Boundary conditions for the electromagnetic field

The boundary conditions for the electromagnetic field are the conditions connecting the values of the intensities and inductions of the magnetic and electric fields on different sides of surfaces characterized by a certain surface density of the electric charge and / or electric current .

The following boundary conditions follow from the Gauss theorem . The equations are given in the GHS system of units.

For normal components of electrical induction :

$D_{2n}-D_{1n}=4\pi \sigma$ ${\ displaystyle D_ {2n} -D_ {1n} = 4 \ pi \ sigma}$ ${\ displaystyle D_ {2n} -D_ {1n} = 4 \ pi \ sigma}$

For tangential (tangent) components of the electric field :

$E_{2t}-E_{1t}=0$ ${\ displaystyle E_ {2t} -E_ {1t} = 0}$ ${\ displaystyle E_ {2t} -E_ {1t} = 0}$

For normal components of magnetic induction :

$B_{2n}-B_{1n}=0$ ${\ displaystyle B_ {2n} -B_ {1n} = 0}$ ${\ displaystyle B_ {2n} -B_ {1n} = 0}$

For tangential (tangent) components of the magnetic field :

$[\mathbf {n} \mathbf {H} _{2}]-[\mathbf {n} \mathbf {H} _{1}]={\frac {4\pi }{c}}\mathbf {j}$ ${\ displaystyle [\ mathbf {n} \ mathbf {H} _ {2}] - [\ mathbf {n} \ mathbf {H} _ {1}] = {\ frac {4 \ pi} {c}} \ mathbf {j}}$ ${\ displaystyle [\ mathbf {n} \ mathbf {H} _ {2}] - [\ mathbf {n} \ mathbf {H} _ {1}] = {\ frac {4 \ pi} {c}} \ mathbf {j}}$

Where $\mathbf {j}$ ${\ displaystyle \ mathbf {j}}$ ${\ displaystyle \ mathbf {j}}$ Is the linear current density, $\mathbf {n}$ ${\ displaystyle \ mathbf {n}}$ ${\ displaystyle \ mathbf {n}}$ - normal to the surface, and $\sigma$ ${\ displaystyle \ sigma}$ $\ sigma$ - surface charge density

Boundary conditions for the electromagnetic field

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