"When considering the Fourier series of a function it is conventional to rescale it so that it acts on [0,2π] (or is 2π periodic). In this section we instead use the somewhat unusual convention taking f to act on [0,1], since that matches the convention of the Fourier transform used here." --https://en.wikipedia.org/wiki/Fourier_inversion_theorem#Square_integrable_functions Isn't there a function from [0,1] to the unit circle, and does it mean [0,2π] is the "same" interval?
Actually, it seems kind of trivial. [0,1] x 2π = [0,2π]. But with Fourier transforms you have either angular frequency or linear frequency, and in each case the transform has to have inverse units (resp. 2π cycles per second and cycles per second).
A sort-of related question: why is the electric component of radiation and not the magnetic component, "active" in energy transfer to charged particles?
By "radiation" I mean electromagnetic radiation. Ok, why are broadcast antennae vertical bars of metal? Because then receiving antennae will "catch" the most signal if they are also oriented vertically. Which suggests that long wavelength "particles" of EM radiation can be emitted by a source with lots of oscillating electrons in it--the power output must be proportional to the number of electrons being accelerated periodically in the antenna. But there is a complex relationship between the number of electrons in motion (i.e. the current), the height of the antenna, and the peak power output. I suppose you could do a thought experiment with an antenna that emits or receives single radio-wavelength photons. Thanks to Krane (3rd ed.) I have: The electric and magnetic components of the field are perpendicular, further, the \( \vec{E} \) field is described purely spatially as the gravitational field is, whereas the \( \vec{B} \) field depends on the existence of a current, i. A plane electromagnetic wave is a special form of wave with plane wavefronts; the wavefronts are where the two perpendicular fields are at a maximum. An electromagnetic wave transmits energy from place to place, specified by the Poynting vector: \( \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \), with units \( W/m^2 \), or power per unit area. The magnetic component of the field vanishes when the relation \( B_0 = E_0/c \) is applied, and is why the intensity is proportional to the square of \( E_0 \).
So we have some receiver with area A, which has to be perpendicular to both \( \vec E \) and \( \vec B \), i.e. perpendicular to the direction of propagation. The magnitude of \( \vec S \) times the area gives the power received by the receiving antenna: \( P = SA = \frac{1}{\mu_0} E_0 B_0 Asin^2(kz - \omega t) \) But, because of \( B_0 = E_0/c \), why don't we have: \( P = \frac{1}{\mu_0 c} E_0^ 2sin^2(kz - \omega t) = \frac{c}{\mu_0} B_0^ 2sin^2(kz - \omega t) \) Or, we do have that, but the convention is to consider only the electric field? I recall being told in electronics that it's possible to calculate the magnetic fields instead, and you will get the same system response (as for the electric field).
The symbol k in the above equations is the wavenumber. This is given by: \( k = \frac{2\pi}{\lambda} \), k has units of inverse distance. As its name implies, it's connected to the number of waves per unit of distance, in this case multiplied by \( 2\pi \). So if the wavelength is say, 1/3 metres, there are three waves per metre and the wavenumber is 3. With a Fourier transform where the integrand is a function of k, the result is a function of (units of) distance.
Ok so wavenumbers and wavelengths describe waves spatially. But plane electromagnetic waves are a composition of two sine (or cosine) curves, which means the actual distance along the curve between wavefronts is \( 2 \pi \) times the wavelength. Sometimes k is used without the factor of \( 2 \pi \). Krane doesn't go into where the speed of propagation comes from, but mentions: " . . .the angular frequency \( \omega \) is found from the frequency f ( \( \omega = 2\pi f \)). Because \( \lambda \) and f are related by \( c = \lambda f \), k and \( \omega \) are related by \( c = \frac{\omega}{k} \)."
My earlier physics text says the electromagnetic field is composed of two related fields, the electric and magnetic fields. Each field can be zero or non-zero in some region of spacetime, and each has an energy associated with it, and a field that describes this energy. Charged particles can store electric energy in an electric field, and they store magnetic energy in a magnetic field, these are "interactions" between the particles (their point charges) and the field itself. Indeed this is exactly what electronic circuits represent: elements such as capacitors and inductors (passive elements), store energy; the circuit 'causes' interactions between electric and magnetic fields, by storing charge, or by "storing current" in a magnetic field (an inductor generates a field around itself, there is a change of phase in the relation between real voltage and real current). A capacitor stores electric energy because all the electrons in the charged plate see a collective field between them, an electrostatic field. There is a kind of pressure, exactly analogous to pressure in a fluid, in the charged plate and a potential between the charged and uncharged plates. An inductor stores magnetic energy because of more complicated reasons involving the existence of a current (coulomb per second) and the path it takes. It's analogous to the reasons vortical flows arise in fluids. In the case of electronic inductors the current is forced to move around a loop, more usually a lot of loops often wrapped around a ferrite core. As the current circulates, a magnetic field circulates perpendicular to the current. Oh hell, it's really about conservation of angular momentum--the electrons are forced to accelerate so they store energy in the magnetic field so they maintain a constant current and charge is conserved (an ideal inductor has no capacitance).
