Maxwell's equations Definition and 245 Threads

Hi guys, I am having hard times in understanding whether or not the longitudinal electromagnetic waves are solutions to Maxwell's equations. In Cohen-Tannoudji "Introduction to QED" it's stated that by writing the fields as the sum of a longitudinal and transverse part one can show that waves...

Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampere's laws). (The currents and charges are not unknowns, being freely...

How would you determine the electric field from Maxwell's equations? One of my friends was asked this when he went for a Master's interview. Thanks.

Homework Statement We are using Gaussian units. To begin, the vector Hk is the magnetic field, which relates to a second rank antisymmetric tensor, Fij. a.) Prove F_{ij} = \frac{\partial A_{j}}{\partial x_{ i }} - \frac{\partial A_{ i }}{\partial x_{j}} = \partial_{ i } A_{j} - \partial_{j}...

I need help solving the vector modal equation for a step index fiber having a constant refractive index in the core and the cladding. (Under the conditions of zero dispersion and absorption.

Hello all - I've been trying to work out an example from a book, and I don't quite understand the math. show that (δT/δV)s = - (δP/δS)v solution (δ/δV (δU(S,V)/δS)v)s = (δ/δS(δU(S,V)/δV)s)v (δ/δV (δ(TdS - PdV)/δS)v)s = (δ/δS(δ(TdS-PdV)/δV)s)v (δT/δV)s = -(δP/δS)v...

Homework Statement Show one can obtain Maxwell's equations from $$\partial^{\mu} F_{\mu \nu} = 0\,\,\,; \,\, \partial_{\mu}F_{\nu \rho} + \partial_{\nu} F_{\rho \mu} + \partial_{\rho}F_{\mu \nu} = 0,$$ where ##F_{ij} = \epsilon_{ijk}B_k## and ##F_{i0} = E_i## with ##F_{\mu \nu} = - F_{\nu...

In my university lecture notes, maxwell's equations in matter are written in the following format: \oint \vec E d \vec L = 0 \oint \vec D \vec dS = \int_V P_f (\vec r)dV \oint_S \vec B d \vec S = 0 \oint_L \vec H \vec dL = \int_S P_f \vec J_f d\vec S I am new to electromagnetism...

Homework Statement Use Maxwell's equations to derive the continuity equation. B=Magnetic Field E=Electric Field ρ=Charge Density J=Current Density Homework Equations Maxwell's Equations: ∇⋅E=ρ/ε0, ∇×E=-∂B/∂t ∇⋅B=0 ∇×B=ε0μ0(∂E/∂t)+μ0J Continuity Equation: ∇⋅J +∂ρ/∂t = 0 The Attempt at...

Definition/Summary Maxwell's equations are a set of four equations which must be satisfied by all electric and magnetic fields throughout all space at all times. They comprise of Gauss' law, Gauss' law for magnetism, Maxwell's modification of Ampère's law, and Faraday's law. Each...

Homework Statement An electromagnetic wave has an electric field \mathbf{E} = E_0 \cos(kz-ωt) \hat{x}. Using Maxwell's equations, find the magnetic field. Homework Equations \mathbf{∇\times E} = \mathbf{\dot{B}} The Attempt at a Solution So this problem appears extremely simple, but other...

Constant for all observers? I have heard that maxwell showed that the speed of light is constant for all observers even before Einstein did. Is that true? If not, then how can we say maxwells equation shows the speed of light is constant?

Hi! I'm having problems with this homework my professor gave us this morning: Prove that Maxwell's equations is invariant under Lorentz Transformation. I'm just currently in third year, and we haven't been taught tensors yet. The extent of what I know mathematically is only until gradient...

Hi all, I'm trying to derive for myself the uniqueness proof for Maxwell's equations, but I'm a little stuck at the end. I've managed to prove the following: \dfrac{A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0 \Rightarrow...

Hi, I'm just beginning to learn relativity, but I have a question about why gravity is so different from other forces of nature in GR. As a start, I read that Einstein tried to find a differential geometric representation of the physical universe which represents the Maxwell equations in a...

Hello Everyone, I am currently reading page 20 of Townsend's Quantum Physics book. Here are a few sentences that I am unsure of: "In general, the magnitude and phase of the probability amplitude are determined from first principles by solving Maxwell's equations. In free space, these...

If I start with the stress-energy tensor T^{\mu\nu} of the electromagnetic field and then apply energy-momentum conservation \partial_\mu T^{\mu\nu}=0, I get a whole bunch of messy stuff, but, e.g., with \nu=x part of it looks like -E_x \nabla\cdot E, which would vanish according to Maxwell's...

