Andrea Macchi Giovanni Moruzzi - Problems in Classical Electromagnetism

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Springer International Publishing AG 2017

Andrea Macchi , Giovanni Moruzzi and Francesco Pegoraro Problems in Classical Electromagnetism

Chapter 1 Basics of Electrostatics

Andrea Macchi 1 , Giovanni Moruzzi 1 and Francesco Pegoraro 1

(1)

Department of Physics Enrico Fermi, University of Pisa, Pisa, Italy

Andrea Macchi

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Topics. The electric charge. The electric field. The superposition principle. Gausss law. Symmetry considerations. The electric field of simple charge distributions (plane layer, straight wire, sphere). Point charges and Coulombs law. The equations of electrostatics. Potential energy and electric potential. The equations of Poisson and Laplace. Electrostatic energy. Multipole expansions. The field of an electric dipole.

Units. An aim of this book is to provide formulas compatible with both SI (French: Systme International dUnits ) units and Gaussian units in Chapters . This is achieved by introducing some system-of-units-dependent constants.

The first constant we need is Coulombs constant , , which for instance appears in the expression for the force between two electric point charges and in vacuum, with position vectors and , respectively. The Coulomb force acting, for instance, on is

(1.1)

where is Coulombs constant, dependent on the units used for force, electric charge, and length. The vector is the distance from to , pointing towards , and the corresponding unit vector. Coulombs constant is

(1.2)

Constant F/m is the so-called dielectric permittivity of free space, and is defined by the formula

(1.3)

where H/m (by definition) is the vacuum magnetic permeability, and c is the speed of light in vacuum, m/s (this is a precise value, since the length of the meter is defined from this constant and the international standard for time).

Basic equations The two basic equations of this Chapter are, in differential and integral form,

(1.4)

(1.5)

where is the electric field, and is the volume charge density, at a point of location vector at time t . The infinitesimal volume element is . In () is evaluated over an arbitrary closed path C . Since , it is possible to define an electric potential such that

(1.6)

The general expression of the potential generated by a given charge distribution is

(1.7)

The force acting on a volume charge distribution is

(1.8)

As a consequence, the force acting on a point charge q located at (which corresponds to a charge distribution , with the Dirac-delta function) is

(1.9)

The electrostatic energy associated with a given distribution of electric charges and fields is given by the following expressions

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