Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

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Physics : Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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Boundary conditions for the normal components of the fields

When an electromagnetic field faces an abrupt change in the permittivity and permeability, certain conditions on electric and magnetic fields on the interface are to be respected for the continuity. These conditions of continuity are known as the boundary conditions for the electromagnetic field. Consider the pillbox in the following figure where two different media are characterised by their permittivities and permeabilities, viz Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevThe interface is shown with a curved surface. The height of the pillbox is h and the two flat surfaces of the pillbox in two different media are shown byBoundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev We start with the Maxwell’s equation, Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevand integrate it over the

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Fig. 11.1: Pillbox on the interface of two media
 

pillbox.

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

where S is the total surface area of the pillbox. Now the left hand side of the above equation is

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev+ flux through the curved surface of the pillbox, (11.3)
where the superscripts identify the fields in different media We now reduce the height of the pillbox eventually making it to zero. In this case the area of the curved surface reduces to zero and hence the flux through it is also zero.
Since there is a finite charge inside the box,  Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev So in this limitBoundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevfinite and

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

where Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is surface charge density on the interface.
In this situation,  Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev The right hand side of the equation for small Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevUsing the above relations in (11.2), we have the condition,

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(11.4)

The above condition says that there is an abrupt jump in the normal component of the displacement vector while crossing the medium if there is a non zero surface charge density on the interface. Similarly we proceed with the Maxwell equation Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevand obtain the following boundary condition,

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(11.5)

which says the normal component of the magnetic filed is always continuous.

 

Boundary conditions for the tangential components of the fields

Let us consider the following small closed curve, P QRS , across the interface of two media. The area, PQRS, has the unit vector Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev  which is normal to the surface P QRS .
Now we start with the Maxwell’s equation, (3.2),

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Fig. 11.2: Closed curve across the interface of two media 

The scalar product of the above equation with Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevis integrated over the surface PQRS .

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The left hand side of equation (11.9) is

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev contributions from QR and S P ,

whereBoundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev are the unit vectors along P Q and RS respectively. Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevand Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRevare the lengths of P Q and RS respectively.

Now in the limit QR → 0 and S P → 0 the right hand side of the equation (11.9) vanishes(as the area of P QRS vanishes) as well as in the left hand side the contributions from QR and S P also vanish and then we have (with Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Since the Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev arbitrary we get the following condition for the tangential component of the electric field (for Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev, normal to the surface)

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(11.10)

Similarly proceeding with the Maxwell’s equation (3.4), we get the condition on the tangential component of the H field as

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev(11.11)

Boundary Conditions on the Fields at Interfaces - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is the surface current density.

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