We have the retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'##
And its curl ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...
This page shows solutions for Maxwell's equations of the electric and magnetic potentials (Eqn.s (509) and (510)):
http://farside.ph.utexas.edu/teaching/em/lectures/node50.html
They are derived with the aid of a Green's function: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html...
We have a surface function z = f(x,y) ; f(x,y) only contains dimensionless constants, and is itself dimensionless.
If we convert it to cylindrical co-ordinates, z = f(r,θ) , does z only depend on θ?
Meaning we can remove r from the equation, literally.
Photons deviate from the above energy-momentum transformations under certain circumstances while still in flat space-time, I'm wondering what set of transformations would more accurately describe them over as wide a range of circumstances as possible, still in flat space-time, I've searched and...
When you look at an object, say, for example, 30 feet away, your eyeballs adapt to focus at that distance of 30 feet.
When you look at the same object through a pair of binoculars, at what distance do your eyeballs adapt to focus at?
Is it more or less than 30 feet?
This is not a question...
The Biot-Savart law which describes a magnetic field created by a displacement current: $$\frac{dB}{dV}=\frac{\mu_0\epsilon_0}{4\pi}\frac{\frac{∂E}{∂t}×r}{r^2}$$
What's the relativistically co-variant form of this equation?
Is the introduction of speed of light propagation delays enough, or...
We all know a moving charge generates a magnetic field.
A moving charge also generates a displacement current ∂E/∂t.
Is the magnetic field generated entirely due to the presence of the displacement current, or is there an independent, separate effect which contributes to the magnetic field?
I'm trying to figure out this volume integral, a triple integral, of a 9-variable function.
3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates.
After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...
In the Wikipedia page of the moving magnet and conductor problem, it asserts "This results in: E' = v x B", but does not elaborate why.
What's the full derivation?
In Total Internal Reflection, the beam can still lose some energy as it reflects off the boundary of the medium it is within.
This isn't covered under the Fresnel equations. The energy loss is probably lost in the form of evanescent waves.
I tried searching for the formulae describing the...
This animation demonstrates a longitudinal wave by means of moving bars.
I realized that if we increase the amplitude of the wave, the bars will eventually start passing through each other, which sounds (no pun intended) like an unphysical scenario.
Does this mean that there is a cap, a...
When a laser beam reflects during total internal reflection, how much of its intensity is lost?
I can't the use Fresnel equations as this is for total internal reflection.
If you don't know the answer to the above question, what about the same question, but for mirrors instead?
What are the...
Lots of people have wondered about whether we can get FTL communication from entanglement.
My question is slightly different.
Is it impossible to get FTL communication out of entanglement? If it has been shown to be impossible, then we can shut off the entanglement-FTL avenue altogether and...
I'm trying to solve some statistical mechanics. This problem appeared.
##\frac{f\left(0\right)}{f\left(a\right)}f\left(x+a\right)=f\left(x\right)##
Any idea as to which function will satisfy this equation?
Someday, mankind will be able to construct rockets that can move at relativistic speeds.
The acceleration is given by ##a=\frac{F_0}{γ^3m_0}##
##F_0## can be easily measured by placing a force gauge on the rocket itself.
The acceleration is much harder to measure, is has to be measured in a...
The formula for the radiation pressure P in n-dimensional space for a given internal energy density u is ##\frac{u}{n}##.
I would really appreciate it if someone could provide a link that gives a simple derivation of this formula for dummies like me.
The difference between Planck's Law and the Rayleigh-Jeans' Law is, in Rayleigh Jeans, the average energy per mode is ##kT##, whereas in Planck, it is ##\frac{hc}{λ(e^\frac{hc}{λkT}-1)}##.
These average energy formulas are multiplied by another formula to give either Planck's Law or the...
In his seminal paper on Special Relativity: On the Electrodynamics of Moving Bodies, Einstein derives a formula for the Relativistic Doppler effect.
See this section:
§ 7. Theory of Doppler's Principle and of Aberration
The formula is:
In this Wiki article, the same formula is derived.
The...
I was reading a paper that described a vector field in terms of its three components , ##A_σ,A_τ,A_φ##.
with σ, τ and φ being the three bispherical coordinates.
what does ##A_σ## mean? In what direction does the component point? Likewise for the other two components.
It isn't difficult to find the electric field of a dipole.
However, it is tricky to find the field lines. All points of a field line have to be parallel to the electric field at those points. A tangent, which is the derivative, is parallel.
We can hence formulate the equation for a field line...
Imagine you have two identical metal balls, each carrying the same amount of charge, but with one ball being positive and the other negative.
When you bring them close together, there will be polarization, so you can no longer treat the balls as point charges.
How does one find the charge...
I'm looking for the inverse functions of the Maxwell-Boltzmann distribution and Planck's Law.
Planck's Law in terms of the wavelength.
Any of you know of any literature on this topic?
I would like to integrate w.r.t. x, the rest are constants.
This is not homework; it is related to a physics problem regarding single-slit diffraction.
I'm asking the above question as I imagined a current loop made out of a curving wire that cannot fit into a flat plane.
Where does the direction of μ point then?
Also, is there only one μ value for any current loop, independent of the chosen axis of tilting?
Or can a current loop have...
I am trying to find the electric field of a hollow, open-ended, thin-walled cylindrical conductor.
I am trying to solve something regarding LINACs:
Assume we have an isolated, hollow, open-ended, thin-walled cylindrical conductor, with a net charge.
The net electric field within the...
Let's introduce a time-varying scalar field ρ(x,y,z,t) [charge density] and vector field J(x,y,z,t) [current density]
Assuming the system follows Maxwell's equations, what must both fields satisfy such that
##∇⋅(\frac{∂B}{∂t})=0## ?