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Exercicios do livro do Bo Thidé
Tipologia: Exercícios
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Tobia Carozzi Anders Eriksson Bengt Lundborg Bo Thidé Mattias Waldenvik
ELECTROMAGNETIC FIELD THEORY
EXERCISES
PLEASE NOTE THAT THIS IS A
VERY PRELIMINARY DRAFT!
Companion volume to
ELECTROMAGNETIC FIELD THEORY
This book was typeset in LATEX 2 ε on an HP9000/700 series workstation and printed on an HP LaserJet 5000GN printer.
Copyright c
(^) 1998 by Bo Thidé Uppsala, Sweden All rights reserved.
Electromagnetic Field Theory Exercises
ISBN X-XXX-XXXXX-X
PREFACE
LESSON 1
Maxwell’s Equations
1.1 Coverage
1.2 Formulae used
1.3 Solved examples
The most fundamental form of Maxwell’s equations is
∂ t B (1.2c)
c^2
∂ t
E (1.2d)
sometimes known as the microscopic Maxwell equations or the Maxwell-Lorentz equa- tions. In the presence of a medium, these equations are still true, but it may sometimes be convenient to separate the sources of the fields (the charge and current densities) into an induced part, due to the response of the medium to the electromagnetic fields, and an extraneous, due to “free” charges and currents not caused by the material properties. One then writes
The electric and magnetic properties of the material are often described by the electric polarisation P (SI unit: C/m^2 ) and the magnetisation M (SI unit: A/m). In terms of these, the induced sources are described by
To fully describe a certain situation, one also needs constitutive relations telling how P and M depends on E and B. These are generally empirical relations, different for different media.
Show that by introducing the fields
the two Maxwell equations containing source terms (1.2a) and ( ?? ) reduce to
∂ t
known as the macroscopic Maxwell equations.
Solution
If we insert
and
Draft version released 15th November 2000 at 20:
Express Maxwell’s equations in component form.
Solution
Maxwell’s equations in vector form are written:
∂ t
c^2
∂ t
In these equations, E , B , and j are vectors, while ρ is a scalar. Even though all the equations contain vectors, only the latter pair are true vector equations in the sense that the equations themselves have several components. When going to component notation, all scalar quantities are of course left as they are.
where the last step assumes Einstein’s summation convention: if an index appears twice in the same term, it is to be summed over. Such an index is called a summation index. Indices which only appear once are known as free indices, and are not to be summed over. What
b. On the other hand, the
vector. The three E (^) j are the components of the vector E in the coordinate system set by the three unit vectors x ˆ (^) j. The E (^) j are real numbers, while the x ˆ (^) j are vectors, i.e. geometrical objects. Remember that though they are real numbers, the E (^) j are not scalars. Vector equations are transformed into component form by forming the scalar product of both sides with the same unit vector. Let us go into ridiculous detail in a very simple case:
x ˆ k (1.32)
x ˆ k (1.33)
This is of course unnecessarily tedious algebra for an obvious result, but by using this careful procedure, we are certain to get the correct answer: the free index in the resulting equation necessarily comes out the same on both sides. Even if one does not follow this complicated way always, one should to some extent at least think in those terms. Nabla operations are translated into component form as follows:
Draft version released 15th November 2000 at 20:
∂ xi
∂ φ ∂ xi
∂ xi
∂ Vi ∂ xi
∂ x (^) j
∂ Vk ∂ x (^) j
where V is a vector field and φ is a scalar field.
Remember that in vector valued equations such as Ampère’s and Faraday’s laws, one must be careful to make sure that the free index on the left hand side of the equation is the same as the free index on the right hand side of the equation. As said above, an equation of the
With these things in mind we can now write Maxwell’s equations as
ρ ε 0
∂ Ei ∂ xi
ρ ε 0
∂ Bi ∂ xi
∂ t
∂ Ek ∂ x (^) j
∂ t
Bi (1.41)
c^2
∂ t
∂ Bk ∂ x (^) j
c^2
∂ Ei ∂ t
Derive the continuity equation for charge density ρ from Maxwell’s equations using (a) vector notation and (b) component notation. Compare the usefulness of the two systems of notations. Also, discuss the physical meaning of the charge continuity equation.
Solution
looks like this:
Compute
∂ t E^ in two ways:
∂
ε 0
∂ t
ρ (1.43)
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above:
It is sometimes difficult to see what one is calculating in the component system. The vector system with div, curl etc. may be closer to the physics, or at least to our picture of it.
memory or to consult a math handbook, while with the component system you need only the definitions of ε i jk and δ i j.
unambiguous) when dealing with tensors of higher rank, for which vector notation becomes cumbersome.
any coordinate system, while in the component notation, the components depend on the unit vectors chosen.
∂ t
is known as a continuity equation. Why? Well, integrate the continuity equation over some volume V bounded by the surface S. By using Gauss’s theorem, we find that
d Q d t
V
∂ t
V
S
j
d S (1.52)
which says that the change in the total charge in the volume is due to the net inflow of electric current through the boundary surface S. Hence, the continuity equation is the field theory formulation of the physical law of charge conservation.
Draft version released 15th November 2000 at 20:
