Electromagnetic effects of a moving charge

by Oliver Heaviside

PART I.

IN connection with the letters of Profs. Poynting and Lodge in The Electrician, Nov. 2, 1888, I believe that the following extract from a letter from Sir William Thomson (which I have permission to publish) will be of interest [see Postscript, p. 483, vol. II., to elucidate]:

"I don't agree that velocity of propagation of electric potential is a merely metaphysical question. Consider an electrified globe, A, moved to and fro, with simple harmonic motion, if you please, to fix the ideas. Consider very quickly-acting electroscopes B, B', at different distances from A. If the indications of B, B' were exactly in the same phase, however their places are changed, the velocity of propagation of electric potential would be infinite; but if they showed differences of phase, they would demonstrate a velocity of propagation of electric potential.

"Neither is velocity of propagation of 'vector-potential' metaphysical. It is simply the velocity of propagation of electromagnetic force the velocity of electromagnetic waves', in fact."

Taking the second point first, it is, I think, clear that if by the propagation of vector-potential is to be understood that of electric and magnetic disturbances, it is merely the mode of expression that is in question. I am myself accustomed to mentally picture the electric and magnetic forces or fluxes, and their propagation, which takes place at the speed of light or thereabouts, because they give the most direct representation of the state of the medium, which, I think, must be agreed is the real physical subject of propagation. But if we regard the vector-potential directly, then we can only get at the state of the medium by complex operations, and we really require to know the vector-potential both as a function of position and of time, for its space-variation has to furnish the magnetic force, and its time-variation the electric force; besides which, there is sometimes the space-variation of a scalar potential in addition to be regarded, before we can tell what the electric force is. Besides this roundaboutness, it implies a knowledge of the full solution, and if we do not possess it, it is much simpler to think of the propagation of the electric and magnetic disturbances, and I find that this method works out much more easily in the solution of problems.

The other question will, I believe, be found to be ultimately of precisely the same nature. Start with the sphere A at rest, and the field steady, and consider two external points, P and P', at different distances. The electric force at them has different values, and the whole field has a potential. But now give the sphere a displacement, and bring it to rest again in a new position. Is the readjustment of potential instantaneous? I should say, Certainly not, and describe what happens thus. When the sphere is moved, magnetic force is generated at its boundary (lines circles of latitude, if the axis be the line of motion), and with it there is necessarily disturbance of electric force. The two together make an electromagnetic wave, which goes out from the sphere at the speed of light, and at the front of the wave we have E=μvH, where E is the electric and H the magnetic force intensity. Before the front reaches P or P' we have the electric field represented by the potential function, but after that it cannot be so represented until the magnetic force has wholly disappeared, when again we have a steady field representable by a potential function. It is difficult to see how to plainly differentiate any propagation of potential per se.

If the motion is simple-harmonic, there is a train of outward waves and no potential. I imagine that an electroscope, if infinitely sensitive and without reactions, would register the actual state of the electric field, irrespective of its steadiness. By an electroscope, as this is a purely theoretical question, I understand the very simplest one, a very small charge at a point; or, say, the unit charge, the force on which is the electric force of the field.

When these things are closely examined into, if the facts as regards the propagation of disturbances (electric and magnetic) are agreed on, the only subject of question is the best mode of expressing them, which I believe to be in terms of the forces, not potentials.

But there really is infinite speed of propagation of potential sometimes; on examination, however, it is found to be nothing more than a mathematical fiction, nothing else being propagated at the infinite speed.

It will be understood that I preach the gospel according to my interpretation of Maxwell, and that any modification his theory of the dielectric may receive may involve a fresh kind of propagation at present not in question.

⁠Nov. 5, 1888.

Part II.

