Unit 1 Relevant Electrostatics and Magnetostatics (Old and New) The whole of classical electrodynamics is encompassed by a set of coupled partial differential equations (at least in one form) bearing the name “Maxwell’s Equations”. On the basis of material already encountered and some new concepts, it is the purpose of this course to develop these equations and to make some elementary applications. To facilitate these intentions, we shall briefly review a little of the prerequisite elec- trostatic and magnetostatic material and, in so doing, revisit some of the necessary mathematics. It should be remembered that much of what now may be written as concise mathematical formalism was, in fact, often developed initially by painstaking experimental investigation. We shall seek to not stray too far from the physical sig- nificance and practicalities of the results “derived”. For the level and length of this course, it will not be possible to explore in detail the scope of the usefulness of the equations considered. In fact, in all but the simplest cases, solutions in closed form do not exist. It will quickly become apparent, especially in future studies based on this course, that numerical methods and approximation techniques are essential in most engineering applications of Maxwell’s equations. Still, there will be plenty of “interesting” problems which may be tackled with the arsenal of tools developed in this and earlier terms. 1
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Unit 1
Relevant Electrostatics andMagnetostatics (Old and New)
The whole of classical electrodynamics is encompassed by a set of coupled partial
differential equations (at least in one form) bearing the name “Maxwell’s Equations”.
On the basis of material already encountered and some new concepts, it is the purpose
of this course to develop these equations and to make some elementary applications.
To facilitate these intentions, we shall briefly review a little of the prerequisite elec-
trostatic and magnetostatic material and, in so doing, revisit some of the necessary
mathematics. It should be remembered that much of what now may be written as
concise mathematical formalism was, in fact, often developed initially by painstaking
experimental investigation. We shall seek to not stray too far from the physical sig-
nificance and practicalities of the results “derived”. For the level and length of this
course, it will not be possible to explore in detail the scope of the usefulness of the
equations considered. In fact, in all but the simplest cases, solutions in closed form
do not exist. It will quickly become apparent, especially in future studies based on
this course, that numerical methods and approximation techniques are essential in
most engineering applications of Maxwell’s equations. Still, there will be plenty of
“interesting” problems which may be tackled with the arsenal of tools developed in
this and earlier terms.
1
A Few Preliminary Observations on Notation
(1) We shall use �� to represent the position vector in space for some observation point
(relative to an origin) and ��′ to indicate the position of sources (relative to an origin).
(2) Based on note (1), �� − ��′ will thus be the displacement vector from the source
to the observation position. This vector will assume various labels throughout the
course (eg., 𝑅, 𝑅12, etc.) as may be convenient.
Illustration:
(3) The text uses bold letters to represent vector quantities. We shall use vector signs
or ⇀ (eg., �� or⇀
𝐸≡ electric field intensity). We’ll use ˆ to indicate unit vectors (eg.,
�� ≡ unit vector in the 𝑥 direction).
Reference should be made to the text for various coordinate systems which may
be encountered. Also, the vector operators, divergence, gradient and curl (∇⋅, ∇ and
∇×) are found at the back of the text, while various vector operator identities are
located in Appendix A.3.
1.1 Electrostatics Review
For the sake of completeness and for future reference, we shall write down the following
previously encountered results.
Coulomb’s Law:
With reference to the accompanying diagram, the force, ��21, on charge 𝑄2 due to
charge 𝑄1 is given by
2
��21 = −��12 =
= (1.1)
where we have assumed that the charges are in free space and 𝜖0 (=8.85×10−12 F/m)
is the permittivity of free space. The definition of unit vector is obvious.
Electric Field Intensity:
The electric field intensity, ��, is by definition the force per unit charge on a small
positive test charge, 𝑞𝑡, brought into the field and is in the direction of the force; i.e.
