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1 Fundamentals 1.1 Overview Physics 704 Spring 2018 1 Fundamentals 1.1 Overview The objective of this course is: to determine and fields in various physical systems and the forces and/or torques resulting from them. The domain of applicability is the classical regime in which the field picture is appropriate. In QM the photon is the particle that mediates the E-M force. The field picture is appropriate when interactions involve lots of photons. This regime has widespread applicability. Mostly we’ll be working with continuous charge distributions in which the discrete nature of electrons is also ignored. Idealizations we’ll make include: 51 1. The photon has zero mass: the experimental limit is 10− g. (See link on web site for details.) The mass of the photon is linked to the form of the point charge potential Φ and the corresponding inverse square force law. (Most experiments test∝ for the inverse square law rather than trying to measure mass directly.) If the photon had mass, then the potential would be of the form 1 Φ exp ∝ − ¯ Think about what the force law looks like³ in this´ case, and how it differs from apureinverse-squarelaw. 2. Perfect conductors exist. 3. An extended surface can be maintained at a constant potential, and two different surfaces can be maintained at the same potential ("common ground"). This idealization is closely related to #2. 4. Surface charge and current layers have zero thickness. The actual thick- ness is of order a few atomic diameters, so this idealization is equivalent to size of system few atomic diameters À 1.2 Units: The first half of Jackson’s book uses SI units, the second half uses Gaussian. Read the appendices on units! SI is an engineering system and is theoretically corrupt. In the Gaussian unit system, and are dimensionally equivalent; in SI they are not. This creates difficulties when and are different components of the same tensor! (See Chapter 11.) In the Gaussian system all units are derived from the three fundamental units of mass (gm), length (cm), and time (s). For example, the Coulomb force law is used to define the charge unit. 2 = = √ 2 ⇒ 1 andthustheesu(electro-static unit of charge or stat-coulomb) is gm cm 32 12 1 1 esu = cm · = cm gm s− (1) s2 r In SI a pseudo-fundamental unit of current is also included. (This is the corruption.) In cgs-Gaussian units there is no need for constants such as 0 and 0 The only constants that appear in Maxwell’s equations are and (the speed of light). 1.3 Fundamental relations: The fundamental relations of electromagnetic theory are Maxwell’s equations. In SI units, we have Gauss’ law: = = (2) ∇ · 0 ∇ · where is the total charge density and isthefreechargedensity. =0 (no magnetic monopoles) (3) ∇ · Faraday’s law = (4) ∇ × − Ampere’s law = + 0 = + (5) ∇ × 0 0 ∇ × where isthefreecurrentdensity.WealsohavetheLorentzforcelaw = + (6) × We will spend the entire semester learning³ how´ to solve these five equations, and understand the solutions. Charge conservation + =0 (7) ∇ · is a useful additional relation, but it is not independent of Maxwell’s equations. To see this, start with Gauss’ law = 0 ∇ · ³ ´ = 0 and are independent variables, 0 is constant ∇ · µ ¶ = ∇ × Ampere’s law ∇ · à 0 − ! = divergence of a curl is zero −∇ · 2 You can also derive this equation from the principle of charge conservation, as in Lea Problem 1.6. 1.4 Nature of the mathematical problem Maxwell’s equations are linear, 2nd order, elliptic PDEs. Classes of problems we’ll be interested in include: 1. Given sources ( ) everywhere, find the fields. 2. Given boundary conditions on the bounding surface of a finite volume, find the fields in the volume. (Boundary conditions show the effect of charges and currents outside the volume.) 3. Given the fields, find their effect on These are well-posed problems with unique solutions. Difficulties can arise because the fields due to sources act back on the sources, changing the position and velocity of charges and thus changing the sources that produce the fields. For the most part we will not get involved with these complications. (But see plasma physics course (Phys 712) , and also J Chapter 16). Linearity is nice. It gives us the principle of superposition. We can compute the fields due to individual sources and simply add the results. The technique of using Green’s functions depends on the linearity of Maxwell’s equations. Structure of the equations. The equations may be divided into two groups in two ways: 1. Divergence equations and curl equations. This is useful when finding boundary conditions. 2. Equations with sources and equations without sources. This is actually more fundamental— See Ch 11. If we write the equations in 4-vectors, there are only two equations. The lack of symmetry is due to non-existence of magnetic monopoles. 