The question raised by Prof. S. P. Thompson (in The Electrician, Nov. 16, 1888, p. 54) as to whether the motion of an uncharged dielectric through a field of electric force produces magnetic effects must, I think, be undoubtedly answered in the affirmative. As the distribution of displacement varies, its time-variation is the electric current, with determinable magnetic force to match. When the speed of motion is a small fraction of that of light, we may regard the displacement as having at every moment its proper steady distribution, so that there is no difficulty in estimating the magnetic effects, except, it may be, of a merely mathematical character. For instance, the case of a sphere moving in a field which would be uniform were the sphere absent, may be readily attacked, and does perfectly well to illustrate the general nature of the action.

But if the moved dielectric have the same electric permittivity as the surrounding medium, so that there is no difference made in the steady distribution, the question which may be now raised as to the possible production of transient disturbances is one to which the above theory does not present any immediate answer. I believe that the body will be magnetized transversely to the electric displacement and the velocity. [The motional magnetic force is referred to.]

Another question, somewhat connected, is contained in Prof. Poynting's suggestion (in letter to Prof. Lodge, The Electrician, p. 829, vol. xxi.) that electric displacement may possibly be produced without magnetic force by the agency of pyroelectricity. But, whatever the agency, it would, I conceive, be a new fact—quite outside Maxwell's theory legitimately developed. We may have subsidence of electric displacement without magnetic force; but I cannot see any way to produce it.

But the main subject of this communication is the electromagnetic effect of a moving charge. That a moving charge is equivalent to an electric current-element is undoubted, and to call it a convection-current. as Prof. S. P. Thompson does, seems reasonable. The true current has three components, thus,

$\scriptstyle {\mathrm {curl} \ \mathbf {H} =4\pi (\mathbf {C} +\mathbf {\dot {D}} +\rho \mathbf {u} )}$,

where H is the magnetic force, C the conduction-current, D the displacement, and ρ the volume-density of electrification moving with velocity u. The addition of the term ρu is, I presume, the extension made by Prof. Fitzgerald to which Prof. S. P. Thompson refers. At any rate, I can at present see no other.

There are several ways of arriving at the conclusion that a moving charge must be regarded as an electric current; but, when that is admitted, we are very far from knowing what its magnetic effect is. No cut-and-dried statement of it can be made, because it varies according to circumstances. The magnetic field, whatever it be in a given case, is not that of a current-element (supposing the charge to be at a point), for that is anti-Maxwellian, but is that of the actual system of electric current, which is variable.

Thus, in the case of motion at a speed which is a small fraction of that of light, the magnetic field (as found by Prof. J. J. Thomson) is the same as that of Ampère's current-element represented by ρu; that is, a current-element whose direction is that of u and whose moment is ρu, if u is the tensor of u (understanding by "moment," current-density × volume); but the true current to correspond bears the same relation to the current-element as the induction of an elementary magnet bears to its magnetic moment. The magnetic energy due to the motion of a charge q upon a sphere of radius a in a medium of inductivity μ, at a speed u which is only a very small fraction of that of light, is expressed by $\scriptstyle {{\frac {1}{3}}\mu q^{2}u^{2}/a}$. But if the speed be not a small fraction of that of light, the result is very different. Increasing the speed of the charge causes not merely greater magnetic force but changes its distribution altogether, and with it that of the electric field. It is no use discussing the potential. There is not one. The magnetic field tends to concentrate itself towards the equatorial plane, or plane through the charge perpendicular to the line of motion. When the speed equals that of light itself this process is complete, and the is simply a plane wave (electromagnetic).

Since a charge at a point gives infinite values, it is more convenient to distribute it. Let it be, first, of linear density q along a straight line AB, moving in its own line at the speed of light. Then the field is contained between the parallel planes through A and B perpendicular to AB, and is completely given by

$\scriptstyle {E/\mu v=H=2qu/r}$,

where E and H are the intensities of the electric and magnetic forces at distance r from AB. The lines of E radiate uniformly from AB in all directions parallel to the planes; those of H are everywhere perpendicular to those of E, or are circles centred upon AB. Outside this electromagnetic wave there is no disturbance. I should remark that the above is a description of the exact solution. It is, of course, nothing like the supposed field of a current-element AB.

