Maxwell's Equations and Electrostatic Boundary Conditions

Key Takeaways

• In a homogeneous medium, electromagnetic quantities vary smoothly and continuously whereas, in a non-homogeneous medium, field quantities become discontinuous.

• Electrostatic boundary conditions are the fundamental equations involving field quantities that represent the changes in the electrostatic field at the boundary conditions.  

• The electrostatic boundary conditions used in electromagnetic field problems involving two mediums are:

a) The tangential component of the electric field is continuous across an interface between two mediums.

b) The normal component of electric field intensity is discontinuous across the interface between two mediums.

Maxwell’s equations

Consider electromagnetic wave propagation in wireless communications, where the wave travels through free-space. The free-space, or vacuum, can be considered a homogenous medium. Now, imagine the electromagnetic wave in a conducting cavity or a wave interfacing between two mediums. In the first example, the electromagnetic quantities vary smoothly and continuously. In the second example, the variation of electromagnetic quantities is not continuous. 

In the boundaries where an electromagnetic wave crosses from one medium to another, the field quantities become discontinuous. These discontinuities can be described mathematically as electrostatic boundary conditions and magnetostatic boundary conditions corresponding to the electric fields and magnetic fields, respectively. These boundary conditions are used to constrain solutions for associated electromagnetic quantities obtained by solving Maxwell’s equations limited to the two different mediums.

The Importance of Electrostatic Boundary Conditions

There are two methods to solve any electromagnetic problem:  

1. Find out the charge and current distribution everywhere in the space and then solve Maxwell’s equations everywhere. 

In a homogeneous medium, this can be possible, as the electromagnetic quantities are continuous over the region. In a non-homogeneous medium, electromagnetic quantities become discontinuous at the boundaries between the two mediums, say, for example, at the boundary between two dielectrics or between free-space and a conductor. In such cases, electromagnetic problems can be solved using the second method, which involves boundary conditions.

2. Solve Maxwell’s equations under the boundary conditions defined on the boundaries of the mediums. 

The boundary conditions can be related to the electric field and magnetic field. Maxwell’s equations are decoupled into electrostatic and magnetostatic equations in this method. The electrostatic boundary conditions are the fundamental equations involving field quantities that represent the changes in the electrostatic field at the boundary conditions. The method involving boundary conditions is often used when solving electromagnetic problems involving different mediums. 

The Application of Boundary Conditions in Electromagnetic Wave Problems

Boundary conditions are essential to deriving the solution in an electromagnetic wave problem. When an electromagnetic wave interfaces between two mediums, some of the electromagnetic energy gets transmitted over the boundary and some parts get reflected. The boundary conditions of the electric and magnetic fields can be utilized to determine the direction and intensities of the incident, transmitted, and reflected waves. 

Consider an electromagnetic wave problem involving two different mediums (1 and 2). In the boundaries, the derivatives of the field quantities are infinite due to discontinuity. The problem solution can be obtained by solving Maxwell’s equations in the two mediums separately. This procedure is similar to the first method. Maxwell’s equations are solved in homogenous mediums 1 and 2 separately. The solutions obtained by doing so are connected via the boundary conditions. In electromagnetic wave problems involving two mediums, boundary conditions for tangential electric fields and normal electric fields are applied to constrain the solutions.

Electrostatic boundary conditions and magnetostatic boundary conditions are important to understand when working with electromagnetic fields in conducting cavities and waveguides. 

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