An electromagnetic signal is generated by a Hertzian dipole located at a point P

Question

An electromagnetic signal is generated by a Hertzian dipole located at a point P, which has the position vector r = (100 m) ez. The signal is detected by a small wire loop located at the origin. Apart from the dipole and the loop, the nearby space is empty.

Experimentation reveals that the detected signal is induced by a changing magnetic field:

Bphys(t) = B0 sin (2f t) ex,

where B0 = 0.1 T and f = 30 MHz.

(d) Using the Amp`ere–Maxwell law show that the associated electric field is E = i cB0 exp[i(kz t)] ey. (13 marks)

Explanation / Answer

Maxwell’s equations Maxwell’s equations may be written in differential form as follows: · D~ = , (1) · B~ = 0, (2) × H~ = J~ + D~ t , (3) × E~ = B~ t . (4) The fields B~ (magnetic flux density) and E~ (electric field strength) determine the force on a particle of charge q travelling with velocity ~v (the Lorentz force equation): F~ = q E~ + ~v × B~ . The electric displacement D~ and magnetic intensity H~ are related to the electric field and magnetic flux density by the constitutive relations: D~ = E, ~ B~ = µH. ~ The electric permittivity and magnetic permeability µ depend on the medium within which the fields exist. The values of these quantities in vacuum are fundamental physical constants. In SI units: µ0 = 4 × 107 Hm1 , 0 = 1 µ0c 2 , where c is the speed of light in vacuum. The permittivity and permeability of a material characterize the response of that material to electric and magnetic fields. In simplified models, they are often regarded as constants for a given material; however, in reality the permittivity and permeability can have a complicated dependence on the fields that are present. Note that the relative permittivity r and the relative permeability µr are frequently used. These are dimensionless quantities, defined by: r = 0 , µr = µ µ0 . (5) arXiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Fig. 1: Snapshot of a numerical solution to Maxwell’s equations for a bunch of electrons moving through a beam position monitor in an accelerator vacuum chamber. The colours show the strength of the electric field. The bunch is moving from right to left: the location of the bunch corresponds to the large region of high field intensity towards the left hand side. (Image courtesy of M. Korostelev.) That is, the relative permittivity is the permittivity of a material relative to the permittivity of free space, and similarly for the relative permeability. The quantities and J~ are respectively the electric charge density (charge per unit volume) and electric current density (J~·~n is the charge crossing unit area perpendicular to unit vector ~n per unit time). Equations (2) and (4) are independent of and J~, and are generally referred to as the “homogeneous” equations; the other two equations, (1) and (3) are dependent on and J~, and are generally referred to as the “inhomogeneous” equations. The charge density and current density may be regarded as sources of electromagnetic fields. When the charge density and current density are specified (as functions of space, and, generally, time), one can integrate Maxwell’s equations (1)–(3) to find possible electric and magnetic fields in the system. Usually, however, the solution one finds by integration is not unique: for example, as we shall see, there are many possible field patterns that may exist in a cavity (or waveguide) of given geometry. Most realistic situations are sufficiently complicated that solutions to Maxwell’s equations cannot be obtained analytically. A variety of computer codes exist to provide solutions numerically, once the charges, currents, and properties of the materials present are all specified, see for example References [1–3]. Solving for the fields in realistic systems (with three spatial dimensions, and a dependence on time) often requires a considerable amount of computing power; some sophisticated techniques have been developed for solving Maxwell’s equations numerically with good efficiency [4]. An example of a numerical solution to Maxwell’s equations in the context of a particle accelerator is shown in Fig. 1. We do not consider such techniques here, but focus instead on the analytical solutions that may be obtained in idealized situations. Although the solutions in such cases may not be sufficiently accurate to complete the design of real accelerator components, the analytical solutions do provide a useful basis for describing the fields in (for example) real RF cavities and waveguides. An important feature of Maxwell’s equations is that, for systems containing materials with constant permittivity and permeability (i.e. permittivity and permeability that are independent of the fields present), the equations are linear in the fields and sources. That is, each term in the equations involves a field or a source to (at most) the first power, and products of fields or sources do not appear. As a consequence, the principle of superposition applies: if E~ 1, B~ 1 and E~ 2, B~ 2 are solutions of Maxwell’s equations with given boundary conditions, then E~ T = E~ 1 + E~ 2 and B~ T = B~ 1 + B~ 2 will also be so- 2 lutions of Maxwell’s equations, with the same boundary conditions. This means that it is possible to represent complicated fields as superpositions of simpler fields. An important and widely used analysis technique for electromagnetic systems, including RF cavities and waveguides, is to find a set of solutions to Maxwell’s equations from which more complete and complicated solutions may be constructed. The members of the set are known as modes; the modes can generally be labelled using mode indices. For example, plane electromagnetic waves in free space may be labelled using the three components of the wave vector that describes the direction and wavelength of the wave. Important properties of the electromagnetic fields, such as the frequency of oscillation, can often be expressed in terms of the mode indices. Solutions to Maxwell’s equations lead to a rich diversity of phenomena, including the fields around charges and currents in certain basic configurations, and the generation, transmission and absorption of electromagnetic radiation. Many existing texts cover these phenomena in detail; for example, Grant and Phillips [5], or the authoritative text by Jackson [6]. We consider these aspects rather briefly, with an emphasis on those features of the theory that are important for understanding the properties of RF components in accelerators.

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