ABSTRACT
G− obeys Newton's second law (2.9), which reads
vEe dt vdme
GGG α−−= (12.1) where represents a damping force, acting against the velocity, due to collisions with the ions. If there are N electrons in a volume V, that is a free charge density
vGα− ,V/ neNe −=−=ρ all moving with the same velocity ,vG the magnitude of the electric
current is defined as the time rate of charge through the normal cross section of area S, that is
dqI neSv dt
Svρ= = − = (12.2) and Eq.(12.1) becomes
dI ne SE dII d dt m m mI ne SE
α α α= − = −− t
Integrating gives
ln e
ne SEI t m α
α ⎛ ⎞
C− = − +⎜ ⎟⎝ ⎠
By applying the condition of initial zero current 0=I at 0=t we obtain
( )2 /1 et mne SEI e αα −= − (12.3) which shows an exponential rise, approaching the steady current
V l SneISEneI α=α=
22 or (12.4)
where the electric field has been expressed in terms of the potential difference V applied across the ends of a conductor of finite length l, according to Eq.(11.8). If we write
2 l lR S Sne
α σ= =
the equation (12.4) becomes RIV = (12.5)
This equation is known as Ohm's law. The constant R is called the resistance of the conductor, and the electrical conductivity σ is defined as
2neσ α= (12.6)
The charge flow is always tangential to the electric field lines so that it is convenient to define a current density field ,j
G of magnitude given by the current flow
per unit normal area, as
ordIj v j dS
vρ ρ= = =G G (12.7)
The current I then represents the flux of current density across an arbitrary section of the conductor
I j dS= ⋅∫ GG (12.8) For an elementary volume of the conductor, illustrated in Figure 12.1, substituting Eqs.(12.6) and (12.7) into Eq.(12.4) gives ordI EdS j dS E dSσ= ⋅ =σ ⋅G GGG (12.9) which is true however small the cross-sectional area ,Sd
G irrespective of its orientation. It
is therefore clear that is proportional to j G
E G
with σ being the constant of proportionality, that is
Ej GG σ= (12.10)
dS
j
Figure 12.1. A tube of current flow in a conductor The differential form (12.10) of Ohm's law, which is formally similar to Eq.(11.28) defining the proportionality of ED
GG and in a dielectric, is often regarded as a
constitutive relation for a conductor. The law of energy conservation in a conductor, under a stationary field ,E
G is
exactly analogous to the form of the circulation law (11.6) for electrostatics. Its alternative form (11.8), upon multiplication of both sides by dq, shows that the decrease
of potential energy along the current flow line, in a time interval can be expressed by
(qVd− ) ,dt
( ) 2V VJ dq E drdP d E v j E d E ddt ρ ⋅= = ⋅ = ⋅ = G
Vσ G G GGG (12.11)
which gives the power per unit volume as 2/ VJP Eσ= (12.12) This is Joule's law, which can also be written in an integral form by considering a finite volume of the conductor. It follows that Sl=V
( )( ) ( ) 22J VP E El S jS V IV RI Rσ= = = = = (12.13) where V stands for the potential difference applied across the ends of the current flow line. Let us now consider the flux of the current density ,j
G that is the current crossing a
closed surface S, enclosing a volume V (Figure 12.2). The total current flowing out of S must be equal to the time rate of decrease of charge inside S, that is
V S
j dS d d t t
ρρ∂ ∂⋅ = − = −∂ ∂∫ ∫ ∫ V GG
(12.14)
The left hand side transforms according to Gauss's divergence theorem and so
V Vj d d t ρ∂∇ ⋅ = − ∂∫ ∫
G
which is equivalent to
0j t ρ∂∇ ⋅ + =∂
G (12.15)
Equation (12.15) is called the equation of continuity for currents, on account of its similarity with Eq.(6.11) for fluid flow.
