Charge Density in a Current-Carrying Wire
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(December 23, 2010)
1 Problem
Discuss the volume densities
+
and
< 0 of positive and negative electric charges in a
wire that carries a steady current, assuming that in the lab frame the positive charges are
at rest and the current is due to negative charges (electrons) that all have speed v.
2 Solution
Discussions of the force on a charged particle outside a current-carrying wire often assume
that the wire is electrically neutral. This problem explores how this assumption is not quite
correct.
We give solutions both in the lab frame and in the rest frame of the conduction electrons,
using both Maxwells equations and special relativity.
We note that for current to ow in a resistive wire, there must be an axial electric eld
inside the wire, which requires a surface charge distribution that varies with position along
the wire. See, for example, sec. 17 of [1], and [2, 3]. The surface charge distribution could
include a uniform term of any magnitude. These surface charges are kept from leaving the
surface by quantum eects often summarized by the term work function. Likewise, the
positive charges in the interior of the wire are held together in a lattice by quantum eects.
We suppose that the positive charge density
+
is uniform in the lab frame.
In this problem we assume that the conduction electrons can be described classically.
Then, for steady axial motion, there must be zero radial force on these electrons.
We use a cylindrical coordinate system (r, , z) whose axis is the axis of the wire. We
suppose that the ow of conducting electrons is purely axial, and azimuthally symmetric.
The negative charge density,
, could depend on the radius r. The electric eld E has no
azimuthal component, while the magnetic eld B has only an azimuthal component.
2.1 Lab Frame
For steady ow of current, the axial component of the electric eld must be independent of
z. Then, assuming that there is no azimuthal component to the electric eld, Maxwells rst
equation tells us that
E =
1
r
r
(rE
r
) = 4(r) = 4[
+
+
(r)] (1)
(in Gaussian units). The radial component of the electric eld vanishes at r = 0, so we nd
that
E
r
(r) =
4
r
_
r
0
r
[
+
+
(r
)] dr
(2)
1
(in Gaussian units).
The azimuthal magnetic eld is due to the motion of the negative charges with velocity
v z, and Amp`eres law tells use that
B
(r) =
4v
cr
_
r
0
r
(r
) dr
. (3)
The Lorenz force density on the negative charges is
f
(r) =
(r)
_
E(r) +
v
c
B(r)
_
=
(r)
_
4 r
r
_
r
0
r
+
+
(r
)
_
1
v
2
c
2
__
dr
+ E
z
(r) z
_
. (4)
For steady axial motion of the conduction electrons, the radial force on them must vanish
at all r inside the wire, which implies that the negative charge density is uniform with value
=
+
1 v
2
/c
2
=
2
+
, (5)
where
=
1
1 v
2
/c
2
. (6)
The positive charge density is less than the negative by one part in 10
21
for v = 1 cm/s. This
corresponds roughly to 10 more electrons than protons in each cubic millimeter of copper
wire.
Using eq. (5) in eqs. (2)-(3), the radial electric eld and azimuthal magnetic eld inside
the wire are
E
r
(r) = 2(
2
1)
+
r, B
(r) =
2pi
2
v
+
r
c
. (7)
Because of the small dierence between the positive and negative charge densities, the
positive charges experience a small inward radial force density (in addition to the axial force
due to collisions with the conduction electrons),
f
+,r
=
+
E
r
(r) = 2(
2
1)
2
+
r =
2
2
v
2
2
+
r
c
2
. (8)
2.2 Rest Frame of the Conduction Electrons
We denote quantities in the rest frame of the conduction electrons with a
. Thus, the
velocity of the charged particles in the
frame is v
+
= v z.
If the radial electric eld E
r
were nonzero the conduction electrons would experience a
radial force. Hence, we expect the radial electric eld, and the bulk charge density
, to
vanish in the
frame.
2
The Lorentz contraction of a moving stick results in an observer of a charge stick
reporting a charge density larger by a factor than that in the rest frame of the stick.
Thus, the density
of negative charge in the lab frame is larger than that in the
frame,
, (9)
while the density
+
of positive charge in the
frame is larger than that in the lab frame,
+
=
+
. (10)
The total charge density in the
frame is
+
+
=
+
= 0. (11)
Thus, the bulk charge density of a current-carrying wire vanishes in the rest frame of the
conduction charges, rather than in the lab frame [4].
The positive charges experience a magnetic Lorentz force density in the
frame,
f
+,r
=
+
vB
c
=
2
+
2
v
2
r
c
2
=
2
2
v
2
2
+
r
c
2
= f
+,r
. (12)
This transverse Lorentz force is the same in the lab and the
frames, as expected for the
transverse spatial component of a 4-vector.
1
References
[1] A. Sommerfeld, Electrodynamics (Academic Press, New York, 1952).
[2] J.D. Jackson, Surface charges on circuit wires and resistors play three roles, Am. J.
Phys. 64, 855 (1996),
http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_64_855_96.pdf
[3] K.T. McDonald, Hidden Momentum in a Coaxial Cable (Mar. 28, 2002),
http://puhep1.princeton.edu/~mcdonald/examples/hidden.pdf
[4] P.C. Peters, In what frame is a current-carrying conductor neutral?, Am. J. Phys. 53,
1165 (1985),
http://puhep1.princeton.edu/~mcdonald/examples/EM/peters_ajp_53_1165_85.pdf
1
The Lorentz force f = E + J/c B on a charge-current density 4-vector j
= (, J/c) is the spatial
component of the 4-vector f
Lorentz,
= (J E/c, f ). Thus, the Lorentz force density is more relativistic
than the so-called Minkowski force based on the Newtonian relation F = ma, for which the 4-vector relation
is f
Minkowski,
= (F , F) for the force on a particle of velocity v = c, where = 1/
_
1 v
2
/c
2
, in that
the Minkowski force has the awkward factor of in its spatial components. Of course, the (Minkokwski)
force F = f dVol on a charged volume element, dVol, is not a component of a 4-vector; however, F = f
dVol is.
3
