186
6 Electric and Magnetic Fields in Life
∇ × E +
1
c
∂B
∂t
= 0
(6.2)
∇ · B = 0
(6.3)
∇ × B −
1
c
∂E
∂t
= 4π
1
c
J
(6.4)
together with the equations of motion for the particle with charge q . In the classical
case, these are
dp
dt
= q
(E + (v
/c) × B) .
(6.5)
The right-hand side express the Lorentz force law and p is the momentum of the
particle with charge q . 8
One can move a couple of “test charges” through the region of interest with two
different velocities, and determine by their observed motion what the strength of E
and B must have been. Because of how the electric and magnetic fields are defined,
and the linear dependence of the electric and magnetic forces on the size of the test
charge, E and B do not depend on the size of the test charge, but rather the electric
and magnetic fields determine a property we assign to the space around the source
of these fields. Moreover, because Maxwell’s equations have only first powers of the
electric and magnetic field, solutions for the fields outside the regions where there
are charges and currents will satisfy the principle of linear superposition: The sum
of two solutions will again be a solution. This property leads to great simplifications
in the solutions to the equations. For example, you can always express the
general solution as a sum of solutions found for individual point charges and
currents.
It can be shown with the Special Theory of Relativity that if an interaction is
transmitted by a field at the fastest allowed speed, c, then the force produced by
interaction must diminish as the square of the distance between the interacting
particles. Maxwell’s theory is consistent with the Special Theory, so that the solution
of the first of Maxwell’s equations for a point charge also gives an inverse-square
law for how the electric field drops with the distance from that charge. Quantum
field theory can be used to show that the quantum of the electromagnetic field, the
photon, carries a unit quantum of spin, and that in consequence 9 electromagnetic
interactions are both attractive and repulsive.
8 In quantum theory, one uses a relativistic wave equation for the ‘primed’ particle, with the
electromagnetic potential field entering with the particle’s energy and momentum operators. From
the wave function, average positions, velocities, energies, etc. can be determined.
9 A. Zee, Quantum Field Theory in a Nutshell, 2nd ed [Princeton University Press] (2007).
6 Electric and Magnetic Fields in Life
∇ × E +
1
c
∂B
∂t
= 0
(6.2)
∇ · B = 0
(6.3)
∇ × B −
1
c
∂E
∂t
= 4π
1
c
J
(6.4)
together with the equations of motion for the particle with charge q . In the classical
case, these are
dp
dt
= q
(E + (v
/c) × B) .
(6.5)
The right-hand side express the Lorentz force law and p is the momentum of the
particle with charge q . 8
One can move a couple of “test charges” through the region of interest with two
different velocities, and determine by their observed motion what the strength of E
and B must have been. Because of how the electric and magnetic fields are defined,
and the linear dependence of the electric and magnetic forces on the size of the test
charge, E and B do not depend on the size of the test charge, but rather the electric
and magnetic fields determine a property we assign to the space around the source
of these fields. Moreover, because Maxwell’s equations have only first powers of the
electric and magnetic field, solutions for the fields outside the regions where there
are charges and currents will satisfy the principle of linear superposition: The sum
of two solutions will again be a solution. This property leads to great simplifications
in the solutions to the equations. For example, you can always express the
general solution as a sum of solutions found for individual point charges and
currents.
It can be shown with the Special Theory of Relativity that if an interaction is
transmitted by a field at the fastest allowed speed, c, then the force produced by
interaction must diminish as the square of the distance between the interacting
particles. Maxwell’s theory is consistent with the Special Theory, so that the solution
of the first of Maxwell’s equations for a point charge also gives an inverse-square
law for how the electric field drops with the distance from that charge. Quantum
field theory can be used to show that the quantum of the electromagnetic field, the
photon, carries a unit quantum of spin, and that in consequence 9 electromagnetic
interactions are both attractive and repulsive.
8 In quantum theory, one uses a relativistic wave equation for the ‘primed’ particle, with the
electromagnetic potential field entering with the particle’s energy and momentum operators. From
the wave function, average positions, velocities, energies, etc. can be determined.
9 A. Zee, Quantum Field Theory in a Nutshell, 2nd ed [Princeton University Press] (2007).