge density corresponding to conduction currents is zero.
In a medium with constant permittivity \(\epsilon\), magnetic permeability \(\mu_0\) and conductivity \(\sigma=0\) derive that every component of the magnetic field \(\mathbf{\mathcal{H}}\) satisfies the wave equation.
In a medium with constant \(\epsilon\), magnetic permeability \(\mu_0\) and conductivity \(\sigma\), derive that in a region without external sources the electric field satisfies:
Write the expressions for the real and the complex electric and magnetic field of a time-harmonic plane wave with frequency \(\omega\) and wave number \(k\) which propagates in the \((x,z)\)-plane with angle of \(45^0\) with the positive \(x\)-axis and has electric field parallel to the \(y\)-direction with unit amplitude and such that it is maximum for \((x=z=0\) at time \(t=0\).
Derive the Fresnel equations (141), (142) for a p-polarised plane wave.
Derive the Fresnel equation (143), (144), (145) and (146) for the case that \(k_z^i\) and \(k_z^t\) are real.
Let a p-polarised plane wave be incident on an interface \(z=0\) from \(z<0\). Let there be vacuum i n \(z<0\): \(\epsilon_i=\epsilon_0\) and let the permittivity in \(z>0\) be \(\epsilon_t>0\). The reflected field can be considered to be radiated in vacuum by dipoles in \(z>0\) with density given by (189):
where \(\mathbf{\mathcal{E}}^t(\mathbf{r})\) is the transmitted electric field in point \(\mathbf{r}\) with \(z>0\).
a) According to (75) the field radiated by a dipole vanishes in the line of sight parallel to the direction of the dipole. Derive from this the relationship between the angle of reflection \(\theta_r\) and the incident angle \(\theta_i\) for which the reflected field vanishes.
b) Show that this relationship is satisfied by the Brewster angle for a \(p\)-polarised incident wave but that it can not be satisfied for a \(s\)-polarised incident wave.
Consider two electric plane waves with wave vectors \(\mathbf{k}_1=k_x\hat{\mathbf{x}}+k_z \hat{\mathbf{z}}\) and \(\mathbf{k}_2=k_x\hat{\mathbf{x}}-k_z \hat{\mathbf{z}}\), where \(k_z=\sqrt{k_0^2\epsilon-k_x^2}\), where \(k_0\) is the wave number in vacuum and \(\epsilon\) is the permittivity which is assumed to be real. Let both plane waves be polarised parallel to the \(y\)-direction and let them have real amplitudes \(A_1\) and \(A_2\) and suppose that they are in phase for \(\mathbf{r}=\mathbf{0}\) at \(t=0\).
a) Write the expressions for the total complex and real electric field \(\mathbf{E}\), \(\mathbf{\mathcal{E}}\) and magnetic field \(\mathbf{H}\) and \(\mathbf{\mathcal{H}}\).
b) Compute the square modulus \(|\mathbf{\mathcal{E}}|^2\) of the total electric field.
c) There is a standing wave as function of \(z\). What is the period of this standing wave as function of the angle \(\theta=2 \arctan(k_z/k_x)\) between the wave vectors? Make a sketch of this period as function of \(\theta\). Note: the standing wave occurs in the direction in which the wave vectors are opposite, i.e. in this case the \(z\)-direction.
d) Compute the time-averaged Poynting vector. Show that its \(z\)-component is the sum of the \(z\)-components of the time-averaged Poynting vectors of the individual plane waves. This result proves that the net energy flow in the \(z\)-direction for the case of an incident plane wave that is partially reflected at an interface is the difference between the incident and the reflected intensities.
e) The \(x\)-component of the Poynting vector is a function of \(z\). Show that nevertheless the total flux through the boundary of any cube with faces that are perpendicular to one of the unit vectors \(\hat{\mathbf{x}}\), \(\hat{\mathbf{y}}\) and \(\hat{\mathbf{z}}\), vanishes.
Consider a wave in \(z<0\) that is incident on an interface \(z=0\) between glass with \(n_i=1.5\) in \(z<0\) and air with \(n_t=1\) in \(z>0\). Let the wave vector of the incident wave be \(\mathbf{k}^i=k_x\hat{\mathbf{x}}+k_z^i\hat{\mathbf{z}}\) and let \(k_x>k_0n_i\), so that the angle of incidence is above the critical angle.
a) If the incident wave is s-polarised, derive expressions for the complex electric \(\mathbf{\mathcal{E}}^t\) and magnetic \(\mathbf{\mathcal{H}}^t\) field in \(z>0\).
b) Compute the time averaged Poynting vector in a point \(\mathbf{r}\) with \(z>0\).
c) What is the direction of the energy flow in \(z>0\)?
d) Explain that when the Poynting vector would have a nonzero \(z\)-component, this would contradict the conservation of energy.
* Let an electric dipole be at the origin and let its dipole vector be parallel to the \(z\)-direction: \(\mathbf{p}=p\hat{\mathbf{z}}\). Then \(\hat{\mathbf{R}}=\hat{\mathbf{r}}\) and \(R=r\) in (74) and (75). Let the frequency be \(\omega\) and let the surrounding medium have real permittivity \(\epsilon\).
a) Show that at large distance to the dipole, the time-averaged Poynting vector is given by
where \(n=\sqrt{\epsilon}\) is the refractive index.
b) Show that the totally radiated power is:
Hint: integrate over a sphere \(r=\text{constant}\) using spherical coordinates \(\mathbf{r}=r\sin \theta \cos\varphi \hat{\mathbf{x}}+ r\sin\theta \sin \varphi \hat{\mathbf{y}}+r \cos \theta \hat{\mathbf{z}}\).
c) For a given dipole vector, the radiated power increases with the fourth power of the frequency. Explain with this property why the clear sky is blue.