�� =𝐹
𝑞𝑡(1.2)
The unit is N/C or, more conveniently, volt/metre (V/M). For a volume charge
distribution with charge density 𝜌𝑣(��′),
��(��) = (1.3)
Note that 𝑑𝑣′ is the elemental volume at ��′. Clearly, the integral sums the effects of
all the charges of the form 𝜌𝑣(��′)𝑑𝑣′. In Cartesian coordinates, 𝑑𝑣′ = 𝑑𝑥′𝑑𝑦′𝑑𝑧′. This
integral can be “nasty” and is usually explicitly carried out only for simple cases.
(There are often better ways of finding ��, eg., Gauss’ Law).
Electric Flux Density and Gauss’ Law:
Gauss’ Law states that the electric flux, Ψ, measured in coulombs (C), passing
through any closed surface is equal to the total charge enclosed by that surface.
The electric flux density, ��, is measured in C/m2.
Ψ = 𝑄 =∮𝑆�� ⋅ 𝑑�� =
∫vol
𝜌𝑣𝑑𝑣 (1.4)
3
Note that
𝑑�� = 𝑑𝑆𝑆
where 𝑆 is the outward unit normal to the elemental surface 𝑑𝑆. For example, in
cartesian coordinates, 𝑑𝑆𝑧 = 𝑑𝑥𝑑𝑦.
We also note a so-called constitutive relation between �� and ��. In general, for an
isotropic space with permittivity 𝜖,
�� = 𝜖�� , (1.5)
and for free space, where 𝜖 = 𝜖0 it is
�� = 𝜖0�� .
Gauss’ Integral Theorem (or the Divergence Theorem):
This theorem states that
∮𝑆�� ⋅ 𝑑�� =
∫vol
∇ ⋅ ��𝑑𝑣 (1.6)
∇ ⋅ �� is the divergence of �� throughout the volume. Equation (1.6) is a general
statement for vector fields, not just ��-fields. Note that (1.4) and (1.6) give
∇ ⋅ �� = 𝜌𝑣 (1.7)
4
which happens to be the “point form” of one of Maxwell’s equations alluded to earlier.
Aside: Recall
Equation (1.7) holds for time-varying fields also, and it will be used in this context
later in the course.
Work, 𝑊 , Done on a Charge, 𝑄, Moved in an ��–Field:
𝑊 = −𝑄∫ 𝐴
𝐵�� ⋅ 𝑑�� (1.8)
where 𝑑�� is a differential displacement vector along the path, 𝐶, from 𝐵 to 𝐴 as
illustrated.
Not surprisingly, only the component of �� along the path contributes to energy
expenditure or work done.
Potential Difference, 𝑉𝐴𝐵, and Potential at a Point and Potential Gradient:
From equation (1.8), we define the potential difference between points 𝐴 and 𝐵
as
𝑉𝐴𝐵 =𝑊
𝑄= −
∫ 𝐴
𝐵�� ⋅ 𝑑�� = 𝑉𝐴 − 𝑉𝐵 (1.9)
where 𝑉𝐴 ≡ potential at point 𝐴, 𝑉𝐵 ≡ potential at point 𝐵. The potential at a
point is the potential difference measured with respect to some arbitrarily chosen
zero reference point. For example, if 𝑉∞ = 0, then 𝑉𝐴 is the work done per unit
charge in moving the charge from ∞ to point 𝐴.
5
Conservative Field:
Note that, by definition, a conservative field is one which satifies the condition “
a closed line integral of the field is zero”. For example, the electrostatic field, ��, is
conservative so ∮𝐶�� ⋅ 𝑑�� = 0 (1.10)
Note that the path is closed. Equation (1.10) is not valid for time-varying fields –
more later!!
Also, recall, importantly,
�� = −∇𝑉 (1.11)
which provides a convenient way of finding the (vector) electric field, ��, if the (scalar)
potential field, 𝑉 , is known.
