I'm looking at Griffith's Electrodynamics, problem 4.20 (a sphere of linear dielectric material has embedded within it a uniform free charge density.) I solved the potential just fine, but I'm stuck on this one small conceptual point that doesn't really have much to do with the solution itself but is something I was thinking about while working out the math because I like to visualize things in my head.
As far as I know, the total bound charge density should always sum to zero, and since the bound volume charge density in the case of constant P is zero, that must mean that the total surface-bound charge density also equals zero.
Yet, to the best of my knowledge, P is directed radially outward, and so the bound surface charge density of the sphere seems to me, at a glance, to be non-zero.
I know I'm missing something silly. Can someone explain what's wrong with my reasoning?
EDIT: I'm not sure if it matters, but I'm self-studying physics (I took a three-semester college physics course and the required calculus / linear algebra before graduating many years ago and am now trying to continue the journey on my own.)
EDIT 2: Solved! Thank you, u/pinkpanzer101! P is not constant, so there's a divergence that results in a volume bound charge density that cancels out the surface bound charge.
https://www.google.com/amp/s/phys.org/news/2022-01-physicists-secret-sauce-exotic-properties.amp
"At 100 Kelvin, the kagome material studied by Comin and collaborators exhibits yet another strange kind of behavior known as charge density waves. In this case, the electrons arrange themselves in the shape of ripples, much like those in a sand dune. "They're not going anywhere; they're stuck in place," Comin says. A peak in the ripple represents a region that is rich in electrons. A valley is electron-poor. "Charge density waves are very different from a superconductor, but they're still a state of matter where the electrons have to arrange in a collective, highly organized fashion. They form, again, a choreography, but they're not dancing anymore. Now they form a static pattern.""
Hello everyone, I'm preparing for the physics exam (engineering degree in Italy), to be more confident in my abilities I'm doing physics exercises proposed in other degrees. The drama begins with this exercise proposed for chemistry students (attached figure).
I reported the original text of the problem, my observations and the solution formula I came up with but:
(1) I have no idea where to start for the resolution of the integral;
(2) I am pretty sure that this type (difficulty level) of integrals are not given to chemistry students to solve.
I fear it is one of those situations where the solution is simpler than it seems, but the problem is being overanalyzed.
Thanks in advance
https://preview.redd.it/prdf0ohs44881.png?width=2312&format=png&auto=webp&s=c32f12e913e854bfbd2f67630d08b69853d442ca
Hey everyone, I have yet another question about one of the problems in Griffith's Electrodynamics (this time 3.25.) The problem asks me first to find the potential outside a cylindrical shell (no dependence on z) in a uniform electric field using separation of variables in cylindrical coordinates. This is very similar to example 3.8, though the shape of the conductor is different.
Finding this potential wasn't a problem for me. Where I did get stuck was at the point where I had to calculate the surface charge density. I know that the surface charge density can be found by subtracting the normal derivative of the inner potential from the normal derivative of the outer potential, evaluated at s = R (radius of the cylinder), and multiplying by negative epsilon 0. But I don't have the inner potential, only the outer one.
After looking up the solution (I'm self studying, not taking a course), I found that the answer only uses the outer potential I found in the first part of the answer. Why? Isn't there a discontinuity on the surface? In the very next problem, 3.26, we use both the inner and outer potential to calculate the surface charge density.
What am I missing?
(I think the title is pretty self explanatory)
It is intuitive how one can construct a density functional, take the argmin to get the ground state charge density parameters, but it does not seem trivial to decompose this charge density into wavefunctions. How is that done?
Introductory electrodynamics loves to make use of objects with (rho = constant).
It makes the theory simple, but is there a way to do this in practice?
https://preview.redd.it/ag84jgrtn4n71.png?width=1280&format=png&auto=webp&s=e690ce95576e6fa3ddd295da15af42da4fecdc14
So the question asks me to find the charge density for each side of the conducting slab. I simply thought I could just treat this like a sphere problem and do a little Gaussian surface that just gets one side of the conductor and the sheet of charge. Since electric charge and field in side a conductor is always 0, I just figured that side a was the same as the charge density of the sheet. I was wrong. They I factored in the net charge of the slab and just added half of the net charge to my answer, still wrong. Took a step back, redid the math and just added up everything and halfed it, also wrong. Now I'm just confused and my professor offered no helpful advice.
Also, this homework's due date has already passed so I'm really just asking for the sake of understanding how to do this.
Here's an abstract of a series of lectures that I gave about the introduction to Voynichian Toroids. After several requests, here it is for you!
The dot product (β§) of the electric field and the current density gives us the electrical power density. Electric displacement current density in the vacuum of free space is j = (dE/dt) * Ξ΅_0, so its electrical power is E β§ j = E β§ (dE/dt) * Ξ΅_0, while its total energy stored is the corresponding time integral (1/2)Ξ΅_0(E β§ E).
Just as forces may be exerted on currents consisting of electrical charges, can displacement currents (the time derivative of electrical fields) have forces exerted on them? If so, what is their acceleration? We cannot know unless we know what their mass is. But displacement currents cannot have mass. Or can they? Can we actually ascribe an "energy density" and "mass density" to a displacement current?
Just as electric current I times one-half of the magnetic flux linkage (1/2)LI gives us the magnetic energy (1/2)LI^(2), could we have magnetic flux linkage imposed on a displacement current, thereby ascribing to it the properties of energy, and therefore mass, then acceleration, and then velocity? If so, the implication is that we can then calculate the charge density of the "vacuum" displacement current by simply dividing the displacement current density by the calculated velocity.
