5 votes

Lossless beam splitter relations

Write $$ \left(\begin{array}{c} E_c\\ E_d\end{array}\right)= U \left(\begin{array}{c} E_a\\ E_b\end{array}\right)\, . $$ If the beamsplitter is lossless, then $$ (E_c^*,E_d^*)\left(\begin{array}{c} ...
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4 votes

Lossless beam splitter relations

I use a slightly different notation. This is your beam splitter and this is you system \begin{equation*} \begin{cases} \tilde{\boldsymbol{E}}_a = T\boldsymbol{E}_a +R\boldsymbol{E}_b \\ \tilde{\...
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  • 556
4 votes
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Lossless beam splitter relations

$$\left(r_{ac}t_{bc}^{*}+r_{bd}^{*}t_{ad}\right)E_{a}E_{b}^{*}+\left(r_{ac}^{*}t_{bc}+r_{bd}t_{ad}^{*}\right)E_{a}^{*}E_{b}=\Re\left[\left(r_{ac}^{*}t_{bc}+r_{bd}t_{ad}^{*}\right)E_{a}^{*}E_{b}\right]=...
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3 votes

Local gauge transformations and Noether Current

A global/$n$-parameter quasisymmetry is the realm of Noether's 1st theorem. Example: Global gauge symmetry in E&M with matter sectors implies electric charge conservation, cf. e.g. this Phys.SE ...
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3 votes
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What does it mean to give the fields energy and momentum?

Fields can exert forces on charged particles. Force by definition is the rate at which momentum changes. Meaning when the field exerts a force on charged particles it increases its momentum, this ...
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  • 4,471
3 votes

Why is thermal noise Gaussian distributed in voltage, but Rayleigh distributed in amplitude?

The amplitude is $\rho^2=x^2+y^2$, which is by definition positive, and which will obey Reyleigh distribution, if $x,y$ are a bivariate normal. Note also that detector is always a quadratic detector - ...
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2 votes
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On the motion of a charged particle in a magnetic field

Yes indeed particles oscillate around their equilibrium trajectory in a particle accelerator. The emitted radiation is due to the circular orbit itself and is named the synchrotron radiation. What you ...
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2 votes

Magnetic potential in spherical coordinates

$\nabla\cdot{\bf B}$ does not equal zero for the $l=0$ term. $\nabla\cdot{\bf B}=4\pi\delta({\bf r}$) (in Gaussian units).\
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2 votes
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Magnetic potential in spherical coordinates

The equations $\nabla \cdot {\bf B}=0$ and $\nabla \times {\bf B}=0$ imply that ${\bf B}$ can be written as the gradient of a scalar potential $V$ satisfying Laplace's equation in the current-free ...
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Motivation for Electric Displacement Field

$\mathbf P$ is not the electric dipole moment of anything; it is, instead, the electric dipole moment density. So when you write $\rm {curl} \mathbf H = \mathbf J + \frac{\partial \mathbf P}{\partial ...
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  • 9,735
1 vote

Local gauge transformations and Noether Current

Funnily enough this has only really been understood in recent years in the connection to large gauge transformations and soft theorems. See https://arxiv.org/abs/2112.05289 for a recent in depth ...
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1 vote

Local gauge transformations and Noether Current

Noether's theorem is about global symmetries. Gauge "symmetries" are not physical symmetries. They are local redundancies. In quantum field theory, a gauge transformation is a do-nothing ...
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1 vote

Charge distribution and Shape

Same free electrical charges (e.g. for example negative charge electrons) in a medium like a conductor metal repel each other with the Coulomb force and get attracted to opposite charges when ...
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  • 3,202
1 vote

Charge distribution and Shape

Excessive charge is subjective to the Coulomb force. This force is repelling for like-charged particles, such as a bunch of electrons. If exessive charge was to be gathered in the bulk of a conductor (...
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  • 947
1 vote

What does it mean to give the fields energy and momentum?

What does it mean to give the fields energy and momentum? The concept of a field in physics In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value ...
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1 vote

What does it mean to give the fields energy and momentum?

As you mentioned, the energy and momentum stored in the fields can be "given back" to particles. This does make it seem like we defined field energy and momentum just for the sake of it and ...
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1 vote

What does it mean to give the fields energy and momentum?

Perhaps the most direct consequence is radiation pressure. While small in everyday life, it's measurable on Earth with a Nichols radiometer. In space, it's a big deal, with a substantial effect on ...
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1 vote
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Why is induced electric current moving clockwise/counterclockwise?

Consider the Maxwell equation equivalent to Faraday's law: \begin{equation} \boldsymbol{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \end{equation} If you perform a surface ...
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  • 819
1 vote
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Why is thermal noise Gaussian distributed in voltage, but Rayleigh distributed in amplitude?

Maybe this will help a bit. Consider two normally distributed variables $X_1,X_2\sim N(\mu=0,\sigma=1) $. Then $X_1,X_1^2,X_1^2+X_2^2$ and $\sqrt{X_1^2+X_2^2}$ will produce the following plots. Notice ...
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1 vote

Why is this happening to the milk?

It probably is a surface tension effect with the milk not wetting the plastic surface. Creating surface requires the input of energy and the reverse is also true. The milk starts off with a very large ...
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1 vote
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Matching conditions for perfect conductors in electro*dynamics*

If we take the definition of a "perfect conductor"1 to be a medium in which $\mathbf{E}(\mathbf{r}, t) = 0$, then the magnetic field in a perfect conductor must be constant in time: $$ \frac{...
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1 vote

UV Wavelengths Sparkd

When you have a spark in air you get an emission spectrum that will look something like this. So you see a emission from the nitrogen and oxygen atoms. You can see that there is some UV from in the ...
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