Continuity of Current:
Consider a current density, 𝐽 , measured in amperes per square metre (A/m2). The
principle of charge conservation leads to the point form of the equation of continuity
of current as
∇ ⋅ 𝐽 = −∂𝜌𝑣∂𝑡
(1.12)
Boundary Conditions:
(1) Perfect Conductor
Recall that within a perfect conductor no free charge and no electric field may
exist. Using this fact, it is easily argued from equation (1.10) that at the surface
between a perfect conductor and free space
𝐸𝑡 = 0 =⇒ 𝐷𝑡 = 0
and (1.13)
𝐷𝑁 = 𝜌𝑠 (or 𝐸𝑁 =𝜌𝑠𝜖0)
6
where 𝑡 ≡ tangential, 𝑁 ≡ normal, and 𝜌𝑠 is a surface charge density (charge may
exist on the conductor surface).
(2) Perfect Dielectric
Now, in general, �� may be non-zero on both sides of the boundary. Again, arguing
from equation (1.10), it may be verified that for the case at hand
𝐸𝑡1 = 𝐸𝑡2
and (1.14)
𝐷𝑁1 = 𝐷𝑁2 (or 𝐸𝑁1 =𝜖2𝜖1
𝐸𝑁2)
with 𝜖1 and 𝜖2 being the material permittivities.
Poisson’s and Laplace’s Equations:
Finally, in the electrostatics portion of the previous course, the following important
results were derived from equations (1.7), (1.5), and (1.11):
(1) Poisson’s Equation:
∇ ⋅ ∇𝑉 = −𝜌𝑣𝜖
(1.15)
where 𝜌𝑣 is the usual volume charge density and 𝜖 is the permittivity of the region
under consideration.
(2) Laplace’s Equation:
A special case of (1.15) occurs in a region for which 𝜌𝑣 = 0. The result, known as
Laplace’s equation is clearly given by
∇ ⋅ ∇𝑉 = 0
commonly abbreviated as
∇2𝑉 = 0 (1.16)
At this point it would be a good idea to review a few relevant electrostatic
examples encountered in Engineering 5812.
7
1.2 Magnetostatics Review
We have seen that static electric charges give rise to static electric fields. We have
also defined direct current (dc current) to be constituted of charges which flow with
a constant speed in one direction. These dc currents are a source of steady magnetic
fields. We shall have opportunity in the next course to see how time-varying magnetic
fields may be generated.
In this unit we will introduce the magnetic field intensity, �� , which is measured
in amperes per metre (A/m). There will be analogies observed between the equations
involving the electric field intensity, ��, and those involving ��. There is one significant
difference: while eletric fields may be generated by point charges, there are no known
point magnetic ‘charges’ – although to make calculations easier we sometimes assume
they exist (but NOT in this course)!
The Biot-Savart Law:
The Biot-Savart law is the magnetic field counterpart (sort of) to Coulomb’s law.
This law, which relates the magnetic field intensity, �� , in amperes per metre (A/m)
to a source current, 𝐼, may be written, with reference to the diagram which follows,
in differential form as
𝑑�� =𝐼𝑑��′ × ��
4𝜋𝑅2=
𝐼𝑑��′ × ��
4𝜋𝑅3
Illustration:
Clearly, from the cross product, the �� field element (𝑑��) produced by the current,
𝐼, flowing along the differential displacement, 𝑑��′, of the path, 𝐶, is perpendicular to
the plane containing 𝑑��′ and the unit vector �� from 𝑑��′ (i.e. the source region) to
the point of observation of the �� field. 𝑅 is simply the distance from 𝑑��′ to the place
8
where �� is observed. The right hand rule for the cross product gives the orientation
of the contribution, 𝑑��, due to the current in element 𝑑��′.
A current element may not be isolated to check the law in its differential form, but
considering 𝐶 to be a closed path we may write an experimentally verifiable integral
form as
(1.17)
Equation (1.17) may also be written for surface current densities �� in A/m or
(volume) current density 𝐽 in A/m2 as
(1.18)
Illustration:
or
(1.19)
To emphasize and illustrate the importance of coordinate system representation of
these expressions consider (1.19):
Illustration:
9
In view of the above illustration, (1.19) becomes
(1.20)
Again, note that the primed coordinates represent the source position and the un-
primed coordinates represent the field (observation) position. Comparing equation
(1.20) (i.e. the Biot-Savart law for the magnetic field) with equation (1.3) (i.e.