I'm trying to get a sense of how widely applicable Maxwell's equations really are. I've read that electrodynamics becomes non-linear in the Schwinger limit where electric field strengths get high enough, but are there other situations where Maxwell's equations are insufficiently accurate? What...

I am solving a question that asks me to find an H field in phasor form from the given E field in phasor form Es = j30(beta)(I)(dl)sin(theta)e^(-j(beta)r) a(theta) V/m Given that the EM wave propagates in free space. Why do I get different answers if I : 1) Divide Es by the magnitude...

I have some copper enamel wire, winded up into something of a circle/ellipse, with about 20 turns, and the purpose for it is to give me a decent magnetic field (which it does) using a DC source. The magnetic field is picked up by a hall effect sensor on a nearby circuit, which is connected to...

Hi All, Thanks again to all the great mentors and contributors to this forum. I wanted to ask a question about the Gauss's law/Ampere's law equations in Maxwell's Equations: \nabla \bullet \textbf{E} = \frac{\rho}{\epsilon_0} \\ \\ \nabla \times \textbf{B} = \mu \left( \textbf{J} + \epsilon...

My question is essentially about Ampere's law. I went the long way about and evaluated the curl of the magnetic field, \vec{B}, of a point charge, q, located at position \vec{r_{0}}, and moving with velocity \vec{v}: \vec{B} =...

Can one show that strict charge conservation ##\nabla_{a}J^{a} = 0## follows directly from ##\nabla_{a}F^{ab} = 4\pi J^{b}## alone? Also, how does ##d^{\star}F = 4\pi ^{\star}J## follow directly from that same equation where ##\star## is the Hodge dual operator?

Hello there Ladies and Gents! This question is (mostly) related to problem 10.2 in Wald which is to show that the source-free Maxwell's equations have a well posed initial value formulation in curved space-times. We start off with a globally hyperbolic space-time ##(M,g_{ab})## and a spacelike...

what changes will take place in maxwell's equations if the space-time was curved?

Homework Statement Show that the general relationship from Maxwell's equations for the conservation of energy \nabla \cdot \textbf{S} + \frac{\partial u}{\partial t} = 0, where u = \frac{1}{2} \epsilon _{0} \left| \textbf{E} \right| ^{2} + \frac{1}{2 \mu _{0}} \left| \textbf{B}...

Homework Statement I have to take the curved space - time homogenous and inhomogeneous maxwell equations, \triangledown ^{a}F_{ab} = -4\pi j_{b} and \triangledown _{[a}F_{bc]} = 0, and show they can be put in terms of differential forms as dF = 0 and d*F = 4\pi *j (here * is the hodge dual...

Homework Statement Consider the following representation of Maxwell's eqns: $$\nabla \cdot \underline{E} =0,\,\,\, \nabla \cdot \underline{B} = 0,\,\,\, \nabla \times \underline{E} = -\frac{\partial \underline{B}}{\partial t}, \,\,\,\frac{1}{\mu_o}\nabla \times \underline{B} = \epsilon_o...

I understand that one is able to derive the inhomogenuous pair of Maxwell's equations from varying the field strength tensor Lagrangian. Now implying the U(1) gauge invariance, how is one led to the Maxwell's equations?

Homework Statement Condensed/simplified problem statement \vec{E} = f_{y}(x-ct)\hat{y} + f_{z}(x-ct)\hat{z} \\ \vec{B} = g_{y}(x-ct)\hat{y} + g_{z}(x-ct)\hat{z} \\ All the f and g functions go to zero as their parameters go to ±∞. Show that gy = fz and gz = -fy Homework Equations \nabla...

What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity? I'm working through Einstein's original 1905 paper(available here), and I'm having trouble with the section on the transformation of Maxwell's equations from rest to moving frame. The...

Homework Statement Let F_{ab} be the Faraday Tensor and \xi ^{a} a killing vector field. Suppose that the lie derivative \mathcal{L} _{\xi }F_{ab} = \xi ^{c}\triangledown _{c}F_{ab} + F_{cb}\triangledown _{a}\xi ^{c} + F_{ac}\triangledown _{b}\xi ^{c} = 0. Show that F_{ab}\xi ^{b} =...