To still further realize, we may substitute a cylindrical distribution for the linear, and then, again, terminate the lines of E on another cylindrical surface between the bounding planes. To find the resulting distributions of E and H (always perpendicular) may be done by super-imposition of the elementary solutions, or by solving a bidimensional problem in a well-known manner.

Those who are acquainted with my papers in this journal will recognise that what we have arrived at is simply the elementary plane wave travelling along a distortionless circuit. All roads lead to Rome!

Returning to the case of a charge q at a point moving through a dielectric, if the speed of motion exceeds that of light, the disturbances are wholly left behind the charge, and are confined within a cone, AqB. The charge is at the apex, moving from left to right along Cq. The semi-angle, θ, of the cone, or the angle AqC, is given by

$\scriptstyle {\sin \ \theta =v/u}$,

where v is the speed of light, and u that of the charge. The magnetic lines are circles round the axis, or line of motion. The displacement is away from q, of course, and of total amount q, but not uniformly distributed within the cone. The electric current is towards q in the inner part of the cone, and away from q in the outer.

It will be seen that the electric stress tends to pull the charge back. Therefore, applied force on q in direction Cq is required to keep up the motion. Its activity is accounted for by the continuous addition at a uniform rate which is being made to the electric and magnetic energies at q. For the motion at the wave-front, at any point on Aq or Bq, is perpendicularly outward, not towards q. Whilst the cone is thus expanding all over, the forward motion of q continually renews the apex, and keeps the shape unchanged.

Steady motion alone is assumed.

To avoid misconception I should remark that this is not in any way an account of what would happen if a charge were impelled to move through the ether at a speed several times that of light, about which I know nothing; but an account of what would happen if Maxwell's theory of the dielectric kept true under the circumstances, and if I have not misinterpreted it. [See footnote on p. 516, later.]

⁠Nov. 18, 1888.

PART III.

All disturbances being propagated through the dielectric ether at the speed of light, when, therefore, a charge is in motion through the medium, the discussion of the effects produced naturally involves the consideration of three cases, those in which the speed u of the charge is less than, or equal to, or greater than v, that of light.

In a previous communication [Part II. above], I gave the complete and very simple solution of the intermediate case of equality of speeds. A formal demonstration is unnecessary, as the satisfaction of the necessary conditions may be immediately tested.

But I was not then aware that the case u<v admitted of being presented in a nearly equally-simple form. That such is the fact is rather surprising, for it is very exceptional to arrive at simple results, and these now in question are sufficiently free from complexity to take a place in text-books of electricity.

Let the axis of z be the line of motion of the charge q at speed u. Everything is symmetrical with respect to this axis. The lines of electric force are radial out from the charge. Those of magnetic force are circles about the axis. The two forces are perpendicular. Having thus settled the directions, it only remains to specify their intensities at any point P distant r from the charge, the line r making an angle θ with the axis. Let E be the intensity of the electric, and H of the magnetic force. Then, if c is the permittivity and μ the inductivity, such that μcv²=1, we have

$\scriptstyle {(u<v)}{\begin{cases}\scriptstyle {cE={\frac {{\frac {q}{r^{2}}}\left(1-{\frac {u^{2}}{v^{2}}}\right)}{\left(1-{\frac {u^{2}}{v^{2}}}\sin ^{2}\theta \right)^{\frac {3}{2}}}},\quad \cdots \cdots \cdots \cdots \cdots \cdot \cdot }&\scriptstyle {({\text{A}})}\\\scriptstyle {H=cEu\ \sin \theta .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {({\text{B}})}\end{cases}}$

That (A), (B) represent the complete solution may be proved by subjecting them to the proper tests. Premising that the whole system is in steady motion at speed u, we have to satisfy the two fundamental laws of electromagnetism:—

(1). (Faraday's law). The electromotive force of the field [or voltage] in any circuit equals the rate of decrease of the induction through the circuit (or the magnetic current × -4π).

