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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: ISTE Invoiced
- Sprache: Englisch
Introduction to Classical Electrodynamics 1 introduces the fundamentals of electromagnetic field theory.
This book begins with electrostatics, focusing on the concepts of point charges, electric fields, electric potential and Gauss’s theorem. It then examines conductors, influence phenomena and capacitors, before exploring electrokinetics, detailing the concepts of electric current, circuits and resistance, as well as Kirchhoff’s laws.
Next, this book analyzes the notion of magnetic field, Lorentz and Laplace forces, Biot-Savart’s law, Ampère’s theorem, electric and magnetic moments, and multipole developments. Boundary problems for static potentials in vacuum are also studied.
The book is aimed at students of physics and mathematics, as well as engineering students interested in electromagnetic theory, providing an overview of electromagnetic theory, with numerous practical applications.
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Seitenzahl: 459
Veröffentlichungsjahr: 2025
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Dedicated to the memory of my parentsDedicated to the memory of Daniel ArnaudonDedicated to Prof. Amitabha Chakrabarti
Series EditorFrédérique de Fornel
Introduction to Classical Electrodynamics 1
Boucif Abdesselam
First published 2025 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2025The rights of Boucif Abdesselam to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2025933055
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-83669-000-9
Preface
There is no place in this new kind of physics both for the field and matter, for the field is the only reality.
Albert Einstein
An electric charge is an abstract concept, comparable to the concept of mass, introduced to explain certain behaviors in physics. However, unlike mass, an electric charge can take two experimental forms that are arbitrarily called positive and negative. Two charges of the same nature, both positive or both negative, for instance, repel each other, while two opposite charges attract each other. This phenomenon is called electrostatic interaction.
The history of electric charges goes back to the ancient Greeks who observed that rubbing certain objects against fur produced an imbalance in the total electric charge. These “electrified” objects would also attract other light objects, like hair. What is more, if the objects are rubbed against each other for long enough, it is even possible to create a spark.
Electric charges can be experimentally measured. Particles observed in nature have charges that are integer multiples of an elementary charge, which is a fundamental physical constant. An elementary charge is the electric charge on a proton or, equivalently, the opposite of the electric charge on an electron. It is expressed in Coulombs (C), or in A × s in the international system of units. The value of an electric charge, considered to be invisible, and its discrete nature were first discovered in 1909 by Robert Andrews Millikan.
The elementary charge, denoted by e, has an approximate value e = 1.602176634 × 10−19C. The exact value of the elementary charge is currently defined by: , where h is Planck’s constant, with h = 6.626070040 × 10−34 Js; α is the fine structure constant, with α = 7.2973525664 × 10−3; μ0 is the vacuum magnetic permeability, with μ0 = 4π × 10−7Hm−1; and c is the speed of light in a vacuum, with c = 299, 792, 458 m/s. The existence of quarks was postulated in the beginning of 1960 and they are believed to possess a fractional electric charge or but they are confined within hadrons, whose charge and may be zero, and may be equal to the elementary charge, or may be an integer multiple of the elementary charge. Quarks have not yet been separately detected, but they are assumed to have existed in a free state in the very first instants of the Universe (the quark epoch). Charge is an invariant in the theory of special relativity. A particle of charge q, whatever its velocity, will always retain the same charge q.
There are two branches of study:
Electrostatics
is the branch of physics that studies the phenomena created by static electric charges for the observer. In physics, an electrically charged particle creates a vector field that is called an
electric field
. In the presence of a charged particle, the local properties of a space are therefore modified. This is therefore an accurate translation of the concept of a field. If there is another charge in this same field, it will be subject to the action of the electric force (one of the four forces of nature) exerted at a distance by the particle: the electric field is in a way the mediator of this distant action. If there is a movement of electric charges, generally electrons, within a conductive material under the effect of a potential difference at the ends of the material, an electric current is produced. The concept of electric current was first introduced by Benjamin Franklin. He was the first to imagine electricity as a type of invisible fluid present in all matter. He postulated that friction of an insulating surface causes the fluid to move and that a flow of this fluid constitutes an electric current. A material is made up of a large number of electric charges, but these charges balance out among themselves, that is, there is the same number of electrons (negative) as protons (positive). At normal temperatures, the material is electrically neutral. When a static electricity effect is produced, this means that there has been a displacement of charges from material A to material B. This is the
electrification phenomenon
. Excess or missing charges, that is,
uncompensated charges
, are responsible for electrical effects on a body (the example of the rubbed stick). There are two kinds of materials:
A material is said to be a
perfect conductor
if, when electrified, the uncompensated charges move freely through the material.
