Maybe my question was simpler than it seems. In fiber optics the light source must generate a signal whose angle of incidence is not greater than the acceptance angle, in order for the signal to be guided.
As regards dielectric slab guides, instead: is the condition about the angles more...
Hello!
Let's consider a plane wave represented by a ray, propagating in a 2D dielectric slab. It has a medium with refractive index n_1 as its core and a medium with refractive index n_2, n_2 < n_1, as its cladding. In order for this ray to represent a mode, it must satisfy two conditions:
-...
Thank you both. Your examples provide a static magnetic field.
The requirements seem to be (observing \nabla \times \mathbf{H} = \epsilon \partial \mathbf{E} / \partial t):
- a constant \partial \mathbf{E} / \partial t and so a linear time-dependency for the \mathbf{E} components.
But this will...
Hello!
In this thread, in this answer, my statement "A time-varying electric field creates a magnetic field which is time-varying itself" was refuted.
Because I never observed this before, I would like to discuss about it. As far as I know, Maxwell's equations are valid always together, that...
A time-varying electric field creates a magnetic field which is time-varying itself.
What you are saying is valid only for fields that vary with time. But not all the electric and the magnetic fields vary with time. When they are fixed in time, they are called static. For static fields, we can...
Not new conclusions, but: is there a textbook where similar computations are showed? Or another pdf from the web, with the same subject matter?
I need the derivation of the longitudinal fields for a step-index optic fiber, with the considerations about the choice of the (r,\phi) dependence: when...
Hello!
In this document a solution of Maxwell's equations in cylindrical coordinates is provided, in order to determine the electric and magnetic fields inside an optic fiber with a step-index variation. The interface between core and cladding is the cylindrical surface r = a.
For example, the...
Are you meaning that the integral in the RHS
\displaystyle \int_{0}^{+ \infty} f_1(y) f_2(y)dy
is 0 by itself? I know that sine and cosine are orthogonal functions, but it is difficult to figure out how this integral can vanish when the k_1, k_2 in f_1 are different from the k_1, k_2 in f_2...
A typical mode in a dielectric slab like this, with propagation along x, uniformity along z and refractive index variation along y, is represented by the following function:
f (y) = \begin{cases} \displaystyle \frac{\cos (k_1 y)}{\cos (k_1 d)} && |y| \leq d \\ e^{-j k_2 (y - d)} && |y| \geq d...
This is reasonable. But why?
Anyway, there is a direct approach to express E_x, E_y, H_x, H_y as functions of E_z, H_z only.
For example, let's combine the first and the fifth equations in order to cancel H_y: we will obtain
E_x = -j \displaystyle \frac{1}{k^2 - \beta^2} \left( \omega \mu...
In a source-free, isotropic, linear medium, Maxwell's equations can be rewritten as follows:
\nabla \cdot \mathbf{E} = 0
\nabla \cdot \mathbf{H} = 0
\nabla \times \mathbf{E} = -j \omega \mu \mathbf{H}
\nabla \times \mathbf{E} = j \omega \epsilon \mathbf{E}
If we are looking for a wave...
Thank you.
My original question anyway was not about this. I was not arguing against the wave impedance definition, but about the physical behaviour of the wave. Why in this problem just the tangential components E_x, H_y of the fields are considered? (Tangential components mean: components that...
Yes, I definitely agree. So, just for the sake of completeness, what is the quantity that you said is continuous? Maybe
\displaystyle \frac{\left| E_x^+ + E_x^- \right|}{\left| H_y^+ + H_y^- \right|} = \displaystyle \frac{\left|E_2^+ \right|}{\left|H_2^+ \right|}
?
Several times I have seen computations involving two media with \epsilon_1 \neq \epsilon_2 but \mu_1 = \mu_2 = \mu_0, it is a (maybe purely mathematic) simplification and it is useful to focus only on the dielectric constants. You may know much more than me these materials, but I don't believe...
Yes, but in this particular case \mu_1 = \mu_2 = \mu_0: the two media differ only for the refractive index; \mu is the same. Take a look at how I computed the wave impedances \eta_1, \eta_2 in the first post.
Sorry, but I can't understand what you mean with this statement. This is a stratified...
I think you are confusing the plane of incidence and the interface plane. The (x,y) plane is the interface plane, which separates the two media, the left one with \epsilon_1, \mu_0 and the right one with \epsilon_2, \mu_0. H_y is the magnetic field component which is tangential to the interface...
