For a given electromagnetic field, that spreads across two regions with varying constitutive parameters, how electric and magnetic fields will change across the interfaces of the two regions. A) Describe relations, mathematical or otherwise, between the electric fields across the interface and magnetic field across the interface. Assume that there are no charges (source-free) on the interface itself. B) How these relations change if one of the media is metallic or a perfect electrical conductor (PEC)?

A) In electromagnetism, when an electromagnetic wave encounters an interface between two different media, the electric and magnetic fields at the interface are governed by the boundary conditions derived from Maxwell's equations. Assuming the interface is source-free (no free charges or currents):

1. **Continuity of the Tangential Components of the Electric Field (`E`):** The tangential component of the electric field is continuous across the interface. This means that if you draw a line on the interface, the component of the electric field that is parallel to this line is the same on both sides. Mathematically, it is expressed as:

`E1_tangential = E2_tangential`

where `E1_tangential` and `E2_tangential` are the tangential electric field components in regions 1 and 2, respectively.

2. **Continuity of the Tangential Components of the Magnetic Field (`H`):** Similarly to the electric field, the tangential component of the magnetic field is also continuous across the interface between two non-magnetic media (assuming the media are non-magnetic and there is no surface current density). It is given by:

`H1_tangential = H2_tangential`

3. **Discontinuity of the Normal Components of the Electric Flux Density (`D`):** The normal component of the electric flux density can be discontinuous across an interface, depending on the permittivities of the two media. The electric flux density is given by `D = εE`, where `ε` is the permittivity of the medium. The relationship across the interface is:

`ε1 * E1_normal = ε2 * E2_normal`

where `E1_normal` and `E2_normal` are the normal components of the electric field on each side of the interface, and `ε1` and `ε2` are the permittivities of media 1 and 2, respectively.

4. **Discontinuity of the Normal Components of the Magnetic Induction (`B`):** The normal component of the magnetic induction (magnetic flux density) is continuous across an interface. This is because magnetic monopoles do not exist, and therefore, the magnetic lines are always continuous. The relationship is:

`B1_normal = B2_normal`

where `B1_normal` and `B2_normal` are the normal components of the magnetic flux density on each side of the interface.

B) When one of the media is a Perfect Electric Conductor (PEC), the scenario changes drastically because a PEC completely reflects the incident electromagnetic wave. The internal fields within the PEC are zero because the perfect conductor does not allow the electromagnetic fields to penetrate. The boundary conditions in this case are:

1. **Tangential Electric Field:** At the surface of a PEC, the tangential component of the electric field is zero because the free charges in the conductor rearrange themselves to cancel any external electric field.

`E_tangential = 0` on the surface of the PEC.

2. **Normal Electric Flux Density:** Since the internal electric field is zero, by the earlier relation `D = εE`, the normal component of the electric flux density is also zero:

`D_normal = 0` inside the PEC.

3. **Tangential Magnetic Field:** The tangential components of the magnetic field are such that they induce a surface current on the PEC that exactly cancels the internal field:

`H_outside_tangential = Surface Current Density / μ` on the PEC surface.

4. **Normal Magnetic Induction:** The normal component of the magnetic induction is zero just inside the surface because magnetic field lines cannot penetrate a perfect conductor:

`B_normal = 0` inside the PEC.

Extra: When discussing the interactions of electromagnetic fields with matter, it's important to understand some key concepts in electromagnetism that inform the boundary conditions. Maxwell's equations are the fundamental principles that describe how electric and magnetic fields behave. From Maxwell's equations, one can derive the conditions that electrical and magnetic fields must satisfy at the boundaries between different media.

In the real world, most conductors are not perfect, but the concept of a PEC helps in understanding limiting behavior. For practical conductors, the electromagnetic field does penetrate, but its intensity decreases exponentially with depth. This is known as "skin effect." The degree of penetration is characterized by the skin depth, which depends on the frequency of the electromagnetic wave and the electrical properties of the conductor.

For non-magnetic media, where the relative permeability is close to one, the change in magnetic fields across the interface is determined primarily by the magnetic field's tangential component, as explained earlier.

Knowing how electric and magnetic fields behave at interfaces is crucial for designing electrical and electronic devices, such as waveguides, antennas, and high-frequency circuits, where control over wave propagation and reflection is necessary for proper functionality.