1.1 Classical electrodynamics
1.1.2 Electromagnetic boundary conditions
ε0E(r, t) +P(r, t)
=%free(r, t) (1.24)
after some rearrangement. We refer to (1.24) as the free source Maxwell divergence equation for the electric field. Because in the free source Maxwell equations (1.21) and (1.24) the medium’s information is absorbed in the electric and magnetic polarization we now need to supplement the modified set of Maxwell’s equations (1.8), (1.9), (1.21) and (1.24) with the material equations for the electric and magnetic polarization instead of the charge and current density. One example, where the material information is given in terms of the electric polarization, is the material model for non-magnetic Raman-active materials according to section 5.1.
1.1.2 Electromagnetic boundary conditions
Usually, we do not consider an infinitley extended material but rather a material confined within certain boundaries. For example, if we have a particle with a certain geometry sitting in a contrasting material, the problem is not fully described by Maxwell’s equations (1.7) to (1.10) and a supplementary set of material equations. In addition, one needs boundary conditions, which determine the behaviour of the macroscopic fields at the interface between the particle and its surrounding. In the following we determine the boundary conditions for the electric and magnetic field at the interface between two different materials denoted by 1 and 2. Note, however, that in addition one needs to choose appropriate boundary conditions for the material quantities, e.g. the the charge and current density, which are specific to the material model at hand. This is done, for instance, for the hydrodynamic material model in section 4.4. We, first, focus on the boundary condition for the normal component of the electric and magnetic field, respectively. Therefore, we imagine a closed pillbox volume V at the interface with the basis area F along the tangential plane of the interface and height h (see fig. 1.2). To explore the behaviour of the normal component of the electric field we integrate Maxwell’s divergence equation for the electric field (1.7) over the pillbox volume
Z
V
d3r ∇ ·E(r, t) = 1 ε0
Z
V
d3r %(r, t). (1.25)
In the limit of an infinitesimally small pillbox height (h→0) the integral on the r.h.s. of (1.25) is equivalent to the charge qsurf at the surface
h→0lim Z
V
d3r %(r, t) =qsurf. (1.26)
and the dielectric displacement
D(r, t) =ε0E(r, t) +P(r, t), (1.23) which lead to a version of Maxwell’s equations often referred to as Maxwell’s equations in matter [28, 29]. We abstain from the introduction of these auxiliary fields as we want to describe the electromagnetic field by the electric fieldEandBinstead.
Classical electrodynamics 1.1
We then calculate the l.h.s of (1.25) in the limit of h→0 by using Gauß’s theorem [31]
lim
h→0
V
d3r ∇ ·E(r, t)
Gauß’s theorem
= lim
h→0
∂V
d2r n˘·E(r, t) =F n˘·(E2−E1) (1.27) where the face normal ˘n of the interface and the directional area F are given according to fig. 1.2. Furthermore, E1 denotes the limit of the electric field at the interface coming from material 1 and E2 denotes the limit of the electric field at the interface coming from material 2. Then from (1.26) and (1.27) it follows that the normal component of the electric field has a jump discontinuity across the interface of two different materials according to
˘
n·(E2−E1) = qsurf
ε0F. (1.28)
Following the same argumentation line as for the normal component of the electric field one finds the boundary condition for the normal component of the magnetic field. We, hence, integrate Maxwell’s divergence equation of the magnetic field (1.8) over the pillbox volume V
V
d3r ∇ ·B(r, t) = 0. (1.29) Using Gauß’s theorem the l.h.s. of (1.29) becomes
h→0lim
V
∇ ·B(r, t)
Gauß’s theorem
= lim
h→0
V
d2r n˘·B(r, t) =F n˘·(B2−B1). (1.30) in the limit of h→0. Then from (1.29) and (1.30) it follows that due to the lack of magnetic charges the normal component of the magnetic field is continuous across the interface
˘
n·(B2−B1) = 0. (1.31)
So far, we have only discussed the boundary conditions for the normal component of the electric and magnetic field, respectively. To obtain the boundary conditions for the entire electric and magnetic field we, yet, need to determine the boundary conditions for the tangential
˘ n
F=Fn˘
−F 2
1 h
Figure 1.2: Depiction of an auxiliary volume to determine the boundary con-ditions for the normal component of the electric and the magnetic field at the in-terface of two different materials 1 and 2. Adapted from [31].
components. We, thus, imagine a two-dimensional box formed by the closed pathC=s1−h+ s2+h around an arbitrary point at the interface between material 1 and 2 (see fig. 1.3), where we require the path segmentss1ands2to be in binormal direction with respect to the interface tangential ˘tand the interface normal ˘n. Accordingly,s1 ands2 have the same lengths, yielding s1=−s2≡s(˘t×n). Furthermore, the path˘ Cencloses the directional areaF, where the face normal of F is equivalent to the interface’s tangential. We, therefore, have F≡F˘t. We now integrate Maxwell’s curl equation for the electric field (1.9) over the area F
Z
F
df·[∇ ×E(r, t)] = Z
F
df ·[∂tB(r, t)] (1.32) In the limit of an infinitesimally small box height (h→ 0) the integral on the r.h.s. of (1.32) vanishes identically
lim
h→0
Z
F
df·[∂tB(r, t)] = 0 (1.33) as the partial derivative (∂t B) is bounded at the interface. Using Stokes’ theorem [31] we calculate the l.h.s. of (1.32) in the limit of h→0
h→0lim Z
F
df·[∇ ×E(r, t)]
Stokes’
theorem
= lim
h→0
Z
C
dl·E(r, t) =
= s1·E1+s2·E2 =s(˘t×n)˘ ·(E2−E1) =
= s[ ˘n×(E2−E1) ]·˘t,
(1.34)
where we have used the vector identity (A×B)·C= (B×C)·A in the last line. From (1.33) and (1.34) it follows that the tangential component of the electric field is continuous across the interface, yieling
˘
n×(E2−E1) =0. (1.35)
F=Ft˘
2 1
˘ n
s2
s1
h
Figure 1.3: Depiction of an auxil-iary area to determine the boundary conditions for the tangential compo-nents of the electric and magnetic field at the interface of two differ-ent materials 1 and 2. Adapted from [31].
Classical electrodynamics 1.1
In the same manner we construct the boundary condition for the tangential component of the magnetic field strength. By integrating (1.10) over the area Aone finds
Z Since the temporal derivative of the electric field (∂tE) is bounded at the surface the second integral on the r.h.s. of (1.36) vanishes identically in the limit of h→0
h→0lim Z
A
df·∂tE(r, t) = 0. (1.37) Therefore, only the surface integral of the current density remains on the r.h.s. of (1.36). For h→0 this yields where Isurf is the electric current at the interface. Finally, in the limit of h → 0 the surface integral on the l.h.s. of (1.36) yields
h→0lim Hence, combining (1.36) to (1.39) it follows that the tangential component of the magnetic field is discontinuous at the interface, where the jump discontinuity is determined by
˘
n×(B2−B1) = µ0Isurf s
˘t. (1.40)
Note, that according to (1.21) for non-magnetic materials (M=0) the surface current can only stem from the free excess charges at the surface because limh→0 R
Adf·∂tP=0as long as (∂tP) is bounded at the interface. Thus, if there are no excess charges, the tangential component of the magnetic field is smooth at the interface between two different non-magnetic materials.
We summarize the boundary conditions for the electric and magnetic field at the interface of two different materials 1 and 2 by
˘