Recall the introduction of electric field **E**
from the measurement of electrostatic force **F** on charge q.
These concepts are connected by **F**=q**E**
where **E**=kq**r**/r^{3}. Here force F is often
in [Newtons]≡[kg m/s^{2}], electric charge q in
[Coulombs], and electric field E in
[Newton/Coulomb] ≡ [Volt/meter] = [Franklins?].

The finite value of "lightspeed" c in
relativity requires that any force between
objects depend on the frame of observation, so that for example
a neutral current-carrying wire will not be neutral to a moving
charge. That is, the moving charge experiences a Coulomb force.
Hence magnetism. If we define the charge
vector^{1} **q**_{m} of a moving charge
or current element *in [Coulombs]* as
**q**_{m}=q**v**/c=I**δs**/c,
then one can define a
magnetic field vector *in electric-field units* (e.g. [N/C]) as
**B**'=k**q**_{m}×**r**/r^{3}. For
an isolated charge q this expression works only for low speeds in the
absence of acceleration, but for steady currents in a
closed loop the expression is robust. This field in turn
exerts a Lorentz force on a separately moving charge or current **q**_{m}
equal to **F**=**q**_{m}×**B**'.
The magnitude of q_{m} is the effective *charge deficit* seen by us
as co-moving charges increase their speed^{2},
while the magnitude of B' is the force exerted on
a charge vector perpendicular to **B**', per unit q_{m}.

Thus magnetic
interactions can be fully expressed using only Coulomb's constant k,
your favorite unit for electric charge, and familiar mechanical
units for force, distance, etc. In this way *magnetic fields
might be seen as electric fields that are caused by, and act on,
charge vectors instead of charges*. The form of the equations for
magnetic fields and forces
mirror those for electrostatics (cf. the Figure at right),
as one considers forces between charge vectors **q**_{m}
separately from forces between scalar charges q. The role of
the force-law in each equation is easy to see.
Electric and magnetic expressions for field energy U per unit volume
V also fit together nicely,
i.e. U/V=(E^{2}+B'^{2})/8πk, as do
expressions for scalar and vector potential (last paragraph below).

Two different definitions are
instead used^{3} in Système International
d'Unités (SI), perhaps because magnetism, electrostatics,
and light were first seen as separate phenomena. In particular,
the charge vector **q**_{m}' of a moving charge
or current element is defined as
**q**_{m}'=q**v**=I**δs**=**q**_{m}c
in units of [Coulomb m/s] ≡ [Ampere meter], i.e. as the
charge deficit mentioned above times c. It is
natural then to define magnetic induction
as **B**=**B**'/c in [Ns/Cm] ≡ [Tesla] i.e. as a
force per unit q_{m}'.
This gives rise to the forms of Biot-Savart
**B**=(k/c^{2})**q**_{m}'×**r**/r^{3}
and Lorentz **F**=**q**_{m}'×**B** used in most
texts. Two variations
on the Coulomb constant are also defined to aid in the
definition of non-vacuum fields, namely
ε_{o}≡1/(4πk) and
μ_{o}≡4πk/c^{2}.

To see if B' offers some complementary perspective, try answering these questions. What's the magnetic field strength B' in [Volts/meter] of a record-breaking 14.7 Tesla magnetic field? How does this compare to the 30[kV/cm] field strength often cited as the ionization breakdown value for dry air? Why is ionization less of a concern in the presence of that magnetic field?

*A word of caution as well:* The fact that
"**q**_{m1}×(**q**_{m2}×k**r**/r^{3})
is not always equal to
-**q**_{m2}×(**q**_{m1}×k**r**/r^{3})"
raises a flag with the equations above in context of
action-reaction and momentum conservation. If this is integrated
over a closed current loop, the disparity vanishes. In the process,
however, this observation highlights a shortcoming in what we tell
intro-physics students by ignoring (e.g. in the E-field expression
above) the induced electric field component
(e.g. relevant
to rail guns) expressed as a time derivative (-δ**A**/δt)
of the vector potential (Maxwell's electrokinetic momentum per unit charge)
**A**=(k/c^{2})**q**_{m}'/r=**A**'/c.
As you might imagine, **A**' = k**q**_{m}/r
is like scalar potential Φ=kq/r also in [Volts].
However, vector potential is avoided in most intro texts.
Are the consequences of this omission lessened by asserting that the
Lorentz equation w/o the induced E-field works for an *external*
static magnetic field (e.g. like that from a fixed current loop) but not
for time-varying fields (e.g. like the field from a moving charge)?
These caveats may be another reason that Biot-Savart is generally introduced
with I**δs** but not q**v**.

