Page 57 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 37

Meanwhile, only half of this energy is due to the force acting on a test charge. The other half
comes from the force exerted by the test charge on the charges inside V (recall 3 Newton’s
rd
law). Therefore, we can find the energy stored in electric fields by halving (1.72) and integrating

= ∫ ∘ (1.73)
1

2
Using the moving charge as a test element, we can show that
= ∫ ∘ (1.74)
1

2
These two expressions are the basis for the most concrete definition of the electric and magnetic
field strength based on measured energy. For example, recall that in vacuum = and =

. Then

0 2
= ∫ ||

2 � (1.75)
0

= ∫ ||

2
Assume that the measured energy is 1 J in small enough volume ∆ where E-field destribution
is uniform. Then according to (1.75)
5
⁄
|| = �2 ∆ = �1/∆ ∙ �2 ∙ 1/8.854 ∙ 10 −12 = 4.75 ∙ 10 ⁄ √∆ [V/m]
0

3
Meanwhile, the same 1 J of energy corresponds || = 1.26 ∙ 10 ⁄ √ [A/m]. Why it is so big
difference? Blame the SI units. In general, the ratio ||/|| = � ⁄ ≅ 376.98 [Ω] is called
0
0
the impedance of free space. We will run into this constant many times later.
0
1.7 ELECTRIC CURRENT AND CHARGES AS SOURCES OF EM
FIELDS
1.7.1 Currents and Charges as Sources of EM Fields
The electrical currents had been considered before mainly arose from the electric charges
motion forced by the existing electrical fields in matter. These electrical fields push and
accelerate the charges sharing with them some part of their energy. It is natural to ask whether
we can reverse this process and transfer the energy of moving charges and equivalent electric
currents to the energy of EM fields.
The classic example of such device is the hand-cranked generator demonstrated schematically
in Figure 1.7.1a. This generator converts the mechanical energy into electrical form. Moreover,
practically any known type of energy can be transferred into electrical: solar cells converting
the sun radiant energy, electrical discharging batteries start converting the chemical energy into
electrical form at the moment they are connected to circuits, magnetohydrodynamic (MHD)
11
generator shown in Figure 1.7.1b converts thermal and kinetic energy of the moving through
magnetic field the jet of plasma into electrical energy, etc. All of them have one common feature:

11 Public Domain Image, source: https://commons.wikimedia.org/wiki/File:MHD_generator_(En).png