The Intuitive Derivation of Maxwell s Equations

Transcription

1 The Intuitive erivation of Maxwell s Equations Frank Lee at franklyandjournal.wordpress.com April 14, 2016 Preface This paper will cover the intuitive and mathematical reasoning behind all four of Maxwell s equations. While the goal is to maintain accuracy and keep definitions as formal as possible, there will inevitably be imperfections. ince the primary motivation behind this paper is to offer a more intuitive understanding behind the equations, some sections may be over generalised or ill-defined. Nevertheless, this paper will aid anyone attempting to grasp the meaning behind Maxwell s famous equations. 1 Introduction Before, we have seen many scalar equations for electrical and magnetic fields. These equations work well for constant field strengths and constant electrical charge and so on but are incompatible with a varying field across space or a changing charge density in a wire. Using our previous knowledge of electricity and magnetism and calculus, we can formulate four equations that unites electricity and magnetism and provides the basis for modern technology. These four equations: Gauss Law, Faraday s Law, Gauss Magnetism Law and Ampere s Law are collectively known as Maxwell s equations and will be analysed in this paper in that order. Below are Maxwell s equations in the said order E = ρ ε 0 (1) E = B (2) B = 0 (3) B = µ 0 (J + ε 0 E ) (4) where µ 0 is the permeability of free space constant (related to magnetic fields), ε 0 is the permittivity of free space (related to electrical fields) and E and B are the electrical and magnetic vector fields respectfully. 1

2 2 Gauss Law Let s consider the scalar equation for the electrical field strength for a point charge, E at a distance r from a source with charge q E = q 4πε 0 r 2 (5) equation (5) offers an important insight: the strength of the electrical field depends on the amount of charge present. Now lets consider a closed surface that bounds some volume. We know that the electrical flux passing through is related to the amount of charge in. More precisely, the electrical flux, Φ E is proportional to the total charge and hence Φ E = E d = kq (6) for some constant k. Also, notice that we can rewrite the total charge q as ρ is the charge density. Therefore and applying the divergence theorem, we arrive at ( E) dv = k ρ dv where E d = k ρ dv (7) ρ dv (8) E = kρ (9) and it turns out that k = 1 ε 0, which makes sense since E is proportional to 1 ε 0 from equation (5). This is known as Gauss s law, which is often written as E = ρ ε 0 (10) How do we interpret Gauss; Law? Remember that the divergence of a vector field measures the amount of sinks and sources and so from equation (10) we can conclude that charges act like sinks and sources in a an electric field! In this case, we can think of the presence of charge as the source of electrical fields (though, this isn t entirely correct).this conclusion is particularly useful in considering Gauss magnetism law (3), which we will see later on. 2

3 3 Faraday s Law Faraday s Law may be perhaps the most familiar of Maxwell s Laws. Backed with experimental evidence, Faraday deduced that an emf is produced if there is a changing magnetic flux, which can be written as ε = dφ B (11) dt where ε is the induced electromotive force and the negative sign represents Lenz s Law, which states that the induced ε produces a magnetic field that opposes the movement of a moving magnetic field. Our job now is to define the components in equation (11) and see if we can simplify the relationship. The magnetic flux, just like the electric flux is by definition a line integral given by Φ B = B d (12) Now we focus our attention to the emf or the induced potential difference, V. Remember that the electrical field strength is just the change in potential over the change in displacement or E = V. Therefore, the induced emf is just the product between the electrical r field strength and the displacement and so summing this up for any curve c, which bounds the same surface V = E dr (13) and using toke s theorem, we can arrive at V = E dr = ( E) d (14) and substituting (12) and (14) in equation (11) ( E) d = d dt B d (15) E = B (16) which is known as Faraday s Law. We know that an electric field is a potential field and so we would normally expect E = 0. However from Faraday s Law, we can see that this is only true if E is not changing. Otherwise, if the electrical field is varying the curl is a non-zero value. 3

4 4 Gauss Magnetism Law Gauss Magnetism Law is the magnetic field equivalent of Gauss Law for electrical fields. Remember that Gauss Law states E = ρ ε 0 or that the charges act like sinks and sources for the electrical field. Now let s consider what B is. The divergence of the magnetic field B is also a measure of the amount of sinks and sources in the field, but by considering Gauss Law this must be 0. By symmetry, we can think of magnetic monopoles giving rise to magnetic fields but as far as we know, magnetic monopoles do not exist. Every magnet will have a north and a corresponding south side, even if you begin to split the magnets in half. There will always be two opposite poles and so there are no sources or sinks and therefore the divergence is 0. This is Gauss Magnetism Law and is written as 5 Ampere s Law B = 0 (17) Ampere s Law is perhaps the most difficult of Maxwell s Law to understand and is where Maxwell offered his contribution in uniting Maxwell s equations together. The primary focus is how magnetic fields and electrical currents are related and Ampere based his equation on this proposition: Just as how the electrical field is proportional to the charge, then the magnetic field is proportional to the electrical current, I. With ample experimental evidence, we could deduce that the total amount of B along a closed loop, is proportional to the current or B dr = µ 0 I (18) where µ 0 is the proportionality constant and is known as the permeability of free space. We can apply toke s theorem to equation (18) and so µ 0 I = B dr = ( B) d (19) where is the area region bounded by the closed loop. The next step is to rewrite I in terms of a cross sectional current density vector field, J, which is µ 0 I = µ 0 J d. If we equate this to equation (19) we arrive at B = µ 0 J (20) Very quickly, many realised there was a problem with equation (20), including Ampere himself. To see the problem, let s take the divergence of both sides of the equation ( B) = µ 0 ( J) = 0 (21) 4

5 since the divergence of the curl is 0. At first, this seems to make sense since a 0 divergence implies that the amount of current leaving a region equals the amount of current entering. What if we had a capacitor in an A circuit? The current will still flow along the circuit but clearly no current is flowing between the capacitor plates. There is a build up of charge on one plate and most certainly the electrical field is changing and therefore the amount of charge is changing in the capacitor yet equation (21) claims there is no divergence. There must be a missing term, which accounts for the changing electrical field in the capacitor. This is what Maxwell realised and he added his correction to (21). Let s consider what J and then we can see what this correction is. We know that the total current leaving a surface is given by I = J d but there must an equal decrease of charge within the bounded volume. Therefore we can conclude I = q we have seen in Gauss Law that we can write the total charge in terms of the charge density, ρ as q = ρ dv and so ρ J d = and applying the ivergence theorem we arrive at ( J) dv = J = ρ but from equation (10) we know that ε 0 ( E) = ρ. Therefore (22) dv (23) ρ dv (24) (25) J = ε 0 ( E ) (26) We know however that ( B) will always be 0, since this is an identity and so we must add another term to equation (20) in order to cancel out the µ 0 J term. This was Maxwell s great contribution and he coined this term the displacement current. Now, Ampere s Law states B = µ 0 (J + ε 0 E ) (27) Notice the symmetry in Ampere s equation. Maxwell realised that Faraday s law stated a changing magnetic flux produces an electrical field so he reasoned that a changing electrical flux must then produce a magnetic field. 5