Principles of Engineering Physics 1

Introduction

The phenomenon of electromagnetism in any medium is completely described by a set of four first order partial differential equations called Maxwell's equations. Maxwell's equations are the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in vacuum or in matter. The equations represent one of the most elegant and concise way to describe the fundamentals of electromagnetism. Maxwell's equations are a combination of the works of Gauss, Faraday, Ampère, Biot, Savart, and others. Remarkably, Maxwell's equations are perfectly consistent with the transformation equations of the special theory of relativity. To be more exact, these equations constitute a complete description of the behavior of electric and magnetic fields separately or jointly in any medium.

Vector Calculus

In vector calculus, the spatial derivatives of one types of vector and scalar fields give other types of vector or scalar fields. Depending upon the requirement, the first order differential operator (see Eq. 5.2) may be applied to a scalar function to obtain a vector function or vice versa. It may also give rise to one type of vector field from another type of vector field! The concept of vector calculus was fully exploited by James Clerk Maxwell in a simple way in discovering the missing link between the electric field and the magnetic field, thus establishing the electromagnetic nature of light. Therefore, for complete appreciation of complexities of electromagnetism, at least a brief explanation of vector calculus would be highly beneficial.

The field, in vector calculus, is defined as a region within which every physical quantity can be expressed as a continuous function of the position of a point in the region. The corresponding function is called a point function. Broadly speaking, fields are of two types – scalar fields and vector fields. All the quantities in a scalar field are scalars and all the quantities in a vector field are vectors. All quantities in both scalar and vector fields are functions of positions and times. The vectors in vector fields and the scalars in scalar fields may change with respect to positions and times. First, we shall discuss what are line integrals, surface integrals and volume integrals.

Line integrals

The integration of a vector along a curve in a vector field is called a line integral.