*Footnotes:*

[1] The magnitude of the charge vector around a loop is also used in the magnet-manufacturing industry, to measure the local strength of a magnetic pole. Given that, what is the strength in [Coulombs] of a typical refrigerator magnet?

[2] The perpendicular
component of force between two objects co-moving at constant velocity
is equal to the (object-frame)
proper-force^{4} times
Sqrt[1-(v/c)^{2}], as illustrated in the figure
at right which shows how apparent-force (d**p**/dt), and mass times
acceleration (m**a**),
deviate from the proper-force on a moving object. For the Coulomb force
perpendicular to the motion direction of uniformly co-moving charges,
this gives
F_{⊥}=Sqrt[1-(v/c)^{2}]kq_{1}q_{2}r_{⊥}/r^{3}.
A second factor of
Sqrt[1-(v/c)^{2}] comes into play for a charge co-moving with
current elements in a steadily moving wire, to keep the wire neutral
in the face of length contraction as the element speeds up.
The observed force (perpendicular to velocity) between a moving
charge and co-moving current elements in a neutral wire is thus
the proper force times 1-(v/c)^{2}. Hence the net force
between a charged current element I**δs** and co-moving
charge q**v**
can be seen as the proper force minus a "velocity-sensitive
Coulomb" (aka magnetic) force between the two charges, where
qv/c and Iδs/c are magnitudes of the deficit charges involved
in the magnetic interaction. In a neutral wire, of course,
that proper force itself is canceled by complementary non-moving
charge elements, leaving only magnetic interactions to consider. The
cross-products in the equations above take into account
the fact that each deficit charge
impacts *only* that component of force perpendicular to its
vector direction. Thus rather than drop Biot-Savart out of the
blue on intro-physics students, the relativistic frame-dependence
of force can be used to
explain why and how currents in neutral wires interact
with one another but not with scalar charges, including (from their
deficit character) why parallel currents attract.

[3] The foregoing suggests that a modified set of SI
units, for which E/B is dimensionless as with the
Gaussian and Heaviside-Lorentz unit systems used
by Jackson's *Classical Electrodynamics*,
might give students in an introductory course a more direct path
to understanding. According to the 3rd edition preface,
the tension between SI and Gaussian units
resulted in division of Jackson's book across two unit systems
and the betrayal of a
pact between friends. The simplification described above
could be made by changing only units for magnetic induction **B**,
vector potential **A**,
and magnetic flux Φ_{B} to electric field
units, i.e. [V/m], [V], and [Vm] respectively,
with compensatory c
values added into Maxwell's equations to keep everything
else (including H) the same. What are other pros and cons
of making this shift, at least for intro physics classes
eager for the clearest possible view of nature's interconnections?

^{[4]} *Proper-acceleration*
and *proper-force* are nothing more than
the 3-vector acceleration and net force experienced in the
frame of an accelerated object. Like *proper-time* and
*proper-length*, they describe quantities
from the vantage point of a special frame, in this case the frame
of the object to which the acceleration and forces are
applied. Their magnitude also happens to be the Lorentz-invariant magnitude
of the acceleration and net force four-vectors *seen from all
frames*, since in the rest frame of the accelerated object the
time-like component of these four-vectors is zero. Because
forces might be caused by objects moving with respect to one
another, proper force **F**_{o} doesn't obey
an action-reaction principle
like 4-vector and locally-defined frame-variant forces do.
However, it's an excellent
bridge between locally-measured coordinate-acceleration **a**
and force **F**=d**p**/dt, since **F**_{o} is precisely
m**a**γ^{3}/γ_{⊥}
while **F** differs from **F**_{o} in that its
component parallel to **F**_{o} is divided by
γ_{⊥} on rotation toward the velocity line.
The equivalence principle of general relativity, combined with
the concept of local affine-connection (or geometric) forces,
makes these quantities useful even
in non-inertial settings (like gravitationally-curved spacetime).
The utility of Newton's Laws for describing gravity on
earth as "just one more inter-object force" is living proof
of that, since gravity like magnetism is an example of everyday
relativity.

- R. G. Piccioni's
recent
article in
*The Physics Teacher*. - Dan Schroeder's notes on Purcell simplified.
- More details on charges in motion near a current-carrying neutral wire.
- This note discusses the proper-force figure above in more detail.
- More notes on proper quantities and map-based motion at anyspeed.