Boundary Conditions at Plane Interface between two Media PDF

Boundary Conditions at Plane Interface between two Media Electromagnetics for medical physics ‫الكهرومغناطيسية للفيزياء الطبية‬ Second year ‫المستوى الثاني‬ The seventh lecture ‫المحاضرة السابعة‬ Dr. Faten Monjed Hussein...

Boundary Conditions at Plane Interface between two Media Electromagnetics for medical physics ‫الكهرومغناطيسية للفيزياء الطبية‬ Second year ‫المستوى الثاني‬ The seventh lecture ‫المحاضرة السابعة‬ Dr. Faten Monjed Hussein 1 The relationship between the electric flux density D, electric field intensity E, magnetic flux density B, and magnetic field intensity H can be explained with the help of point form or integral form of Maxwell's equations. The field equations postulated by Maxwell are valid at a point in a continuous medium. Maxwell's equations are useful in determining the conditions at the boundary surface of two different media. The concepts of linear, isotropic, and homogeneous medium can be applied. Consider the boundary between medium 1 with parameters ϵ1, μ1, and σ1 and medium 2 with parameters ϵ2, μ2, and σ2. In general, the boundary conditions for time varying fields are same as those for static fields. (i) Boundary Condition for Electric Displacement Vector : We have from Maxwell’s first equation ⃗∇. 𝐷 ⃗ = 𝜌. We now consider the interface of two dielectric media at which we also take a cylindrical pillbox-like surface 𝑆 composed of 𝑆1 , 𝑆2 , 𝑆3 , 𝑎𝑛𝑑 𝑆4 as shown in figure. Integrating over the pill-box-shaped volume𝑉, we get ∫ 𝑑𝑖𝑣 𝐷 ⃗ 𝑑𝑣 = ∫ 𝜌 𝑑𝑣 𝑉 𝑣 ⃗. 𝑑𝑆 = ∫ 𝜌 𝑑𝑣 ∮𝑠 𝐷 𝑣 i.e. ∫ ⃗⃗⃗⃗ 𝐷1. 𝑛 ̂𝑑𝑠 1 + ∫ ⃗⃗⃗⃗ 𝐷2. 𝑛 ̂𝑑𝑠 2 + ∫ ⃗⃗⃗⃗⃗ 𝐷1 ′. 𝑛 ̂′𝑑𝑠 1 + ∫ ⃗⃗⃗⃗⃗ 𝐷2 ′. 𝑛 ̂′𝑑𝑠 2 = ∫ 𝜌𝑑𝑉 𝑠1 𝑠2 𝑠1 𝑠2 𝑣 Where ⃗⃗⃗⃗ ⃗⃗⃗⃗2 are respective electric displacement vectors in medium 𝐷1. and 𝐷 – 1 and medium – 2, and 𝑛 ̂and 1 𝑛 ̂2 are respective unit normal at those two media with respect to the end cross sections of the pill box as shown. If now𝐷 ⃗⃗⃗⃗1 is bounded, letting the height of the pillbox h, approach zero, the third and fourth terms of the above equation vanishes and S1 approaches S geometrically and the entire surface takes the form of A as shown in figure. Hence in the limit ℎ → 0we get: lim [ ∫ ⃗⃗⃗⃗ 𝐷1. 𝑛̂1 𝑑𝑠 + ∫ ⃗⃗⃗⃗ 𝐷2. 𝑛̂2 𝑑𝑠] = lim ∫ 𝜌𝑑𝑉 ℎ→0 ℎ→0 𝑠1 𝑠2 𝑉 If σ is the surface charge density at that interface then ⃗⃗⃗⃗ 𝐷1. 𝑛̂1 𝑆1 + ⃗⃗⃗⃗ 𝐷2. 𝑛̂2 𝑆2 = σ𝐴 i.e. ⃗⃗⃗⃗⃗⃗ (𝐷1. 𝑛̂1 + ⃗⃗⃗⃗ 𝐷2. 𝑛̂2 )𝐴 = σ𝐴 and then also for arbitrary surface area and 𝑛̂1 = 𝑛̂ , 𝑛̂2 = −𝑛 ̂ we get ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ (𝐷1. 𝑛̂ − 𝐷2. 𝑛̂) = σ i.e. 𝐷1𝑛 − 𝐷2𝑛 = 𝜎 2 were. 𝐷1𝑛 and 𝐷2𝑛 are the normal components of electric displacement vector in the two media. Thus, we conclude that the normal component of electric displacement is not continuous at the interface but changes by an amount equal to the free surface charge density at the interface. (ii) Boundary Conditions for Magnetic Induction : We have from Maxwell’s second equation is. Again by taking volume integral over the entire volume of pillbox, we get Using Gauss’s divergence theorem Considering as in earlier part, we note that third and fourth terms vanish while approach each other; so that in the limit , we get Since surface is arbitrary, we get For in opposite sense and then we get I.e. the normal component of magnetic induction is continuous across the interface. (ii) Boundary Condition for Electric Intensity Vector : We again get from Maxwell’s third equation is At the interface between two media, we now consider a rectangular loop bounding a surface as shown in figure. Integrating over the loop , we get. Now by using Stroke’s theorem we get If the loop is now shrunk by taking then along the interface the contribution to the integral from sides and will vanish 3 And also the surface integral on R.H.S. of above equation tends to zero provided that is finite everywhere. Thus in the limit , we get where are the tangential components of the electric field in the two media. Here represents that tangential components of must be continuous across the interface. (iv) Boundary Condition for Magnetic Field Intensity : Also we have from Maxwell’s fourth equation is Again by taking the surface integral over rectangular loop we get. Using Stoke’s theorem we get If the loop is taken compressed along the interface we get in the limit , and Where represents the components of surface current density perpendicular to the direction of -component which is being matched. The idea of surface current density is closely analogous to that of a surface charge density it represents a finite current in an infinitesimal layer. Then in the limit , 4 Thus the tangential component of magnetic field intensity is not continuous at the interface; but changes by an amount equal to the component of the surface current density perpendicular to tangential component of. If the surface current density is zero unless the conductivity is infinite; hence for finite conductivity That is for one medium has infinite conductivity the tangential component of magnetic field intensity is continuous. Conclusion: Etan1=Etan2 ……………………………………………………….(1) ii- The tangential component of magnetic field intensity H is continuous across the surface except for a perfect conductor. Htan1=Htan2 ………………………………………….……….…(2a) At the surface of perfect conductor, the tangential component of the magnetic field intensity is discontinuous at the boundary. Htan1-Htan2 = K …………………………………….…………...(2b) i) The normal component of electric flux density is continuous at the boundary if the surface charge density is zero. DN1=DN2 ………………………………………………….……(3a) If the surface charge density is not zero, then the normal component of electric flux density is discontinuous at the boundary. DN1-DN2=ρs ……………………………………………………(3b) ii) The normal component of magnetic flux density is continuous at the boundary. BN1=BN2 ………………………………………………………(4) 5