Consider the problem shown in the "wave_incidence_1.png" attached image. An electro-magnetic wave is traveling towards an interface between its current medium and a new medium, which has a refractive index n_2 \neq n_1. The interface is represented by the (x,y) plane.
The electric field...
Let's consider the Magnetic hysteresis loop of a certain material: https://www.nde-ed.org/EducationResources/CommunityCollege/MagParticle/Physics/HysteresisLoop.htm is an example. In many sites and books it is written that its area is proportional to the energy wasted as heat, so A = kE_d.
In...
Suppose that a ferrite bead is put around a cable where a constant current I flows, just like in this image.
The coordinate system has the z axis along the cable. Let's evaluate the current through the (x,y) plane: according to the Ampère's law, the only magnetic field component generated by...
Thank you for your very useful observations.
I tried to compute R_{\square} from the power, as you suggested: with L = W = 1. The result, as you predicted, is
R_{\square} = \displaystyle \frac{2}{\sigma \delta (1 - e^{-2t/\delta})}
It is definitely better than mine, even if not exactly what I...
I will try to add some details, hoping that it will be useful.
Suppose that the thickness is along the x direction and the width is along y. The current density across a section can be expressed as
J(x) = J_0 e^{-(1 + j)x/\delta}
so it is supposed to be uniform along y.
The sheet resistance...
Hello!
I have in my notes an expression for the sheet resistance of a uniform conductor with length L, width W = L and thickness t. It is
R_{\square} = \displaystyle \frac{\sqrt{\displaystyle \frac{\pi f \mu}{\sigma}}}{1 - e^{-t/\delta}} = \displaystyle \frac{1}{\sigma \delta} \frac{1}{1 -...
Thank you for your answers and sorry for the great delay. It was misleading to me the fact that for several days there were no replies.
Thank you jasonRF for your complete and useful answers. For Meir Achuz: as said by jasonRF, I made a little confusion between impressed currents and induced...
Hello!
When considering the boundary conditions for the electromagnetic field \mathbf{E}, \mathbf{H} on the surface of a Perfect Eletric Conductor we have:
\mathbf{E} \times \mathbf{\hat{n}} = 0
\mathbf{J}_S = \mathbf{\hat{n}} \times \mathbf{H}
the tangential electric field should vahish...
Hello!
In http://my.ece.ucsb.edu/York/Bobsclass/201C/Handouts/Chap1.pdf, pages 19-20, the Love's Theorem in Electromagnetism is declared. In presence of some electric sources \mathbf{J} and magnetic sources \mathbf{M} enclosed by an arbitrary geometrical surface S, which produce outside S a...
No, I consider them (density) power quantities only after the multiplication by \mathbf{v}.
Yes, it is essential with phasors. Anyway consider my expressions in the time domain, say in a particular instant (with time-varying fields, the only one that can produce a radiation), not with...
Hello!
In the study of electric and magnetic fields, two equations are called the constitutive relations of the medium (the vacuum, for example):
\mathbf{D} = \mathbf{\epsilon} \cdot \mathbf{E}\\
\mathbf{B} = \mathbf{\mu} \cdot \mathbf{H}
But in a generic medium (non linear, non...
Ok! They are indipendent because in this case only one of them is forced by the source.
Yes, obviously, maybe I wrote in a rush, but I know that it is a power density. My problem was not about the \mathbf{E} \cdot \mathbf{J} product, but about the above procedure to associate it to a power...
Hello!
In an electro-magnetic context, the power that an electric source of field delivers to the field itself may be written as
p_S = - \mathbf{E} \cdot \mathbf{J}
where \mathbf{E} is the electric field produced by the source and \mathbf{J} is the corrent flowing on the source, forced by...
You're right. But if we substitute "reactive" with "storing and giving back energy", we can talk also about static fields.
A charge in the space is subjected to the field and the field can make (theoretically) everywhere a work on this charge: this is a sort of transfer of energy from the...
Ok, that's right, we can establish an analogy between transmission lines and free space.
This is not really about my topic, but is anyway a useful in-depth analysis.
Now I would like to know more about the question: reactive power is that carried from static fields? I observed that a static...
Thank you! What "elements" can you consider as inductive/capacitive in the free space?
And I suppose this time is necessary to "charge" the inductors and/or the capacitors.
Emily
Good morning,
in circuit theory I know that reacting power arise from phasors and represents a power which can't be used, because not delivered to any load, but continuously flows back and forth between the load and the generator with a zero mean during one period.
I can't understand very well...