1549380323-Statistical Mechanics Theory and Molecular Simulation

Time-dependent systems 531

spatial and time dependence of the vector fields. The elegance andbeauty of Maxwell’s
theory lies in the simple connection it establishes between the electricand magnetic
fields. For free fields, Maxwell’s equations take the form (in cgs units)

∇·E= 0, ∇×E=−

1

c

∂B

∂t

∇·B= 0, ∇×B=

1

c

∂E

∂t

, (14.1.1)

wherecis the speed of light in vacuum. While the magnetic field is always divergence-
free (there are no magnetic monopoles), the electric field has zerodivergence only in the
absence of sources. The rates of change of the magnetic and electric fields determine,
respectively, how the electric and magnetic fields “curl” to form closed loops.
Notice that the free-field Maxwell equations constitute a set of eight homogeneous
partial differential equations for the six field components,Ex(r,t),Ey(r,t),Ez(r,t),
Bx(r,t),By(r,t),Bz(r,t). Hence, the set is overdetermined, and there is some re-
dundancy among the field equations. This redundancy arises because of a property
exhibited by this and other types of field theories known asgauge invarianceorgauge
freedom.
In order to manifest the gauge freedom and remove the redundancy, it is convenient
to work with related fieldsA(r,t) andφ(r,t) known as thevectorandscalarpotentials,
respectively. These are related to the electric and magnetic fields by the transformation

B=∇×A, E=−∇φ−

1

c

∂A

∂t

. (14.1.2)

The relationB=∇×Afollows from the fact that any divergence-free field can always
be expressed as the curl of another field since∇·(∇×A) = 0. The relation forEin
eqn. (14.1.2) arises from the fact that ifB= 0, then∇×E= 0, so thatEcan be
expressed as a gradientE=−∇φ. The second term in the equation forEis included
when a nonzero magnetic field is present. Although these relations uniquely define the
electric and magnetic fields, they possess some ambiguity in how the vector and scalar
potentialsA(r,t) andφ(r,t) are defined. Specifically, if new potentialsA′(r,t) and
φ′(r,t) are constructed from the original potentials via

A′(r,t) =A(r,t) +∇χ(r,t)

φ′(r,t) =φ(r,t)−

1

c

∂

∂t

χ(r,t), (14.1.3)

whereχ(r,t) is an arbitrary scalar field, then the same electric and magnetic fields
will result whenA′andφ′are substituted into eqns. (14.1.2). The transformations in
eqn. (14.1.3) are known asgauge transformations, and the freedom to chooseχ(r,t)
arbitrarily is the manifestation of the gauge invariance of the electric and magnetic
fields mentioned above. Moreover, if the definitions in eqns. (14.1.2)are substituted