1.5 Fields in media. In a material medium, an applied polarizes the material, reducing the net in the interior. It is useful to define a new field the electric displacement, = 0 + where is the polarization. Since is generally proportional to the net (in which case the material is said to be linear and isotropic) = 0 where is the susceptibilty, then = 0 (1 + ) = (8) Then = 0 + ∇ · ∇ · ∇ · = = − 3 where = −∇ · is the bound charge density and isthefreechargedensity.SeeCh4for more details. (We’ll get to this in April.) This description is most useful in the so-called LIH materials (linear, isotropic, homogeneous) in which is a constant scalar. However, is always a function of the frequency of the applied fields, and strictly equation (8) is true in the Fourier transform domain. ()= () () (9) and this leads to an interesting relation between the time-dependent fields. (See Lea Ch 7 P 12 and Jackson Ch 7.) In static situations we use = lim () 0 → For non-isotropic materials is a tensor, and the relation is = indicating that one component of depends on more than one component of See http://www.physics.sfsu.edu/~lea/courses/grad/plasmawavesi.PDF for an example. Similarly, atoms have magnetic moments which can be modified and/or aligned by applied fields. (see e.g. Ch29inLeaandBurkeandalsoJp 191-194) In these cases we have a magnetization = and = 0 + = ³ ´ In situations where this description is useful (LIH) we find that 1 The 0 ' interesting cases with 0 substantially different from one are not LIH, and the non-linearity gives rise to some very interesting phenomena (hysteresis etc). We won’t get into these phenomena much in this course. 2 Boundary conditions Here we investigate the relations between the fieldcomponentsoneithersideofa boundary between different materials. We start with the divergence equations, and integrate over a pillbox placed across the boundary. The dimensions of the pillbox are chosen as follows: 2 = length scale over which any property changes ' ¿ ¿ 4 Then, from Gauss’s law, we have = = ∇ · Zpillbox Zpillbox Z Thefreechargeisusuallyconfined to the surface (remember: this means within a few atomic diameters of the surface), so defining the surface charge density as = and applying the divergence theorem on the LHS, we have R ˆ = = · I Z Z Z With the sides of the pillbox give a negligible contribution to the integral ¿ on the LHS. With both and are approximately constant over the area. Then, with ˆ¿pointing from medium 2 to medium 1, as shown in the diagram, 1 2 ˆ = − · Thus ³ ´ 1 2 ˆ = (10) − · For the electric field ³ ´ 1 2 ˆ = (11) − · 0 ³ ´ and since satisfies the equation =0 then ∇ · 1 2 ˆ =0 (12) − · ³ ´ The normal component of is continuous across any surface. To avoid sign errors, always draw the pillbox and define the normal nˆ. Next we look at the curl equations and integrate around a rectangle that crosses the boundary. We choose the dimensions such that the rectangle mea- sures where, as before, We start with Ampere’s law. × ¿ ¿ 5 ˆ = ˆ + ˆ ∇ × · · · Z ³ ´ Z Z Here the unit vector ˆ is tangent to the surface, and the normal ˆ is perpendic- ular to the plane of the rectangle, according to the RH rule. Then using Stokes’ theorem, and for ¿ = ˆ + ˆ · · · I Z Z Then as we let 0 the short sides of the rectangle make a negligible contri- bution to the line→ integral, and 1 ˆ + 2 ˆ = ˆ · − · · where h ¡ ¢ i = Z is the free surface current density. Note that as 0 we trap all the sur- face current inside our rectangle. (Effectively, is a→ delta function and thus right at the surface.) However, the time derivative of remains finite everywhere,→∞ and so the 2nd integral on the RHS always goes to zero as 0 . We may rewrite the LHS as follows → 2 1 ˆ ˆ = 2 1 ˆ ˆ = ˆ − · × − × · · ³ ´ ³ ´ h³ ´ i Now we may orient our rectangle arbitrarily in the sense that ˆ can be any vector in the plane perpendicular to ˆ For example, if ˆ =ˆ we can choose ˆ =ˆ or ˆ =ˆ Thus we get all components of the equation 2 1 ˆ = =ˆ 1 2 (13) − × × − ³ ´ ³ ´ or, for ˆ 1 2 = (14) × − 0 Similarly for ³ ´ ˆ 1 2 =0 (15) × − The tangential component of is³ continuous´ across any surface. 6 3 Derivation of equations from experimental re- sults Physics is an experimental science, so Maxwell’s equations have their roots in experiment. In this section we shall assume that all fields and charges change slowly, and all speeds ¿ 3.1 Coulomb’s Law Cavendish (in England) and Coulomb (in France, 1785) independently deter- mined the electostatic force law. The force exerted on one point charge by another is ˆ on 2 due to 1 12 ∝ 2 The constant of proportionality depends on the unit system.
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