In order to assess the properties of B stat, we determine its divergence and curl. With this definition of B stat, Equation (1.8) on the previous page may we written If we generalise expression (1.12) to an integrated steady state current distribution j( x), we obtain Biot-Savart's law:Ĭomparing Equation (1.5) on page 3 with Equation (1.13), we see that there exists a close analogy between the expressions for E stat and B stat but that they differ in their vectorial characteristics. The SI unit for the magnetic field, sometimes called the magnetic flux density or magnetic induction, is Tesla (T). Which expresses the small element d B stat( x) of the static magnetic field set up at the field point x by a small line element d l 0 of stationary current J 0 at the source point x 0. I turns out that B stat can be defined through In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a vectorial magnetic field B stat. This clearly exhibits the expected symmetry in terms of loops C and C 0. Recognising the fact that the integrand in the first integral is an exact differential so that this integral vanishes, we can rewrite the force expression, Equation (1.8) on the previous page, in the following symmetric way Let F denote such a force acting on a small loop C carrying a current J located at x, due to the presence of a small loop C 0 carrying a current J 0įormula (F.54) on page 155, we can rewrite (1.8) in the following way While electrostatics deals with static charges, magnetostatics deals with stationary currents, i.e., charges moving with constant speeds, and the interaction between these currents.Įxperiments on the interaction between two small current loops have shown that they interact via a mechanical force, much the same way that charges interact.
Which is Gauss's law in differential form. Of the Dirac delta function, Equation (M.73) on page 174, we find that Taking the divergence of the general E stat expression for an arbitrary charge distribution, Equation (1.5) on the preceding page, and using the representation Since, according to formula Equation (M.78) on page 175, r 0 for any 3D 3 scalar field ( x), we immediately find that in electrostatics “For instance, Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actions going on in the medium, they were satisfied that they had found it in a power of action at a distance impressed on the electric fluids.”Ģ In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation effects manifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classically this nonlinearity is negligible.ĭraft version released 13th November 2000 at 22:01. We emphasise that Equation (1.5) above is valid for an arbitrary distribution of charges, including discrete charges, in which case can be expressed in terms of one or more Dirac delta functions.ġIn the preface to the first edition of the first volume of his book A Treatise on Electricity and Magnetism, first published in 1873, James Clerk Maxwell describes this in the following, almost poetic, manner: Where, in the last step, we used formula Equation (M.68) on page 172. If the discrete charges are small and numerous enough, we introduce the charge density located at x 0 and write the total field as In the presence of several field producing discrete charges q 0 i, at x 0 i, i =1 2 3 :::, respectively, the assumption of linearity of vacuum 2 allows us to superimpose their individual E fields into a total E field Using formulae (1.1) and (1.2), we find that the electrostatic field E stat at the field point x (also known as the observation point), due to a field-producing charge q 0 at the source point x 0, is given by This means that we can say that any net electric charge produces an electric field in the space that surrounds it, regardless of the existence of a second charge anywhere in this space.
Purpose of the limiting process is to assure that the test charge q does not influence the field, the expression for E stat does not depend explicitly on q but only on the charge q 0 and the relative radius vector x x 0.