(2). (Maxwell's law). The magnetomotive force of the field [or gaussage] in any circuit equals the electric current × 4π through the circuit.

Besides these, there is continuity of the displacement to be attended to. Thus:—

(3). (Maxwell). The displacement outward through any surface equals the enclosed charge.

Since (A) and (B) satisfy these tests, they are correct. And since no unrealities are involved, there is no room for misinterpretation.

When u/v is very small, we have, approximately,

$\scriptstyle {cE={\frac {q}{r^{2}}},\quad \quad H={\frac {qu}{r^{2}}}\sin \theta }$,

representing Prof. J. J. Thomson's solution—that is, the lines of displacement radiate uniformly from the charge, and the magnetic force is that of the corresponding displacement-currents together with the moving charge regarded as a current-element of moment qu. Instantaneous action through the medium is involved—that is, to make the solution quite correct.

That the lines of electric force should remain straight as the speed of the charge is increased is itself a rather remarkable result. Examining (A), we see that the effect of increasing u is to concentrate the displacement about the equatorial plane $\scriptstyle {\theta ={\frac {1}{2}}\pi }$. Self-induction does it. In the limit, when u=v, the numerator vanishes, making E=0, H=0 everywhere except at the plane mentioned, where, by reason of the denominator becoming infinitely small in comparison with the numerator, the displacement is all concentrated in a sheet, and with it the induction, forming a plane electromagnetic wave, as described (and realized) in my previous communication.

If we terminate the field described in (A) and (B) on a spherical surface of radius a, instead of continuing it up to the charge q at the origin, we have the case of a perfectly conducting sphere of radius a possessing a total charge q, moving steadily at speed u through the dielectric ether. As the speed is increased to v, the charge all accumulates at the equator of the sphere. [See footnote on p. 514, later.]

But after that? This brings us to the third case of u>v, and here I have so-far failed to find any solution which will satisfy all the necessary conditions without unreality. The description at the close of Part II. must therefore be received as a suggestion, at present unconfirmed. I hope to consider the matter in a future communication.

P.S.—In a recent number Mr. W. P. Granville raised the question of action through a medium being only action at a short distance instead of a long one, and asked for instruction. His inquiry has elicited no response. This is not, however, because there is nothing to be said about it. The matter did not escape the notice of the "anti-distance-action sage." My own opinion is that the question involved is, if not metaphysical, dangerously near to being so; consequently, whole books might be devoted to it. At present, however, I think it is more useful to try to find out what happens, and to construct a medium to make it happen; after that, perhaps, the matter referred to may be more advantageously discussed. The well of truth is bottomless.

Part IV.

In previous communications [above] I have discussed this matter. Referring to the case of steady rectilinear motion, I gave a description of the result when the speed of the charge exceeds that of light, obtained mainly by general reasoning, and stated my inability to find a solution to represent it. The displacement cannot be outside a certain cone of semi-vertical angle whose sine equals the ratio v/u of the speed of light to that of the charge, which is at the apex.

In the Phil. Mag. for July, 1889, Prof. J. J. Thomson has examined this question. Like myself, he fails to find a solution within the cone; but concludes that the displacement is confined to its surface. If so, it must form, along with the magnetic induction, an electromagnetic wave. But it may be readily seen that such a wave is impossible, having no stability.

For as the charge moves from A to B, a given surface-element, C, would move to D. In doing so its area would vary directly as its distance from the apex, and the energy in the element would therefore vary inversely as its distance from the apex, and the forces, electric and magnetic, would therefore vary inversely as the square root of the distance from the apex, instead of inversely as the distance, which is obviously necessary in order that the displacement may be confined to the surface. This conflict of conditions constitutes instability. In the Phil. Mag. for April, 1889, I suggested that whilst there must be a solution of some kind, one representing a steady state was impossible. This conclusion is confirmed by the failure of Prof. Thomson's proposed surface-wave to keep itself going.