If the uncompensated charges do not move freely and remain fixed in the place where they were deposited, we say that it is a
perfect insulator (or dielectric)
.
In reality, a real material is obviously lies between these two limit states:
Magnetism
represents all physical phenomena in which objects exert attractive or repulsive forces on other materials. The electric currents and magnetic moments of the fundamental elementary particles are at the origin of the magnetic field generated by these forces. The magnetic field is therefore a vector quantity (characterized by an intensity and a direction) defined for each point in space. It is determined by the position and orientation of magnets, electromagnets and the displacement of electric charges. The presence of this field is seen in the existence of a force acting upon the moving electric charges (called the
Lorentz force
), and various effects that affect certain materials (paramagnetism, diamagnetism, or ferromagnetism, depending on the case). Magnetic susceptibility is the quantity that determines how much a magnetic field interacts with a material.
Electricity and magnetism are intimately related. In 1820, Ørsted discovered the relationship between electricity and magnetism: a wire carrying an electric current is capable of deflecting the magnetized needle of a compass. Subsequently, Ampère, based on Ørsted’s work, discovered and formulated a number of laws governing the relationship between magnetism and electrodynamics. In 1831, Faraday discovered that if an electric current produces a magnetic field, the reverse is also true. An electric current can be produced by setting a magnetic field in motion, according to Lenz’s law.
This marked the birth of electromagnetism. It is therefore the field of physics that studies the interactions between electrically charged particles, whether at rest or in motion, using the concept of the electromagnetic field. Electromagnetism can be defined as the branch that studies the electromagnetic field and its interactions with charged particles. The electric field and the magnetic field are the two components of the electromagnetic field described by electromagnetism. The equations that describe the evolution of the electromagnetic field are called Maxwell equations. Electric and magnetic fields waves can propagate freely in space and in most materials. These are called electromagnetic waves and correspond to all manifestations of light across wavelengths (radio waves, microwaves, infrared, visible domain, ultraviolet rays, X-rays and gamma rays). Maxwell wrote: The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws. From this point of view, optics can be seen as an application of electromagnetism. The electromagnetic interaction is also one of the four fundamental interactions. Together with quantum mechanics, it enables us to understand the existence, cohesion and stability of atoms and molecules.
From the point of view of fundamental physics, the theoretical development of classical electromagnetism led to the theory of special relativity at the beginning of the 20th century. The need to reconcile electromagnetic theory and quantum mechanics led to the construction of quantum electrodynamics, in which electromagnetic interaction is interpreted as an exchange of particles called photons. In particle physics, electromagnetic interaction and weak interaction are unified within the electroweak theory.
Note to readers
Electromagnetic field theory is often poorly or insufficiently taught in university physics programs. The heavy dependence on mathematics, especially vector calculus, integral calculus, complex functions and special functions, etc., add to the difficulty in teaching this subject and may obscure certain phenomena such that students get mired in mathematical difficulties and lose sight of practical applications and vice versa. This modest contribution aims to lay out the essential ideas of the theory in the most logical order. Special importance is given to the mathematical developments of this theory. The experimental side of the theory is only lightly approached. We will not insist on units of measurement. These three volumes will contain many examples (the most classic). I have tried to give as much detail as possible in the calculations. Across the three volumes, the discussion is essentially divided into three main topics: (1) charges and/or dipoles, as the source of the electrostatic field; (2) currents and/or magnetization as sources of the magnetic field; (3) electrodynamics, where the electric field and magnetic field are of equal importance. This book discusses many applications of electromagnetic theory. For instance, there are the solutions of Laplace and Poisson equations in various coordinate systems, dielectric medias, magnetized medias, waveguides, transmission lines, electromagnetic radiation, radiation by point-like charges, Kirchhoff diffraction, special relativity, etc. For the reader’s convenience, we have listed all of the mathematical tools used in the book into six appendices, which will be available to download online: vector analysis, tensors, special/hypergeometric functions, etc. References are presented at the end of each volume. Formulas are given here in the international system of units. We have tried to adopt the notations most commonly used in the literature. Graphs and 3D representations have been produced using Maple.
