Recall the introduction of electric field E from the measurement of electrostatic force F on charge q. These concepts are connected by F=qE where E=kqr/r3. Here force F is often in [Newtons]≡[kg m/s2], 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 vector1 qm of a moving charge or current element in [Coulombs] as qm=qv/c=Iδs/c, then one can define a magnetic field vector in electric-field units (e.g. [N/C]) as B'=kqm×r/r3. 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 qm equal to F=qm×B'. The magnitude of qm is the effective charge deficit seen by us as co-moving charges increase their speed2, while the magnitude of B' is the force exerted on a charge vector perpendicular to B', per unit qm.
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 qm 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=(E2+B'2)/8πk, as do expressions for scalar and vector potential (last paragraph below).
Two different definitions are instead used3 in Système International d'Unités (SI), perhaps because magnetism, electrostatics, and light were first seen as separate phenomena. In particular, the charge vector qm' of a moving charge or current element is defined as qm'=qv=Iδs=qmc 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 qm'. This gives rise to the forms of Biot-Savart B=(k/c2)qm'×r/r3 and Lorentz F=qm'×B used in most texts. Two variations on the Coulomb constant are also defined to aid in the definition of non-vacuum fields, namely εo≡1/(4πk) and μo≡4πk/c2.
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 "qm1×(qm2×kr/r3) is not always equal to -qm2×(qm1×kr/r3)" 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 (-δA/δt) of the vector potential (Maxwell's electrokinetic momentum per unit charge) A=(k/c2)qm'/r=A'/c. As you might imagine, A' = kqm/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 Iδs but not qv.
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-force4 times Sqrt[1-(v/c)2], as illustrated in the figure at right which shows how apparent-force (dp/dt), and mass times acceleration (ma), 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]kq1q2r⊥/r3. 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 Iδs and co-moving charge qv 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 ΦB 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 Fo 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=dp/dt, since Fo is precisely maγ3/γ⊥ while F differs from Fo in that its component parallel to Fo 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.