Coulomb’s law for the electric field), the similarities and differences are obvious: (1)
Both are inverse square laws, but (2) �� is in the direction of �� = (��− �� ′)/∣��− �� ′∣while �� is perpendicular to both the source vector, 𝐽 , and ��.
Ampere’s Circuital Law:
An analogy to Gauss’ law for electrostatics is Ampere’s circuital law for magne-
tostatics. Instead of “talking about” electric flux density and charge enclosed by a
surface, 𝑆, we consider a steady magnetic field intensity and a current, 𝐼𝑒, enclosed
by a contour, 𝐶. Ampere’s law states that the line integral of �� around any closed
path is exactly equal to the direct current, 𝐼𝑒, enclosed by the path. That is
∮𝐶�� ⋅ 𝑑�� = 𝐼𝑒 . (1.21)
Illustrations: (1) Current Filament:
10
(2) “Thick” dc-Current-Carrying Wire:
Note, in general, that
𝐼𝑒 =∫𝑆𝐽 ⋅ 𝑑�� (1.22)
(Similar ideas exist for surface current density, ��).
If it is possible to choose a contour with the properties (1) �� is everywhere tan-
gential to the contour and (2) ∣��∣ is constant along the contour, equation (1.21)
(Ampere’s circuital law) may be used to find �� when 𝐼𝑒 is known. For this purpose,
the nature of �� may be observed from the Biot-Savart law (without actually doing
the integration) and symmetry arguments. For example, in the previous course, we
used symmetry and the Biot-Savart law to show that, for an infinitely long current
filament along the 𝑧-axis, the field had the form
�� = 𝐻(𝜌)𝜙
(that is, the �� field is a FUNCTION of 𝜌 and has only a 𝜙 COMPONENT) and that
a contour that led to a simple integration was a circle centred on the 𝑧-axis.
At this point it would be a good idea to review the magnetostatic examples
encountered in Engineering 5812.
11
Another Ampere’s Law Example:
Consider the situation depicted in the illustration associated with surface currents
in relation to equation (1.18). A similar example is addressed in the text on pages
222–223. Suppose that the surface current density is given by �� = 𝐾𝑥�� with 𝐾𝑥 > 0
in the 𝑧 = 0 plane. Let’s find the magnetic field intensity due to this source.
Illustration
(1) We’ll use the Biot-Savart law to establish the field directions: We may
think of the surface current as being composed of infinitely many current filaments
carrying current in the �� direction. From the Biot-Savart law, we know that each of
these filaments produces a magnetic field intensity which follows circular streamlines
with the filaments being at the centre of the circles. It may be easily argued, that
in this case, any two filaments which are symmetrical about a line parallel to and
half-way between them will produce fields whose 𝑧-components cancel, but whose 𝑦
components remain. In addition to this, there is no field component along the line of
the current (again the cross product in the Biot-Savart law ensures this) – i.e. there is
no field component in the ±�� direction. We have thus established that �� = 𝐾𝑥�� has
only components in the ±𝑦 directions, the negative 𝑦 direction being associated with
fields in the halfspace above the current surface and the positive direction associated
with fields below the surface – see illustration on the right.
(2) What about the coordinate dependencies? The current covers the entire
12
𝑧 = 0 plane. Thus, at any point above or below the plane, the field will not depend
on 𝑥 or 𝑦. (In fact, we will see that it doesn’t depend on 𝑧 either, but we will have
to prove this.)
We are now ready to apply Ampere’s law. Since on 𝐻𝑦 components are present
let’s choose an integration contour (or path) whose ‘segments’ are either parallel to
or perpendicular to 𝑦. This contour is labelled above as rectangle 𝑎𝑏𝑐𝑑𝑎. Direct
application of ∮𝐶�� ⋅ 𝑑�� = 𝐼𝑒
gives
Similarly, if we use path 𝑎′𝑏′𝑐𝑑𝑎′ we get
Thus, at any point 𝑧 > 0 above the sheet, �� = 𝐻𝑦𝑦, and symmetry, we can im-
mediately see that at any point 𝑧 < 0 below the sheet, �� = −𝐻𝑦𝑦. Thus, from
∗,−𝐻𝑦 −𝐻𝑦 = 𝐾𝑥
and
which implies below the sheet �� = −𝐻𝑦𝑦 = .