This three-volume set is meant for physics students, engineering students, as well as mathematics students who are interested in problems in electromagnetic theory. It can also be used by anyone interested in physics in general, or who simply wants to learn more out of intellectual curiosity. There are many books on the same subject and therefore it seems reasonable to repeat some classical developments that we find excellent, rather than striving for originality at any cost.
Volume 1 will cover the following topics:
electrostatic field in vacuum (Gauss’ theorem, etc.);
conductor at equilibrium;
electrokinetics;
magnetic field in vacuum (Laplace force, Biot–Savart law, the Ampère theorem, etc.);
electric and magnetic moments;
boundary problems for static potentials in vacuum.
Volume 2 will study the following points:
static potential with a source – Poisson equation (image method, Green function, etc.);
electromagnetic induction phenomena;
electromagnetic waves in vacuum;
electromagnetic energies;
polarization, polarized media (dielectric);
magnetization, magnetized media;
electromagnetic fields in matter.
Volume 3 will analyze the following concepts:
waveguides;
transmission lines;
electromagnetic radiation;
radiation from point-like charges;
Kirchhoff diffraction;
theory of special relativity;
covariant formulation;
The online appendices will be available at the following URL: www.iste.co.uk/abdesselam/electrodynamics1.zip
The appendices will cover:
some vector analysis results (Appendix 1);
tensors (Appendix 2);
the Dirac function (Appendix 3);
orthogonal functions (Appendix 4);
beta and gamma functions (Appendix 5);
special functions (Appendix 6).
Acknowledgments
This work would never have seen the light of day without the encouragement of many people. I am particularly grateful to my students, colleagues and many others. I thank Fréderique de Fornel for her patience and advice. Finally, I dedicate this book to my wife, Nadia, and my children, Abir, Ahmed Nour, Mohammed Sami and Samah. I would like to express my gratitude to them.
Boucif AbdesselamMarch 2025
Chapter 1Electrostatic Field in Vacuum
1.1. Electric charges
We now know that matter is made up of negatively charged electrons and positively charged protons, that is, the Universe contains two types of matter that we can call negative and positive. Particles of the same species repel each other and those of different species attract each other by a force called electrostatic or Coulomb force, similar to the gravitational force, which varies inversely with the square of the distance. However, the electrostatic force is much more intense than the gravitational force. The existence of these two types of charges was revealed in Franklin’s experiments. For example, when a glass rod is rubbed with a dry cloth, some electrons from the glass shift onto the cloth. The cloth is then negatively charged, as it has more electrons than protons. The glass rod becomes positively charged because it has lost electrons (excess protons). The glass rod is then said to undergo an electrification effect. If this operation is repeated with a wax rod, the same electrification phenomenon is observed, except that this time, the electrons shift from the cloth to the wax. The wax rod is thus negatively charged. In addition, if two sufficiently electrified glass rods are brought together, they repel each other, that is, there is repulsion. The same repulsion is observed with two wax rods. On the other hand, if a glass rod is brought near a wax rod, they attract each other. It can thus be concluded that charges with the same sign repel each other and those with opposite signs attract each other. If the positively charged glass rod is brought close to a metallic ball attached to a wire (free to move), the electrons in the ball are attracted to the area on its surface that is close to the glass rod, leaving the other side of the ball positively charged: this is the phenomenon of electrostatic induction. An attractive force then appears between the ball and the glass rod. When it comes into contact with the ball, the positive charges on the rod are neutralized by part of the negative charges on the ball. This allows the ball-rod assembly to conserve a net positive charge. The ball, which is now positively charged, is then repelled by another positively charged rod. This transfer of charges is called the conduction effect. The metallic ball can also be called a conductor, as the charges on the ball are easily induced and conducted.