Homework Statement From Sean Carroll's notes on general relativity (chapter 1, pg. 20): Show that F_{[\alpha\beta,\gamma]} = 0 is equivalent to half of the Maxwell equations. Homework Equations F_{\mu\nu} is the electromagnetic tensor \Phi_{,\nu} \equiv \partial_{\nu}\Phi...

Hi as I'm reading http://www.maths.tcd.ie/~cblair/notes/432.pdf at page 13 I see that he states that the covariant and contravariant field tensors are different. But how can that be? Aren't they related by F_{\mu \nu} = \eta_{\nu \nu'} \eta_{\mu \mu '} F^{\mu ' \nu '} ? and is not the...

Ok, some background: In the static case, the force on a charge is the multiplication of the charge into the electric field {\bf{E}}, defined by Gauss' law, the force on a moving charge with velocity {\bf{v}} is given by the multiplication of the charge (which is Lorentz-invariant) into the...

Einstein postulated that the speed of light in vacuum is constant and is the same for all observers. It this related to the fact that in Maxwell's equations for electromagnetic waves in a vacuum, c = \frac {1} {\sqrt{\mu_0 \epsilon_0}} ? The electric and magnetic constants, which are...

I need some help figuring out what these formulas mean and what they relate to. All I know is the 'upside-down triangle' symbol is known as a "Del", and it's used in vector calculus. Before the list of equations, it says "And God said...". After the list of equations, it says "...And there...

The Maxwell's equations in vacuum leads to the wave equations for the fields of the form \nabla^2 \vec E = \frac{1}{c^2} \frac{\partial ^2 \vec E}{\partial t^2} (the same for the magnetic field) Such equations are Lorentz-invariant. Let's consider now the electromagnetic field in a...

I am specifically talking about differential forms of Maxwell's equations here- I think ( Tell me if I am incorrect here...) since the divergence of the magnetic field is zero, we have to say it is incorrect for my above problem, so the equation should have been the divergence of magnetic field...

By Maxwell's equations, electromagnetic waves seem to come about by means of magnetic fields generating electric fields and in turn electric fields generating magnetic fields (the loop continues.) But how is this not perpetual in the context that energy is not conserved? It seems as if a...

Homework Statement A parallel-plate capacitor with circular plates of radius 1.7 m is being charged. Consider a circular loop centered on the central axis between the plates. The loop has a radius of 2.6 m and the displacement current through the loop is 2 A. (a) At what rate is the...

Homework Statement I am trying to figure out a way to prove mathematically that Maxwell's Equations predict that the speed of light is the same in all reference frames, and therefore are consistent with special relativity. I am having a hard time finding what equations to use for the...

So I've been reading Hehl's Foundations of Classical Electrodynamics - which builds up Electrodynamics from a six of axioms - and their proof that the conservation of charge alone is sufficient to derive the inhomogenous Maxwell equations got me thinking - why don't these extremrly basic...

Basically I couldn't understand Maxwell's equations during my college days mainly because I didn't understand divergence and curl during that time. I need some good book on Electromagnetic Fields (such as Gauss Laws and so on) and also I want to learn about Maxwell's equations. I want to...

Basically I couldn't understand Maxwell's equations during my college days mainly because I didn't understand divergence and curl intuitively. I need some good book on Electromagnetic Fields (such as Gauss Laws and so on) and also I want to learn about Maxwell's equations. I want to...

Here are some questions that have been puzzling me about symmetry and charge. Any answers to any of these questions would be very helpful. Thank you. What does U(1) gauge symmetry mean? Does anyone have a simple explanation? Can Maxwell's equations be derived from the premise of U(1)...

Homework Statement Consider a simple spherical wave, with omega/k=c E(r, theta, phi, t)=((A sin theta)/r)(cos(kr - omega t) -(1/kr)sin(kr - omega t)) phi-hat i) Using Faraday's law, find the associated magnetic field B ii) Show that E obeys the remaining three of Maxwell's equations...

hi i have to deal with a question, that i do not understand fully: in my chemistry lesson, my teacher told me, that if you have an aromatic molecule and you put it into a time-constant magnetic field, this would cause electrons to move. if you are not that familiar with chemistry, all you...

Homework Statement The current I=I0exp(-t) is flowing into a capacitor with circular parallel plates of radius a. The electric field is uniform in space and parallel to the plates. i) Calculate the displacement current ID through a circular loop with radius r>a from the axis of the system...

Homework Statement Hi, this is the first time I post a thread in this forum. I am not sure if I could post this question here since it is not a homework problem. I have trouble understanding two boundary condition between nonconductor and conductor from Maxwell's equations in dynamic case...