Some essential mathematics

Solved Problems in Classical Electromagnetism ◽

10.1093/oso/9780198821915.003.0001 ◽

2018 ◽

Author(s):

J. Pierrus

Keyword(s):

Electric Potential ◽

Vector Fields ◽

Mathematical Form ◽

Simple Derivation ◽

Electricity And Magnetism

This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential Ф and the vector fields E and B are familiar quantities in electricity and magnetism, it is not always known that they are examples of a mathematical form termed a tensor. More sophisticated tensors are required in some descriptions and theories in electrodynamics. As tensor notation is compact and usually allows for a simple derivation of a result, the chapter begins with a series of questions involving Cartesian tensors. The results obtained here will facilitate the solution of many questions in subsequent chapters. Readers who are not familiar with tensors are advised to consult Appendix A before proceeding. Towards the end of this Appendix a ‘checklist for detecting errors when using tensor notation’ is given, which will hopefully provide some help for the uninitiated.

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A Fractional Calculus Perspective in Electromagnetics

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2005-84862 ◽

2005 ◽

Cited By ~ 3

Author(s):

J. A. Tenreiro Machado ◽

Isabel S. Jesus ◽

Alexandra Galhano

Keyword(s):

Twentieth Century ◽

Fractional Calculus ◽

Approximation Method ◽

Electric Potential ◽

Constant Velocity ◽

Order Approximation ◽

Electrical Potential ◽

Late 19Th Century ◽

Electric And Magnetic Fields ◽

Electricity And Magnetism

Some experimentation with magnets was beginning in the late 19th century. By then reliable batteries had been developed and the electric current was recognized as a stream of charge particles. Maxwell developed a set of equations expressing the basic laws of electricity and magnetism, and demonstrated that these two phenomena are complementary aspects of electromagnetism. He showed that electric and magnetic fields travel through space, in the form of waves, at a constant velocity. Maxwell is generally regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics, making contributions to the fundamental models of nature. Bearing these ideas in mind, in this study we apply the concept of fractional calculus and some aspects of electromagnetism, to the static electric potential, and we develop a new fractional order approximation method to the electrical potential.

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