An electron bears an electric charge −e, whereas a proton has an electric charge +e. For a given quantity of ordinary matter, we can proceed with an algebraic sum of the elementary electric charges within the material. We therefore conclude that every body contains an integer quantity of an electric charge, denoted by e. This elementary charge represents the smallest possible value of an electric charge. This is the charge quantification principle. This quantization of electric charges means that any charged body is presented as a discontinuous distribution of its constituent charges. On a large scale, this distribution of constituent charges can be considered to be continuous (the elementary charges are very small). For an insulated system (no material can cross the system’s boundaries), the total electric charge, that is, the algebraic sum of positive and negative charges present in the body at any given moment, remains constant. Charges are neither created nor destroyed.
1.2. Electrostatic force
Between two electric point-like charges q and q′, located, respectively, at points M (x, y, z) and M′ (x′, y′, z′), a force is exerted which is inversely proportional to the square of the distance separating them and is carried by the line joining the two points (central force):
where (respectively, ) is the vector that joins the origin O and the point M (x, y, z) (respectively, M′ (x′, y′, z′)), as depicted in Figure 1.1. The forces applied to q and q′ are of the same magnitude, but opposite in direction. The quantity ε0 is called the electric permittivity of vacuum or vacuum permittivity and is related to the speed of light in vacuum c and the magnetic permeability of vacuum (or vacuum permeability) μ0 through the relation ε0 = 1/μ0c2. (respectively, ) is the force applied on the charge q (respectively, q′). In particular, if qq′ > 0, that is, of the same sign (respectively, qq′ < 0, i.e. of the opposite sign), the force is attractive (respectively, repulsive). This is Coulomb’s law.
According to the superposition principle, if there are several point-like charges located, respectively, at , the total force applied on the charge q is the resultant of all individual forces:
Figure 1.1.Electrostatic force between two point-like charges q and q′
EXAMPLE 1.1. (Multi-charge System).
Two identical electric charges, denoted here by q, located at A and B on the Oz axis, are separated by a distance 2d. The origin O is chosen in the middle of the segment AB. Let us calculate the total force exerted on a test charge q′ located at a point with coordinates (x0, 0, 0) on the Ox axis, as depicted in Figure 1.2. Using the superposition principle, we obtain:
Figure 1.2.Superposition principle for a system with multiple charges
It is seen that the resultant force at (x0, 0, 0) only has a non-zero component along the Ox axis. This property can be explained by the fact that the resulting distribution is rotational symmetrical around Ox. In simpler terms, Ox is a symmetry axis. Note that if the charges at points A and B were of opposite polarities, then axis O would be an antisymmetry axis for the system and the total force exerted on charge q′ would be normal to Ox. We will return to these properties in subsequent sections. At the origin, O (x0 = 0), the force is zero (the forces applied by the charges are mutually annihilated); this is an unstable equilibrium position for the system: when moving away from x0 = 0, q′ tends to move further away. At a point that is very far from the origin, that is, when x0 ≫ d, the force takes the approximate value: ; this corresponds to a force applied by a charge 2q concentrated at O. The force [1.3] reaches its maximum at point , that is, . Let us now add another charge q at the point (x, 0, 0), such that x > x0. The total force becomes:
This force cancels out for . The position x0 is here a stable equilibrium position for the system: when moving away from x0, q′ tends to return to x0.
Consider a differential element of free charges dq′ (located at ) within a macroscopic distribution (of charges). The total force applied by the distribution on a test charge q (located at ) has the value:
The total charge in the distribution is obtained by summing over all differential elements, that is, by integrating over the whole system . In general, there are three types of distribution, which are discussed below.
