Boundary Conditions

Maxwell’s equations may be written in integral form as follows:
Over any closed surface S.

(i)    Φ D. da = Q_fence
(ii)    Φ B. da = 0

For any surface S bounded by the closed loop L,
(iii)    Φ E.  dl = - d/dt ∫ B . da
(iv)    Φ H . dl = I_fenc  + d∫ D.da

The field E, B, D, and H will be discontinuous at a boundary between two different media or at a surface which carries charge density σ or current K.

The boundary conditions that must be satisfied by the electric and magnetic fields at an interface between two media are deduced from Maxwell’s equations.

(i)    We construct Gaussian Pill box of height h (h  0), at the interface between two media.
Applying Maxwell’s first equation,
        D₁.a – D₂.a = σ _fa
Here, σf is the surface charge density on the interface. Thus, the component of D which is perpendicular to the interface is discontinuous is the amount
        D_1| - D_2| = σf                                    … (1)



(ii)    Applying Maxwell’s second equation, we get
         B_1| - B_2| = 0                                    … (2)

(iii)    We construct a small rectangular Amperian loop. The length of the loop is l. Applying Maxwell’s third equation,
         E₁. 1 – E₂. 1 = - d ∫ B.  da

If the loop is now shrunk by letting h1 and h2 go to zero, the flux vanishes. The flux vanishes.
    . :    E_1|| - E_2||  = 0

The components of E parallel to the interface are continuous across the boundary.


(iv)    Applying Maxwell’s fourth equation, we get
H₁. I – H₂. I = I_fenc
Here, I_fence is the free current passing through the Amperian loop.
No volume current density will contribute (in the limit of infinitesimal width) but a surface current can.

Let n be b unit vector perpendicular to the interface pointing from 2 toward 1. So (n x l) is normal to the Amperian loop.
Let K_f be the surface current density. Then,
        I_fence = Kf.(n x l)l = (Kf x n).I
. :    H_1|| = H_2|| = K_f x n
The parallel components of H are discontinuous by an amount proportional to the free surface current density.
In the case of linear media, D = εE, B = μH.

The boundary conditions can be expressed in terms of E and alone.
(i)    ϵ₁ E_1| - ϵ2 E_2| = σ_f
(ii)    B| - B_2| = 0.
(iii)    E_1|| - E_2|| = 0
(iv)    1/μ₁ B_1|| - 1μ₂ B_2|| = 0.

If there is no free charge or free current at the interface, then
(i)    ϵ₁ E_1| - ϵ₂ E_2| = 0
(ii)    B_1| - B_2| = 0.
(iii)    E_1|| - E_2|| = 0.
(iv)    1/μ₁ = B_1|| - 1/μ₂ B_2|| = 0

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