Prof. Thomson, who otherwise confirms my results, has also extended the matter by supposing that the medium itself is set in motion, as well as the electrification. This is somewhat beyond me. I do not yet know certainly that the ether can move, or its laws of motion if it can. Fresnel thought the earth could move through the ether without disturbing it; Stokes, that it carried the ether along with it, by giving irrotational motion to it. Perhaps the truth is between the two. Then there is the possibility of holes in the ether, as suggested by a German philosopher. When we get into one of these holes, we go out of existence. It is a splendid idea, but experimental evidence is much wanting.

But if we consider that the medium supporting the electric and magnetic fluxes is really set moving when a body moves, and assume a particular kind of motion, it is certainly an interesting scientific question to ask what influence the motion exerts on the electromagnetic phenomena. I do not, however, think that any new principles are involved.

The general connections of E and H, referred to fixed space without conductivity, being

${\begin{aligned}\scriptstyle {\mathrm {curl} (\mathbf {e} -\mathbf {E} )=\mu p\mathbf {H} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(1)}}\\\scriptstyle {\mathrm {curl} (\mathbf {H} -\mathbf {h} )=cp\mathbf {E} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(2)}}\\\end{aligned}}$

where p stands for d/dt and e and h are the impressed parts of E and H; if there is also motion of electrification, we have to consider it to constitute a convection-current, a part of the true current, and so make (2) become

$\scriptstyle {\mathrm {curl} (\mathbf {H} -\mathbf {h} )=cp\mathbf {E} +4\pi \rho \mathbf {u} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdot \cdot {\text{(3)}}}$

where ρ is the density of electrification, whose velocity is u. [See Part II.] It now remains to specify e and h. They are zero when the medium supporting the fluxes is at rest. But if it moves, and its velocity is w, there is, first, the electric force due to motion in a magnetic field,

$\scriptstyle {\mathbf {e} =\mu \mathbf {VwH} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(4)}}}$

which is well known; and next the magnetic force due to motion in an electric field,

$\scriptstyle {\mathbf {h} =c\mathbf {VEw} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(5)}}}$

which is not so well known. (First, I believe, given by me in the third Section of "Electromagnetic Induction and its Propagation," The Electrician, January 24, 1885 [vol. I., p. 446]; again, obtained in a different way in Section XXIL, January 15, 1886 [vol. I., p. 546]; see also Phil. Mag., August, 1886 [vol. II., Art. L.], and an example of the use of (4) and (5) in The Electrician, April 12, 1889, p. 683 [vol. II., Art. LI.].)

The mechanical force called by Maxwell the "electromagnetic force" is VCB, where C is the true current and B the induction. It is the force on the matter supporting electric current. Let it move. If w is its velocity, the activity of the force is

$\scriptstyle {\mathbf {wVCB} =\mathbf {CVBw} =-\mathbf {eC} .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot \cdot {\text{(6)}}}$

Similarly, as I obtained in Section XXII. above referred to, there is a mechanical force (the magneto-electric) on matter supporting magnetic current G=μpH/4π, expressed by 4πVDG, and its activity is

$\scriptstyle {4\pi \mathbf {wVDG} =4\pi \mathbf {GVwD} =-\mathbf {hG} .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(7)}}}$

Of course e and h. are reckoned as impressed forces, which is the reason of the change of sign. Their activities are eC and hG.

It should be remarked further, that the above expressions for e and h are not certain. For I have shown that the sources of all disturbances are the lines of curl of the impressed forces (Phil. Mag., Dec., 1887) [vol. II., p. 362], and that the fluxes produced depend solely upon the curls of e and h, both as regards the steady fluxes and the variable ones leading to them. We may, therefore, use any other expressions for e and h which have the same curls as the above. And, in fact, we see that equations (1) and (2) only contain their curls.