1.2.1. Distribution along a line (linear) or a curve (curvilinear)
When the electric charge is distributed along a line or a curve , equation [1.5] is reduced to:
where is the linear, line charge or curvilinear (charge per unit length). Here, dl′ represents an infinitesimal element of the line or curve. The total charge located along the line or curve is obtained from the integral .
EXAMPLE 1.2. (Total Charge).
Let us determine the total charge in each of the following linear distributions: (a) a linear charge λ uniformly distributed along an infinite line; (b) a linear charge distributed along an infinite line; (c) a curvilinear charge (constant) λ distributed over a circle with radius R. For (a), we find ; for (b), ; for (c), .
EXAMPLE 1.3. (Uniformly Charged Infinitely Long Wire).
Consider an infinitely long wire that is uniformly charged (constant
λ
). For reasons of symmetry, we will calculate the force applied to a test charge
q
located in the
Oxy
plane, as shown in
Figure 1.3
. The charge element
λdz
′ exerts the elementary force . After integration, this gives:
The third integral is zero, so , because the function under the sign of the integral is odd. Consequently, we obtain:
If, instead of
λ
, we take the linear charge (b) from
Example 1.2
, then:
To calculate this integral, let us first carry out the change variable z′ = r tan θ′:
Figure 1.3.Uniformly charged infinitely long wire dimensionless: by symmetry, the total force exerted on the test charge q is radial. Differential charges dq1and dq2located symmetrically at z′ and −z′ produce forces whose projections add up in the Oxy plane and cancel each other out in the Oz direction
Now taking t′ = sin θ′, we obtain:
EXAMPLE 1.4. (Uniformly Charged Wire of Finite Length L).
For a uniformly charged wire of finite length L, as in Figure 1.4, the total force experienced by a charge q located at the point has the value:
Figure 1.4.For a uniformly loaded wire of finite length L, the force exerted by a test load q is not radial, except at the middle of the wire
This is reduced in the middle of the wire, for z = 0, at . When L → ∞, equation [1.8] is found.
1.2.2. Distribution over a surface (surface charge density)
When the charge is distributed over a surface S′, the force experienced on the test charge q is written as:
where is the surface charge density (charge per unit area). dS′ is an infinitesimal element of the surface. We have dS′ = dx′ dy′ in Cartesian coordinates, dS′ = r′dr′dφ′ in polar coordinates, etc. The total charge on the surface S′ is .
EXAMPLE 1.5. (Infinite Plane Around a Disc).
1) Consider an infinite plane of negligible thickness, which is charged as follows:
As shown in Figure 1.5, the force applied by dq′ = σ (r′, φ′) dS′ = σr′dr′dφ′ on a charge q located at (0, 0, z) has the value . By summing, we obtain:
The integrals on sin φ and cos φ give 0. Therefore, it comes that:
2) Instead of [1.14], we choose:
We find:
Figure 1.5.Infinite plane distribution around a disc of radius R: By symmetry, the force experienced along the Oz axis is perpendicular to the plane
1.2.3. Distribution over a volume (volume charge density)
When the charge is distributed within a volume τ′, we get:
where is the volume charge density (charge per unit volume) in τ′. dτ′ is an infinitesimal element of the volume. In Cartesian coordinates, dτ′ is equal to dx′dy′dz′; in cylindrical coordinates, dτ′ is r′dr′dφ dz′; in spherical coordinates, it is r′2 sin θ′dr′dθ′dϕ′; etc. The total charge measured inside τ′ is .
EXAMPLE 1.6. (Electron Cloud).
The electronic cloud of charge +e around the hydrogen atom’s nucleus is modeled by the spherically symmetric distribution:
where r0 is called the Bohr radius. The cloud’s total charge is:
EXAMPLE 1.7. (Uniformly Charged Infinite Sheet of Thickness 2a).
(1) Consider an infinite sheet of thickness 2a, which is uniformly charged, as shown in Figure 1.6. The total force experienced by a charge q, located at the point (0, 0, z), is written as:
In cylindrical coordinates, it yields (we integrate with respect to r′ and then with respect to z′):
Figure 1.6.Infinite sheet of thickness 2a: the total force felt on the charge q is perpendicular to the sheet. Outside the sheet, the force does not depend on the distance between the charge and the sheet, while it depends on z inside
We can see that is continuous on the boundaries z = ±a.
