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28 Chapter 1

Let us start off with the equivalent electric field and calculate the work/potential

provided by magnetic field accordingly to (1.15) and (1.26)
= ∮ ∘ = ∮ ( x ) ∘ = − ∮ ( x ) ∘ [V] (1.37)

supposing that E = 0 in
Lorentz’s force equation
(1.15) while B-vector is
time-independent. The
integration curve L in (1.37)
is a closed one, as shown in
Figure 1.6.8 with the surface
area A bounded by this
curve. Eventually, we can Figure 1.6.9 Scalar
Figure 1.6.8 Closed build the infinite number of triple product as
boundary curve L such kind of surfaces volume
bounded by the same curve L
with the only restraint. The orientation of L should be positive meaning that if you keep your
head up walking along the curve, the surface must be on your left.
Choosing the vectors and as the base vectors (see Figure 1.6.9) and recognizing that the
vector product ∘ x = ∘ is the volume of parallelepiped we can transform (1.37) to

( x ) ∘ = ∘ ( x ) = ∘ � x � = ∘ ( x ) = ∘ (1.38)

Therefore, while our man in Figure 1.6.8 travels down the curve L the element moves with
speed along the ribbon on the surface area . As soon as the man made his journey along the
curve, the parallelepiped covered the total surface area . Consequently, in (1.37)

∮ ( x ) ∘ = ∬ ∘ (1.39)

Now recall the two faces of EM field. The moving element can be considered as the train
carrying the current sensor #3 while the static magnetic field is detected by the observer on
the platform. Invoking the principle of relativity, we can put the sensor at rest and start changing
the magnetic field at the same rate. Therefore, in (1.39)

∬ ∘ = ∬ ∘ (1.40)

It means that the expression (1.37) can be rewritten as

∮ ∘ + ∬ ∘ = 0 (1.41)

Pay attention that the subscription m in electric field vector was omitted as a pointless. From

(1.41) and the energy conservation law immediately follows the existence of a unique physics
phenomenon – time-varying magnetic fields induce time-varying electric fields. No charge or
current sources are involved in this process, which forecasts electromagnetic waves
st
propagation. The equity (1.41) is the 1 Maxwell’s equation in Table 1.7 for the EM fields in