Note that using �� as the outward pointing unit normal, �� = 12�� × �� for all 𝑧 ∕= 0.
What would happen to the field structure if another sheet with �� = −𝐾𝑥�� was
placed at position 𝑧 = 𝑧0?
13
1.3 More Magnetostatics
1.3.1 More on Ampere’s Law
The Curl
The del operator, ∇, may also be “crossed” into a general vector, ��. The sym-
bolism is ∇ × �� and the operation is termed “the curl of ��”.
Definition: The curl of a vector function �� (i.e.��(𝑥, 𝑦, 𝑧) in cartesian coordinates) is
defined as
∇ × �� =
(��
∂
∂𝑥+ 𝑦
∂
∂𝑦+ 𝑧
∂
∂𝑧
)× (𝐴𝑥��+ 𝐴𝑦𝑦 + 𝐴𝑧𝑧) .
Therefore, the vector which results is given by
∇ × �� = ��
(∂𝐴𝑧
∂𝑦− ∂𝐴𝑦
∂𝑧
)+ 𝑦
(∂𝐴𝑥
∂𝑧− ∂𝐴𝑧
∂𝑥
)+ 𝑧
(∂𝐴𝑦
∂𝑥− ∂𝐴𝑥
∂𝑦
)
=
∣∣∣∣∣∣∣∣∣
�� 𝑦 𝑧∂
∂𝑥
∂
∂𝑦
∂
∂𝑧𝐴𝑥 𝐴𝑦 𝐴𝑧
∣∣∣∣∣∣∣∣∣(1.23)
In expanding this determinant, the derivative nature of ∇ must be considered. The
determinant in equation (1.23) must be expanded from the top down so that we get
the derivatives shown in the second row. It might be noted that expressions such as
��× ∇ are defined only as differential operators and the normal rules which apply to
the cross products of ordinary vectors do NOT apply here. For example, in general,
��× ∇ ∕= −∇ × ��.
Notice that the curl of a vector is a vector! As with the previous vector operators
encountered, (1.23) will not have such a simple form in cylindrical and spherical
coordinates.
Illustration: �� = 𝑥��+ 𝑦𝑦 + 𝑧𝑧 .
The ∇ × ��, sometimes denoted as curl ��, is
∇ × �� =
14
Interpretation
Some of the finer mathematical details are not included in the following discussion.
However, the basic ideas required for interpretation of the curl are discussed. For
the sake of simplicity, consider the �� component of ∇ × ��, whose magnitude from
equation (1.23) is clearly given by
(∇ × ��
)⋅ �� =
∂𝐴𝑧
∂𝑦− ∂𝐴𝑦
∂𝑧.
With reference to the diagram below, which is a small (i.e. differential) rectangle on
the 𝑦-𝑧 plane, the definition of partial differentiation leads us to
∂𝐴𝑧
∂𝑦= lim
Δ𝑦→0
(𝐴𝑧 on b)− (𝐴𝑧 on d)
Δ𝑦
and
∂𝐴𝑦
∂𝑧= lim
Δ𝑧→0
(𝐴𝑦 on c)− (𝐴𝑦 on a)
Δ𝑧
Thus, getting a common denominator gives
(∇ × ��
)⋅�� = lim
Δ𝑦,Δ𝑧→0
(𝐴𝑦 on a)Δ𝑦 + (𝐴𝑧 on b)Δ𝑧 + (−𝐴𝑦 on c)Δ𝑦 − (𝐴𝑧 on d)Δ𝑧
Δ𝑦Δ𝑧
Using the limits we may legitimately write, on recalling from earlier discussions the
idea of the differential surface area, that
(∇ × ��
)⋅ �� = lim
Δ𝑆𝑥→0
∮𝑎𝑏𝑐𝑑
�� ⋅ 𝑑��Δ𝑆𝑥
(1.24)
15
From this last expression, we see that the curl of �� in the �� direction may be
gotten by traversing the differential rectangle in the 𝑦-𝑧 plane as shown, following
the right-hand rule (i.e. fingers curled in direction of travel with thumb, in this case,
pointing in the 𝑥 direction). Again, it must be emphasized that the rectangle is of
differential proportions and it is assumed that 𝐴𝑦 and 𝐴𝑧 do not vary along a given di-
rection of travel – a more rigorous treatment of these ideas using higher order terms
in a Taylor series expansion would lead properly to the same interpretation. The
above discussion may be summarized as follows:
The “circulation” of the vector field per unit area around a differential area in the
𝑦-𝑧 plane is given by the �� component of ∇ × ��. A similar statement could be made
about the other components.
If we allow for all components, then equation (1.24) may be generalized as
(∇ × ��
)⋅ �� = lim
Δ𝑆→0
∮𝐶Δ𝑆
�� ⋅ 𝑑��Δ𝑆
(1.25)
where �� is the unit normal to the incremental surface, Δ𝑆, and 𝐶Δ𝑆 is the perimeter
of that surface.
To get an intuitive feeling for the curl, consider the following: If we have a paddle
wheel which we dip into some “force” field, the paddle wheel will spin differently
depending on its orientation in the field. If the wheel does not rotate at all, it means
that the curl is zero in the direction of the wheel’s axis. Larger angular “velocities”
of the wheel indicate larger values of the the curl. In order to find the direction of the
vector curl, we hunt around for the orientation that provides the greatest “torque”.
The direction of the curl is then along the axis of the wheel, as given by the right-hand
rule.
16
CURL IN CYLINDRICAL COORDINATES
We state without proof that the curl in cylindrical coordinates is given by
∇ × �� =
(1
𝜌
∂𝐴𝑧
∂𝜙− ∂𝐴𝜙
∂𝑧
)𝜌+
(∂𝐴𝜌
∂𝑧− ∂𝐴𝑧
∂𝜌
)𝜙+
(1
𝜌
∂(𝜌𝐴𝜙)
∂𝜌− 1
𝜌
∂𝐴𝜌
∂𝜙
)𝑧
CURL IN SPHERICAL COORDINATES
We state without proof that the curl in spherical coordinates is given by
∇ × �� =1
𝑟 sin 𝜃
(∂(𝐴𝜙 sin 𝜃)
∂𝜃− ∂𝐴𝜃
∂𝜙
)𝑟 +
1
𝑟
(1
sin 𝜃
∂𝐴𝑟
∂𝜙− ∂(𝑟𝐴𝜙)
∂𝑟
)𝜃 +
1
𝑟
(∂(𝑟𝐴𝜃)
∂𝑟− ∂𝐴𝑟
∂𝜃
)𝜙
Stokes’ Theorem
Consider a surface in space which has been subdivided into incremental pieces as
shown.
Rearranging equation (1.25), for a particular surface increment we get
∮𝐶Δ𝑆
�� ⋅ 𝑑�� =(∇ × ��
)⋅(�� lim
Δ𝑆→0Δ𝑆
)(1.26)
If the line integral here is carried out for every such region in the whole surface, 𝑆,
there will be cancellation along all the cell sides except those which constitute the
outside boundary, 𝐶, of 𝑆. Thus, equation (1.26) becomes, on integrating over the
whole region, ∮𝐶�� ⋅ 𝑑�� =
∫𝑆
(∇ × ��
)⋅ 𝑑�� (1.27)
Here, we have used the fact that limΔ𝑆→0
Δ𝑆 = 𝑑𝑆 and ��𝑑𝑆 = 𝑑��. The very significant
result in equation (1.27) is known as Stoke’s Theorem. The following numerical
example will illustrate the geometry involved.
17
Example: A portion of a spherical surface is specified by 𝑟 = 4, 0 ≤ 𝜃 ≤ 0.1𝜋, 0 ≤𝜙 ≤ 0.3𝜋. The path segments forming the perimeter of this surface are given by