(2) If the charge is now distributed over a maximum radius around Oz with the density (μ is a constant), the total force experienced by the charge q is:
For , equation [1.24] reduces . The total charge held in the sheet is equal to ; this gives and therefore .
The Coulomb force [1.1] can be recovered from equation [1.19] using the Dirac distribution:
1.3. Electric field
The electric field is defined as being the force exerted on a test charge equal to unity. Thus, the electric field created by a charge q′ is equal to:
In the same way, the electric fields created by (a) several point-like charges , (b) a linear charge λ distributed over a curve , (c) a surface charge density σ distributed over an area S′ and (d) a volume charge density ρ distributed within a volume τ′ are, respectively:
If the charge distribution is known, the electric field is obtained by integrating one of the formulas [1.27]. A few simple rules must be observed:
A distinction must be made between the coordinates of the field points and the source points (differential charges). The integration must always be carried out over the source coordinates.
Equations [1.27]
are vector equations: they generally have three components, that is, three integrals to be carried out. Certain symmetries can be used to show that certain field components are zero.
The distance is always positive. When square roots must be taken, it must be ensured that the positive root is taken.
The solution to a particular problem can often be obtained by integrating the contributions of simpler differential systems (see
Example 1.9
).
EXAMPLE 1.8. (Uniformly Charged Infinitely Long Wire (see Figure 1.3)).
The electric field measured at the point (x, y, z) is obtained by dividing the force [1.8] by q:
The field has no component along the Oz axis, since the two charge elements and , symmetrically located at a distance z on either side of the O point, produce fields of equal magnitude, oppositely directed in this direction and therefore cancel each other out, leaving only the radial components, which add to each other. If the wire has a length L (see Figure 1.4), formula [1.28] must be replaced, according to [1.12], by:
EXAMPLE 1.9. (Uniformly Charged Infinite Plane).
Consider a uniformly charged infinite plane located at z = 0, as shown in Figure 1.7. The plane is divided into an infinite number of linear charges along the Oy axis of thickness dx with dλ = σdx. It is also possible to divide the plane into linear charges along the Ox axis of thickness dy; this does not change the result. Each linear charge produces an infinitesimal electric field whose projection along the Oz axis has the value:
We do not take into account the field component along the Ox axis, because the linear charge at −z induces a field in this direction of the same amplitude but in the opposite direction, which thus cancels out the first field.
The total field is then obtained by integration over all linear charges:
Figure 1.7.Uniformly charged infinite plane composed of an infinite number of parallel charged lines (λdx) along Oy: the field created by lines symmetrically placed at x and −x has components that cancel out in the Ox direction and components that add up in the Oz direction
Note that the field does not decrease with distance from the plane.
EXAMPLE 1.10. (An Infinite Plane Around a Disc (see Figure 1.5)).
Along the Oz axis of the disc, a test charge q is subject to the force . It therefore follows that:
When R → 0 (uniformly loaded infinite plane), we return to [1.31].
EXAMPLE 1.11. (Uniformly Charged Infinite Sheet of Thickness 2a (see Figure 1.6)).
EXAMPLE 1.12. (Uniformly Charged Disc with Negligible Thickness).
Let us calculate the electric field, created by a flat disk of negligible thickness, radius R and charge Q (uniformly distributed over the disk), at point M located along axis Oz of the disk at distance z from its center, as shown in Figure 1.8. The charge density is given by for x′2 + y′2 ≤ R2 and 0 elsewhere. It therefore follows that:
Figure 1.8.By symmetry, the electric field created by a uniformly charged disk with radius R and no thickness is normal to the disk along points on the Oz axis
By shifting to cylindrical coordinates, we obtain:
The integral on the variable φ′ eliminates the terms in cos φ′ and sin φ′. Therefore, we have:
When z ≪ R, the electric field is reduced to (i.e. to [1.31]).
1.3.1. Distribution with spherical symmetry
1.3.1.1. Sphere charged in volume
For simplicity’s sake, let us start by considering a sphere of radius R uniformly charged in volume. Let us use a spherical coordinate system whose point of origin is the center of the sphere. We propose to calculate the electric field at a point M (inside or outside the sphere). By symmetry, it is clear that does not depend on the coordinates (θ, φ), as shown in Figure 1.9. We will start by writing the contribution to of the elementary charge ρdr′ contained in a volume element dτ′ (, where Q is the total charge of the sphere) and then integrate over the sphere. The volume element is r′2dr′ sin θ′dθ′dφ′ (for simplicity, the point M is chosen along the Oz axis). The charge contained in this elementary volume creates the differential field at point M. For reasons of symmetry, it is obvious that the electric field is radial. Consequently, we will consider only the radial component:
In order to compute this integral, let us introduce the variable s2 = r2 + r′2 − 2rr′ cos θ′:
Figure 1.9.Electric field created by a uniformly charged sphere in volume
There are two possible cases:
Electric field outside the sphere
: when the point
M
lies outside the sphere, we always have
r
>
r
′:
The electric field of a sphere of radius R and total charge Q coincides outside the sphere with the field of a point-like charge Q placed at the center of this sphere.
Electric field within the sphere
: when the point
M
lies within the sphere, the integration interval must be divided into two intervals, namely, and :
EXAMPLE 1.13.
Let us now choose the spherically symmetric charge density described as follows: if r ≤ R and 0 if r > R. Repeating the same calculation steps, we find the following:
Electric field iside the sphere
:
Electric field inside the sphere
:
1.3.1.2. Sphere charged on its surface
Here, we will consider a charge Q distributed over the surface of the sphere, as shown in Figure 1.10. The radial component of the differential electric field due to the differential charge element σ R2 sin θ′dθ′dφ′ is . Thus, the electric field created by the entire sphere is given as:
Figure 1.10.A sphere of radius R with charge Q uniformly distributed on its surface. Symmetrically located differential charge elements indicate that the electric field is radial. The electric field inside the sphere is zero, while outside the sphere, the field is the same as if all the charge Q were concentrated in a point-like charge at the origin
By introducing the variable change s2 = r2 + R2 − 2r cos θ in the same way, it is possible to rewrite the electric field integral as follows:
1.3.2. Thomson’s model
According to Thomson’s model, a hydrogen atom is seen as a solid sphere with radius r0 and a uniformly distributed positive charge inside. The force acting on the electron placed at (inside the sphere) is:
where . The electrical force acting on the electron is therefore analogous to the restoring force of a spring (of zero length in a vacuum): a central, attractive force proportional to the distance to the center (analogous to the elongation of the spring). The electron is thus elastically linked to the center of the atom. What is more, the moment of this force relative to O is zero. Using the angular momentum theorem, we can deduce that the electron’s angular momentum with respect to O is constant. Since , we can conclude that the vectors and are both perpendicular to a constant vector (i.e. they lie in the same plane): the electron’s motion therefore takes place in a plane. Applying Newton’s second law, that is, , we get:
where . If ω0 (or ν0) corresponds to a frequency of one of the lines in the Lyman spectrum of the hydrogen atom λ0, we write that ; which implies that . Differential equation [1.46] has the solution:
which corresponds to the ellipse with center O and axis .
In addition to the restoring force, if the electron is also subjected to a frictional force, where h > 0 (an opposing force proportional to speed), the electron’s equation of motion becomes:
Let us put and . We obtain the characteristic equation and its discriminant reduced to . There are three possible cases, depending on the sign of Δ:
Δ > 0
or λ
>
ω
0
(high friction)
: the two solutions of the characteristic equation are real and negative: and . The general solution of the differential equation is:
Δ = 0
or λ
=
ω
0
(critical regime)
: we have a double solution for the characteristic equation,
α
= −
λ
. The general solution is:
Δ < 0
or λ
<
ω
0
(low friction)
: we have two distinct complex solutions and . The solution is therefore:
