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2.2. The Maxwell equations: current conservation
2.2 The Maxwell equations: current conservation
Question: Would you distinguish local conservation laws from global conservation laws.
Feynman: If a cat were to disappear in Pasadena and at the same time
appear in Erice, that would be an example of global conservation of cats.
This is not the way cats are conserved. Cats or charge or baryons are
conserved in a much more continuous way. If any of these quantities begin to disappear in a region, then they begin to appear in a neighbouring
region. Consequently, we can identify the flow of charge out of a region
with the disappearance of charge inside the region. This identification of
the divergence of a flux with the time rate of change of a charge density is
called a local conservation law. A local conservation law implies that the
total charge is conserved globally, but the reverse does not hold. However,
relativistically it is clear that non-local global conservation laws cannot
exist, since to a moving observer the cat will appear in Erice before it
disappears in Pasadena.
—From the question-and-answer session following a lecture by R. P. Feynman at the 1964 International School of Physics ‘Ettore Majorana’ (Feynman 1965b).
We begin by considering the basic laws of classical electromagnetism, the
Maxwell equations. We use a system of units (Heaviside–Lorentz) which is
convenient in particle physics (see appendix C). Before Maxwell’s work these
laws were
∇ · E = ρ em
(Gauss’ law)
(2.1)
∂B
∇ × E = −
(Faraday–Lenz laws)
(2.2)
∂t
∇ · B = 0
(no magnetic charges)
(2.3)
and, for steady currents,
∇ × B = j
(Amp` ere’s law).
(2.4)
em
Here ρ em is the charge density and j is the current density; these densities
em
act as ‘sources’ for the E and B fields. Maxwell noticed that taking the
divergence of this last equation leads to conflict with the continuity equation
for electric charge
∂ρ em + ∇ · j = 0.
(2.5)
em
∂t
Since
∇ · (∇ × B) = 0
(2.6)
from (2.4) there follows the result
∇ · j = 0.
(2.7)
em
2.2. The Maxwell equations: current conservation
2.2 The Maxwell equations: current conservation
Question: Would you distinguish local conservation laws from global conservation laws.
Feynman: If a cat were to disappear in Pasadena and at the same time
appear in Erice, that would be an example of global conservation of cats.
This is not the way cats are conserved. Cats or charge or baryons are
conserved in a much more continuous way. If any of these quantities begin to disappear in a region, then they begin to appear in a neighbouring
region. Consequently, we can identify the flow of charge out of a region
with the disappearance of charge inside the region. This identification of
the divergence of a flux with the time rate of change of a charge density is
called a local conservation law. A local conservation law implies that the
total charge is conserved globally, but the reverse does not hold. However,
relativistically it is clear that non-local global conservation laws cannot
exist, since to a moving observer the cat will appear in Erice before it
disappears in Pasadena.
—From the question-and-answer session following a lecture by R. P. Feynman at the 1964 International School of Physics ‘Ettore Majorana’ (Feynman 1965b).
We begin by considering the basic laws of classical electromagnetism, the
Maxwell equations. We use a system of units (Heaviside–Lorentz) which is
convenient in particle physics (see appendix C). Before Maxwell’s work these
laws were
∇ · E = ρ em
(Gauss’ law)
(2.1)
∂B
∇ × E = −
(Faraday–Lenz laws)
(2.2)
∂t
∇ · B = 0
(no magnetic charges)
(2.3)
and, for steady currents,
∇ × B = j
(Amp` ere’s law).
(2.4)
em
Here ρ em is the charge density and j is the current density; these densities
em
act as ‘sources’ for the E and B fields. Maxwell noticed that taking the
divergence of this last equation leads to conflict with the continuity equation
for electric charge
∂ρ em + ∇ · j = 0.
(2.5)
em
∂t
Since
∇ · (∇ × B) = 0
(2.6)
from (2.4) there follows the result
∇ · j = 0.
(2.7)
em