Below I will demonstrate this possibility, with the resulting theoretical object being the superposition of an oscillating magnetic dipole moment "m" and an oscillating toroidal magnetic moment "T" based upon a torus with oscillating toroidal and poloidal electric displacement currents, respectively, in phase quadrature. The energy density of the combined poloidal and toroidal magnetic fields, again respectively, is constant with time, and consequently, it does not radiate.
In line with the above, I will start by considering the case for poloidal displacement currents caused by time-varying toroidal magnetic fields (as in a toroidal transformer). Later on at the end, I will bring up the toroidal displacement currents caused by time-varying poloidal magnetic fields (as in a loop inductor).
magnetic field = curl of A
B = β x A
displacement current density = curl of curl of A / magnetic constant
j = β x (β x A) / Β΅_0
j = (β x B) / Β΅_0
j = (dE/dt) * Ξ΅_0
j = (d(-dA/dt)/dt) * Ξ΅_0
j = (-dΒ²A/dt) * Ξ΅_0
the time-dependent magnetic vector potential
A = A_0 sin(Οt)
the electri
... keep reading on reddit β‘For a single electron inside of a box, what does the equation -e*(the probability density function) mean? I have heard it be referred to as the "charge density", but I hate this term since it only makes sense in the electromagnetics context, where there are many particles (maybe I have to imagine the electron as a single wave instead of a single particle?) and charges present instead of just a single one... I have also heard it being called "the probability of finding the electron in a certain region of the box", but isn't this what the probability density function means anyways? Why multiply it by the electron charge?
https://preview.redd.it/alah84tav7271.png?width=850&format=png&auto=webp&s=b8c18ce1df28802cc8a27b5e0ba85d448b056369
Is my method correct?
A thin rod of length [;L;] has linear charge density that is zero in the middle of the rod and increases linearly (both positively) along the length of the rod in both directions from the middle. What is the electric potential at a distance [;h;] above the middle of the rod? (Directly above where the charge density is 0)
My setup is as follows:
[;\lambda = \alpha|x|;]
[;dq=\lambda dx;]
[;V=\int\frac{k_c}{r}dq=\int\frac{k_c\lambda}{r}dx=\int\frac{k_c\alpha|x|}{\sqrt[](x^2 +h^2)}dx ;]
Since the point of interest is above the middle of the rod I used symmetry to make the integral much easier, so I have
[;V=2k_c\alpha\int_0^{\frac{L}{2}}\frac{x}{\sqrt[](x^2 +h^2)}dx ;]
And that is a standard integral so it becomes
[;V=2k_c\alpha[\sqrt(x^2+h^2)]_0^\frac{L}{2}=2k_c\alpha(\sqrt(\frac{L^2}{4}+h^2)-h) ;]
I thought this was pretty good but I guess I got it wrong so I'm not really sure where to go next. Is something wrong in my setup? Did I not include something that I should have?
When I first started on it I tried to break it into x and y components and then add those, but then I thought that would actually only work for vectors and isn't potential a scalar quantity? So I scrapped it, but was that the right track after all?
Really appreciate any insights.
This will mostly likely be ridiculusly easy but i just can't see how the "area of the inclined face A/costheta", where is the geometry i'm missing here?
I understand the concept of nesting as a geometrical feature of some fermi surface structures, where a phonon wavevector connects two points where the surfaces are parallel to each other. I just can't see the connection to the Peierls instability (which I see as the cause of charge density waves). Can anyone help me with this?
The dot product (β§) of the electric field and the current density gives us the electrical power density. Electric displacement current density in the vacuum of free space is j = (dE/dt) * Ξ΅_0, so its electrical power is E β§ j = E β§ (dE/dt) * Ξ΅_0, while its total energy stored is the corresponding time integral (1/2)Ξ΅_0(E β§ E).
Just as forces may be exerted on currents consisting of electrical charges, can displacement currents (the time derivative of electrical fields) have forces exerted on them? If so, what is their acceleration? We cannot know unless we know what their mass is. But displacement currents cannot have mass. Or can they? Can we actually ascribe an "energy density" and "mass density" to a displacement current?
Just as electric current I times one-half of the magnetic flux linkage (1/2)LI gives us the magnetic energy (1/2)LI^(2), could we have magnetic flux linkage imposed on a displacement current, thereby ascribing to it the properties of energy, and therefore mass, then acceleration, and then velocity? If so, the implication is that we can then calculate the charge density of the "vacuum" displacement current by simply dividing the displacement current density by the calculated velocity.
Below I will demonstrate this possibility, with the resulting theoretical object being the superposition of an oscillating magnetic dipole moment "m" and an oscillating toroidal magnetic moment "T" based upon a torus with oscillating toroidal and poloidal electric displacement currents, respectively, in phase quadrature. The energy density of the combined poloidal and toroidal magnetic fields, again respectively, is constant with time, and consequently, it does not radiate.
In line with the above, I will start by considering the case for poloidal displacement currents caused by time-varying toroidal magnetic fields (as in a toroidal transformer). Later on at the end, I will bring up the toroidal displacement currents caused by time-varying poloidal magnetic fields (as in a loop inductor).
magnetic field = curl of A
B = β x A
displacement current density = curl of curl of A / magnetic constant
j = β x (β x A) / Β΅_0
j = (β x B) / Β΅_0
j = (dE/dt) * Ξ΅_0
j = (d(-dA/dt)/dt) * Ξ΅_0
j = (-dΒ²A/dt) * Ξ΅_0
the time-dependent magnetic vector potential
A = A_0 sin(Οt)
the electric field in terms of the magnetic vector potential
dA/dt = A_0 cos(Οt) Ο
-dA/dt = -A_0 cos(Οt) Ο
E = -A_0 cos(Οt) Ο
the electric disp
... keep reading on reddit β‘