Boundary conditions (field theory)

Boundary conditions are continuity conditions that apply in classical electrodynamics between two different media. They represent the boundary values in the Maxwell equations in the transition area between different materials.

General boundary conditions

The fields in the two media are marked with the indices 1 and 2.

• ${\displaystyle {\vec {n}}\times \left({\vec {E}}_{2}-{\vec {E}}_{1}\right)=0}$

• ${\displaystyle {\vec {n}}\cdot \left({\vec {D}}_{2}-{\vec {D}}_{1}\right)=\sigma _{F}}$

• ${\displaystyle {\vec {n}}\cdot \left({\vec {B}}_{2}-{\vec {B}}_{1}\right)=0}$

• ${\displaystyle {\vec {n}}\times \left({\vec {H}}_{2}-{\vec {H}}_{1}\right)={\vec {J}}_{F}}$

there is a normal vector on the interface, the surface charge density at the interface and the surface current density , which indicates the current per unit area at the interface. ${\displaystyle {\vec {n}}}$ ${\displaystyle \sigma _{F}}$ ${\displaystyle {\vec {J}}_{F}}$

These boundary conditions say: The tangential component of the E field and the normal component of the B field are continuous.

Limit conditions for uncharged isolators

The above relationships are simplified for uncharged insulators , since there are no free charges and therefore no free currents either . ${\displaystyle \sigma _{F}=0}$ ${\displaystyle {\vec {J}}_{F}=0}$

• ${\displaystyle {\vec {n}}\times \left({\vec {E}}_{2}-{\vec {E}}_{1}\right)=0}$

• ${\displaystyle {\vec {n}}\cdot \left({\vec {D}}_{2}-{\vec {D}}_{1}\right)=0}$

• ${\displaystyle {\vec {n}}\cdot \left({\vec {B}}_{2}-{\vec {B}}_{1}\right)=0}$

• ${\displaystyle {\vec {n}}\times \left({\vec {H}}_{2}-{\vec {H}}_{1}\right)=0}$

The continuity conditions in words: The tangential component of the E-field and the normal component of the B-field are continuous. In addition, the tangential component of the H field and the normal component of the D field are continuous.

See also

• Material equations of electrodynamics
• Fresnel formulas

Web links

• Applet for demonstrating and deriving the continuity at interfaces (University of Konstanz)

Footnotes and individual references

1. ↑ ^{a b} The tangential component means that component that is tangential to the interface, analogously the normal component denotes the component in the direction of the normal vector of the interface.