2.5 Maxwell’s Equations and Breaking of Symmetry
27
∇ × E = −
∂B
∂t
,
(2.62)
∇ × H = i +
∂D
∂t,
(2.63)
∇ · D = ρ,
(2.64)
∇ · B = 0.
(2.65)
These are solved with the material relationships:
D = E,
(2.66)
B = μH,
(2.67)
i = σ c E.
(2.68)
In the above, σ c is the electric conductivity and (2.68) is the empirical Ohm’s law.
Comparing (2.62) with (2.63) and (2.64) with (2.65), it is clear that symmetry between
electric and magnetic phenomena does not exist. What is the reason for the breaking
of symmetry?
If we treat the condition in which there is no electric charge and current (ρ = 0 and
i = 0), however, we find that the symmetry appears among them (although note that
the sign factor of the derivative with respect to time is different). This corresponds
to electromagnetic waves. Here we look carefully at how the symmetry breaks by
introducing sources that produce the fields. Since the electric charge is a scalar, the
distortion of the vector field (electric field) is nothing else than the divergence. This
means that the effect of the electric charge can appear only in (2.64). On the other
hand, the current is a vector and the distortion of the vector field (magnetic flux
density) is nothing else than the rotation. Hence, the effect of the current can appear
only in (2.63). This discussion shows that the breaking of symmetry comes from the
mathematical difference between the source being a scalar or vector. This difference
also causes the different natures of the fields, as mentioned above.
Coffee break (2)
Does the Lorentz force do mechanical work?
It is stated in textbooks on electromagnetism that the Lorentz force does not do
mechanical work, since it only changes the direction of motion of an electric charge
in a magnetic field but does not influence the kinetic energy. On the other hand,
when a current is applied to a conductor in a magnetic field, it can happen that the
conductor is driven by the Lorentz force. In this case the mechanical work is done.
How can we understand this phenomenon?
27
∇ × E = −
∂B
∂t
,
(2.62)
∇ × H = i +
∂D
∂t,
(2.63)
∇ · D = ρ,
(2.64)
∇ · B = 0.
(2.65)
These are solved with the material relationships:
D = E,
(2.66)
B = μH,
(2.67)
i = σ c E.
(2.68)
In the above, σ c is the electric conductivity and (2.68) is the empirical Ohm’s law.
Comparing (2.62) with (2.63) and (2.64) with (2.65), it is clear that symmetry between
electric and magnetic phenomena does not exist. What is the reason for the breaking
of symmetry?
If we treat the condition in which there is no electric charge and current (ρ = 0 and
i = 0), however, we find that the symmetry appears among them (although note that
the sign factor of the derivative with respect to time is different). This corresponds
to electromagnetic waves. Here we look carefully at how the symmetry breaks by
introducing sources that produce the fields. Since the electric charge is a scalar, the
distortion of the vector field (electric field) is nothing else than the divergence. This
means that the effect of the electric charge can appear only in (2.64). On the other
hand, the current is a vector and the distortion of the vector field (magnetic flux
density) is nothing else than the rotation. Hence, the effect of the current can appear
only in (2.63). This discussion shows that the breaking of symmetry comes from the
mathematical difference between the source being a scalar or vector. This difference
also causes the different natures of the fields, as mentioned above.
Coffee break (2)
Does the Lorentz force do mechanical work?
It is stated in textbooks on electromagnetism that the Lorentz force does not do
mechanical work, since it only changes the direction of motion of an electric charge
in a magnetic field but does not influence the kinetic energy. On the other hand,
when a current is applied to a conductor in a magnetic field, it can happen that the
conductor is driven by the Lorentz force. In this case the mechanical work is done.
How can we understand this phenomenon?