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Maxwell’s equations can also be written in the integral form as

(1.37)

Consider the boundary between two materials shown in Figure 1.14. Using these equations, we can obtain a number of useful results. For example, if we apply the first equation of Maxwell’s equations in integral form to the boundary between Medium 1 and Medium 2, it is not difficult to obtain [2]:

(1.38)

where is the surface unit vector from Medium 2 to Medium 1 as shown in Figure 1.14. This condition means that the tangential components of an electric field () are continuous across the boundary between any two media.

Figure 1.14 Boundary between Medium 1 and Medium 2

Similarly, we can apply other three Maxwell’s equations to this boundary to obtain:

(1.39)

where J_s is the surface current density and ρ_s is the surface charge density. These results can be interpreted as

 The change in tangential component of the magnetic field across a boundary is equal to the surface current density on the boundary;

 The change in the normal component of the electric flux density across a boundary is equal to the surface charge density on the boundary;

 The normal component of the magnetic flux density is continuous across the boundary between two media, while the normal component of the magnetic field is not continuous unless μ1 = μ2.

Applying these boundary conditions on a perfect conductor (which means no electric and magnetic field inside and the conductivity σ = ∞) in the air, we have

(1.40)

We can also use these results to illustrate, for example, the field distribution around a two‐wire transmission line as shown in Figure 1.15, where the electric fields are plotted as the solid lines and the magnetic fields are shown in broken lines. As expected, the electric field is from positive charges to the negative charges, while the magnetic field forms loops around the current.

Figure 1.15 Electromagnetic field distribution around a two‐wire transmission line