Equations (1) and (3), with e and h defined by (4) and (5), therefore enable us to determine the effect of the moving medium. Prof. Thomson also arrives at (4) and (5), and at the "magneto-electric force," in his paper to which I have referred, by an entirely different method. And to show how well things fit together, he concludes, from the consideration of the moving medium, that a moving electrified surface is a current-sheet, which is another way of saying that a convection current is a part of the true current, as expressed in (3). I must, however, disagree with Prof. Thomson's assumption that the motion must be irrotational. It would appear, by the above, that this limitation is unnecessary.

As an example, and to introduce a new point, take the case of a charge q moving at speed u along the axis of z. It will come to the same thing if we keep the charge at rest, and move the medium the other way. We then use the equations (1) and (2), and in them use (4) and (5) with w=-u. Now when the steady state is arrived at, we have p=0, so (1) and (2) become

${\begin{aligned}\scriptstyle {\mathrm {curl} (\mu \mathbf {VHu} -\mathbf {E} )=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(8)}}\\\scriptstyle {\mathrm {curl} (\mathbf {H} -c\mathbf {VuE} )=0.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(9)}}\\\end{aligned}}$

In addition, the divergence of D must be q at the origin, and the divergence of B must be zero. The latter gives, applied to (9),

$\scriptstyle {\mathbf {H} =c\mathbf {VuE} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(10)}}}$

which gives H fully in terms of E. Eliminate H from (8) by means of (10), and we get

$\scriptstyle {\mathrm {curl} (\mu c\mathbf {VuVEu} -\mathbf {E} )=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(11)}}}$

or

$\scriptstyle {\mathrm {curl} \left[{\frac {u^{2}}{v^{2}}}\left(\mathbf {E} -E_{3}\mathbf {k} \right)-\mathbf {E} \right]=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(12)}}}$

where E₃ is the z-component of E and k a unit vector along z; or, integrating, and writing the three components,

$\scriptstyle {E_{1}=-{\frac {dP}{dx}},\quad E_{2}=-{\frac {dP}{dy}},\quad E_{3}=-\left(1-{\frac {u^{2}}{v^{2}}}\right){\frac {dP}{dz}},\quad \cdot \cdot {\text{(13)}}}$

where P is a scalar potential. Here is the new point. There is a potential, of a peculiar kind. The displacement due to the moving charge is distributed in precisely the same way as if it were at rest in an eolotropic medium, whose permittivity is c in all directions transverse to the line of motion, but is smaller, viz., c(1-u²/v²), along that line and parallel to it. The potential P is given by

$\scriptstyle {P={\frac {q}{c\left[\left(x^{2}+y^{2}\right)\left(1-u^{2}/v^{2}\right)+z^{2}\right]^{\frac {1}{2}}}}.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(14)}}}$

It is a particular case of eolotropy. In general, c₁, c₂, c₃, the principal permittivities, are all unequal. Then, with q at the origin, the potential is

$\scriptstyle {P={\frac {q}{\left(c_{1}c_{2}c_{3}\right)^{\frac {1}{2}}\left({\frac {x^{2}}{c_{1}}}+{\frac {y^{2}}{c_{2}}}+{\frac {z^{2}}{c_{3}}}\right)^{\frac {1}{2}}}}.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot {\text{(15)}}}$

Observe that although the electric force in the substituted problem of a charge at rest in an eolotropic medium is the slope of a potential; yet it is not so when the medium is isotropic, and moves past the fixed charge, or vice versâ, although the distributions of displacement are the same.

When u=v, we abolish the permittivity along the z-axis in the substituted case, so that the displacement must be wholly transverse. We then have the plane electromagnetic wave. When u is greater than v it makes the permittivity negative along z; this is an impossible electrical problem, and furnishes another reason for supposing that there can be no steady state in the corresponding electromagnetic problem.

It now remains to find what would happen if electrification were conveyed through a medium faster than the natural speed of propagation of disturbances. There is the cone; but what takes place within it?

⁠Aug. 25, 1889.