Maxwell's Equation Guide

Electromagnetism Chapter-06 Maxwell's equation We have the basic equation of electricity and magnetism are, 1. Gauss law in electrostatics; ∇. 𝐸⃗ = ∇. 𝜖 𝐸⃗ = 𝜌 ∇. 𝐷⃗ = 𝜌 ………………………………………………….1 Or, Where, 𝐷⃗ = 𝜖 𝐸⃗ is displacement current vector. 2. Gauss law in magnetostatics; ∇. 𝐵⃗ = 0 ……………………………………2 3. Faradays law of e.m. induction; Induced emf, 𝜖=− Or, ∫ 𝐸⃗ . d 𝑙⃗ = − Or, ∮(∇ × 𝐸⃗ ). 𝑑𝑆⃗ = − Or, ∇ × 𝐸⃗ = − (∮ 𝐵⃗. 𝑑𝑆⃗ ) (∮ 𝐵⃗. 𝑑𝑆⃗) ⃗ ……………………………………3 4. Amperes circuital law; ∮ 𝐵⃗. 𝑑𝑙⃗ = 𝜇 𝐼 ∮ ∇ × 𝐵⃗ . 𝑑𝑆⃗ = 𝜇 ∮ 𝐽⃗. 𝑑𝑆⃗ ∇ × 𝐵⃗ = 𝜇 𝐽⃗ ∇× ⃗ = 𝐽⃗ ∇ × 𝐻⃗ = 𝐽⃗ ………………………………………………………4 These four equations are called maxwells equations which are used in electrostatics and magnetostatics. Equation (3) & (4) also relates electric and magnetic fields. Among these four equations, first three are hold for both static and dynamic electric & magnetic field. But, fourth one is valid only for steady current. So, it does not hold for time varying field. Electromagnetism Diaplacement current We have amperes circuital law, ∮ 𝐵⃗. 𝑑𝑙⃗ = 𝜇 𝐼 Here, we know, 𝐼 = ∮ 𝐽⃗. 𝑑𝑆⃗ and using stokes theorem, we get ∮ ∇ × 𝐵⃗ . 𝑑𝑆⃗ = 𝜇 ∮ 𝐽⃗. 𝑑𝑆⃗ ∇ × 𝐵⃗ = 𝜇 𝐽⃗ ∇× ⃗ = 𝐽⃗ ∇ × 𝐻⃗ = 𝐽⃗ ………………………………………………………1 Taking Divergece on both sides, ∇. ∇ × 𝐻⃗ = ∇. 𝐽⃗ 0 = ∇. 𝐽⃗ ∇. 𝐽⃗ = 0 ………………………………………………………..2 But according to continuity equation, ∇. 𝐽⃗ = − ≠ 0 …………………………………………3 Thus, equation (2) & (3) are incompatible for time varying conditions. We must modify equation (1) to agree with equation (3). To do this, we add a term to equation (1), so that it becomes, ∇ × 𝐻⃗ = 𝐽⃗ +𝐽⃗ ………………………………………..4 Where, 𝐽⃗ is to be determined and defined. Now taking divergence on both sides, we get ∇. (∇ × 𝐻⃗ ) = ∇. (𝐽⃗ +𝐽⃗ ) 0 = ∇. 𝐽⃗ + ∇. 𝐽⃗ Or, − + ∇. 𝐽⃗ = 0 Electromagnetism ∇. 𝐽⃗ = ∇. 𝐷⃗ = 𝜌 , so, We know, (∇. ⃗) ∇. 𝐽⃗ = ∇. 𝐽⃗ = ∇. ⃗ 𝐽⃗ = Or, ⃗ Substituiting this value in equation (4), we get ⃗ ∇ × 𝐻⃗ = 𝐽⃗ + Where, 𝐽⃗ = ⃗ ……………………………………………….5 is called displacement current density and 𝐽⃗ is conduction current density Here, equation (5) is maxwells fourth equation for time varying field. Now, on the basis of displacement current density, displacement current is defined as, I = ∫ 𝐽⃗ . d𝑆⃗ = ∫ ⃗ . d𝑆⃗ ⃗ Hence, Displacement current is defined as the rate of change of the electric displacement field (𝑫 ) in a magnetic field. In other words, this is a current produced in a magnetic field due to change in electric electric field with time. So this is the result of time varying electric field. The typical example of this current is the current through a capacitor when an alternating voltage source is applied to its plates. Significance of displacement current Displacement currents play a central role in the propagation of electromagnetic radiation, such as light and radio waves, through empty space. A traveling, varying magnetic field is everywhere associated with a periodically changing electric field that may be conceived in terms of a displacement current. Maxwell's equation in final form We have the maxwells equations; ∇. 𝐷⃗ = 𝜌 ∇. 𝐵⃗ = 0 ∇ × 𝐸⃗ = − ∇ × 𝐻⃗ = 𝐽⃗ + ⃗ ⃗ Electromagnetism Proof: 1. ∇. 𝐷⃗ = 𝜌 We have according to Gauss law, ϕ = ∮ 𝐸⃗ . 𝑑𝑆⃗ = q ∮ 𝐸⃗ . 𝑑𝑆⃗ = ∫ (∇. 𝐸⃗ ) 𝑑𝑣 = q ∫ ρdv For arbitary volume, ∇. 𝐸⃗ = ∇. ϵ 𝐸⃗ = ρ ∇. 𝐷⃗ = ρ Which is first law of Maxwell. This states the relation between the electric field and the charges that produced it. this is the generalization of Coulombs law and valid for static charges as well as charges in uniform motion i.e. if electric field varies with time. 2. ∇. 𝐵⃗ = 0 Since the magnetic lines of force are either close or go of infinity then the number of magnetic force entering any arbitary close surface is exactly same as leaving it. It means the flux of magnetic induction vector B⃗ across any closed surfaces is always zero. i.e. ∫ B⃗. dS⃗ = 0 or, ∫ (∇.B⃗)dv= 0 For arbitary volume 𝑑𝑣, ∇.B⃗ = 0 Which is maxwells second law. 3. ∇ × 𝐸⃗ = − ⃗ According to faradays law of electromagnetic induction, e.m.f. induced in coil is, 𝐸 = − = − (∫ B⃗. dS⃗) …………………………………….1 Since, emf 𝐸 can be also calculated by calculating workdone in carying a unit charge completely around a close loop. i.e. 𝐸 = ∮ 𝐸⃗ . d 𝑙⃗ ……………………………………..2 so, from equation (1) and (2). ∮ 𝐸⃗ . d 𝑙⃗ = − (∫ B⃗. dS⃗)