The simple proof of the Maxwell's vacuum equation (original) (raw)

Basics of Electromagnetics – Maxwell's Equations (Part -I

1. ∮ í µí° ¶ í µí°´⃗. í µí±í µí± ⃗⃗⃗⃗ = ∮ í µí° ¶ __. í µí± ⃗ í µí± [GATE 1994: 1 Mark] Soln. ∮ í µí±¨̅. í µí² í µí² ̅̅̅ = ∬ í µí» × í µí±¨⃗⃗⃗. í µí² í µí² ⃗⃗⃗⃗⃗⃗ í µí°®í µí°¬í µí°¢í µí° §í µí° í µí°í µí°í µí°¨í µí°¤í µí°′í µí°¬ í µí°í µí°¡í µí°í µí°¨í µí°«í µí°í µí°¦ = ∮ í µí±º í µí» × í µí±¨⃗⃗⃗. í µí² í µí² ̅̅̅̅ 2. The electric field strength at distant point, P, due to a point charge, +q, located at the origin, is 100 µ V/m. If the point charge is now enclosed by a perfectly conducting metal sheet sphere whose center is at the origin, then the electric field strength at the point, P, outside the sphere, becomes (a) Zero (b) 100 µV/m (c) – 100 µV/m (d) 50 µV/m [GATE 1995 : 1 Mark] Soln. The point charge +q will induce a charge – q on the surface of metal sheet sphere. Using Gauss's law, the net electric fulx passing through a closed surface is equal to the charge enclosed = + q – q = 0 D = 0, E = 0 at point P. Option (a) 3. A metal sphere with 1 m radius and surface charge density of 10 Coulombs / m 2 is enclosed in a cube of 10 m side. The total outward electric displacement normal to the surface of the cube is (a) 40 π Coulombs (b) 10 π Coulombs (c) 5 π Coulombs (d) None of the above [GATE 1996: 1 Mark] Soln. The sphere is enclosed in a cube of side = 10m. using Gauss's law, the net electric flux flowing out through a closed surface is equal to charge enclosed.

Maxwell's equation revisited

An analytical expression for displacement current density has been derived for both a line current and a current element or a point charge particle in motion. We have shown that the divergence of the inverse square field is zero, in contrast to the accepted notion that it is a delta function. As a result, Maxwell's equations are source-free and the displacement current density is free of divergence; furthermore, both the displacement current and magnetic field are produced simultaneously by a current. Résumé : Une expression analytique pour le courant densité de déplacement a ´ eté tirée pour un courant de ligne et un courantélément ou une particule point-chargée en marche. Nous avons demontré que le divergence de l'inverse square field est zéro, en contraste au notion accepteé que c'est une fonction delta. Conséquemment, leséquations de Maxwell sont source-free et le courant densité de déplacement n'a aucun divergence ; de plus les deux, le courant de déplacement et le champ magnétique sont produit simultanément par un courant.

Maxwell's Equations 1.1 Maxwell's Equations

Maxwell's equations describe all (classical) electromagnetic phenomena: ∇ ∇ ∇ × E = − ∂B ∂t ∇ ∇ ∇ × H = J + ∂D ∂t ∇ ∇ ∇ · D = ρ ∇ ∇ ∇ · B = 0 (Maxwell's equations) (1.1.1) The first is Faraday's law of induction, the second is Ampère's law as amended by Maxwell to include the displacement current ∂D/∂t, the third and fourth are Gauss' laws for the electric and magnetic fields. The displacement current term ∂D/∂t in Ampère's law is essential in predicting the existence of propagating electromagnetic waves. Its role in establishing charge conservation is discussed in Sec. 1.7. Eqs. (1.1.1) are in SI units. The quantities E and H are the electric and magnetic field intensities and are measured in units of [volt/m] and [ampere/m], respectively. The quantities D and B are the electric and magnetic flux densities and are in units of [coulomb/m 2 ] and [weber/m 2 ], or [tesla]. D is also called the electric displacement, and B, the magnetic induction. The quantities ρ and J are the volume charge density and electric current density (charge flux) of any external charges (that is, not including any induced polarization charges and currents.) They are measured in units of [coulomb/m 3 ] and [ampere/m 2 ]. The right-hand side of the fourth equation is zero because there are no magnetic mono-pole charges. Eqs. (1.3.17)–(1.3.19) display the induced polarization terms explicitly. The charge and current densities ρ, J may be thought of as the sources of the electromagnetic fields. For wave propagation problems, these densities are localized in space; for example, they are restricted to flow on an antenna. The generated electric and magnetic fields are radiated away from these sources and can propagate to large distances to 2 1. Maxwell's Equations the receiving antennas. Away from the sources, that is, in source-free regions of space, Maxwell's equations take the simpler form: ∇ ∇ ∇ × E = − ∂B ∂t ∇ ∇ ∇ × H = ∂D ∂t ∇ ∇ ∇ · D = 0 ∇ ∇ ∇ · B = 0 (source-free Maxwell's equations) (1.1.2) The qualitative mechanism by which Maxwell's equations give rise to propagating electromagnetic fields is shown in the figure below. For example, a time-varying current J on a linear antenna generates a circulating and time-varying magnetic field H, which through Faraday's law generates a circulating electric field E, which through Ampère's law generates a magnetic field, and so on. The cross-linked electric and magnetic fields propagate away from the current source. A more precise discussion of the fields radiated by a localized current distribution is given in Chap. 15. 1.2 Lorentz Force The force on a charge q moving with velocity v in the presence of an electric and magnetic field E, B is called the Lorentz force and is given by: F = q(E + v × B) (Lorentz force) (1.2.1) Newton's equation of motion is (for non-relativistic speeds): m dv dt = F = q(E + v × B) (1.2.2) where m is the mass of the charge. The force F will increase the kinetic energy of the charge at a rate that is equal to the rate of work done by the Lorentz force on the charge, that is, v · F. Indeed, the time-derivative of the kinetic energy is: W kin = 1 2 m v · v ⇒ dW kin dt = m v · dv dt = v · F = q v · E (1.2.3) We note that only the electric force contributes to the increase of the kinetic energy— the magnetic force remains perpendicular to v, that is, v · (v × B)= 0.

Maxwell's equations revisited

An elementary solution of the Maxwell equations for a time-dependent source

European Journal of Physics, 2002

We present an elementary solution of the Maxwell equations for a timedependent source consisting of an infinite solenoid with a current density that increases linearly with time. The geometrical symmetries and the time dependence of the current density make possible a mathematical treatment that does not involve the usual technical difficulties, thus making this presentation suitable for students that are taking a first course in electromagnetism. We also show that the electric field generated by the solenoid can be used to construct an exact solution of the relativistic equation of motion of the electron that takes into account the effect of the radiation. In particular, we derive, in an almost trivial way, the formula for the radiation rate of an electron in circular motion.

Challenges to Faraday's flux rule

Faraday's law ͑or flux rule͒ is beautiful in its simplicity, but difficulties are often encountered when applying it to specific situations, particularly those where points making contact to extended conductors move over finite time intervals. These difficulties have led some to challenge the generality of the flux rule. The challenges are usually coupled with the claim that the Lorentz force law is general, even though proofs have been given of the equivalence of the two for calculating instantaneous emfs in well-defined filamentary circuits. I review a rule for applying Faraday's law, which says that the circuit at any instant must be fixed in a conducting material and must change continuously. The rule still leaves several choices for choosing the circuit. To explicate the rule, it will be applied to several challenges, including one by Feynman.

The simple proof of the Maxwell's vacuum equation (original) (raw)

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