A Treatise on Electricity and Magnetism/Part IV/Chapter XI

by James Clerk Maxwell

CHAPTER XI.

ON ENERGY AND STRESS IN THE ELECTROMAGNETIC FIELD.

Electrostatic Energy.

630.] The energy of the system may be divided into the Potential Energy and the Kinetic Energy.

The potential energy due to electrification has been already considered in Art. 85. It may be written where $e$ is the charge of electricity at a place where the electric

potential is $\Psi$, and the summation is to be extended to every place where there is electrification.

If $f$, $g$, $h$ are the components of the electric displacement, the quantity of electricity in the element of volume $dx\,dy\,dz$ iswhere the integration is to be extended throughout all space.

631.] Integrating this expression by parts, and remembering that when the distance, $r$, from a given point of a finite electrified system becomes infinite, the potential $\Psi$ becomes an infinitely small

quantity of the order $r^{-1}$, and that $f$, $g$, $h$ become infinitely small quantities of the order $r^{-2}$, the expression is reduced to where the integration is to be extended throughout all space. If we now write $P$, $Q$, $R$ for the components of the electromotive force, instead of $-{\frac {d\Psi }{dx}}$, $-{\frac {d\Psi }{dy}}$, and $-{\frac {d\Psi }{dz}}$, we find

Hence, the electrostatic energy of the whole field will be the same if we suppose that it resides in every part of the field where electrical force and electrical displacement occur, instead of being confined to the places where free electricity is found.

The energy in unit of volume is half the product of the electromotive force and the electric displacement, multiplied by the cosine of the angle which these vectors include.

In Quaternion language it is $-{\frac {1}{2}}S{\mathfrak {E}}{\mathfrak {D}}$.

Magnetic Energy.

632.] We may treat the energy due to magnetization in a similar way. If $A$, $B$, $C$ are the components of magnetization and $\alpha$, $\beta$, $\gamma$ the components of magnetic force, the potential energy of the system of magnets is, by Art. 389, the integration being extended over the space occupied by magnetized matter. This part of the energy, however, will be included in the kinetic energy in the form in which we shall presently obtain it.

633.] We may transform this expression when there are no electric currents by the following method.

We know that Hence, by Art. 97, if as is always the case in magnetic phenomena where there are no currents,the integral being extended throughout all space, or

Hence, the energy due to a magnetic system

$-{\frac {1}{2}}\iiint (A\alpha +B\beta +C\gamma )\,dx\,dy\,dz$	${}={\frac {1}{8\pi }}\iiint (\alpha ^{2}+\beta ^{2}+\gamma ^{2})\,dx\,dy\,dz$,
	${}=-{\frac {1}{8\pi }}\iiint {\mathfrak {H}}^{2}\,dx\,dy\,dz$.

(11)

Electrokinetic Energy.

634.] We have already, in Art. 578, expressed the kinetic energy

of a system of currents in the form.where $p$ is the electromagnetic momentum of a circuit, and $i$ is the strength of the current flowing round it, and the, summation extends to all the circuits. But we have proved, in Art. 590, that $p$ may be expressed as a line-integral of the form where $F$, $G$, $H$ are the components of the electromagnetic momentum, $A$, at the point ($x\,y\,z$) and the integration is to be extended round the closed circuit $s$. We therefore find If $u$, $v$, $w$ are the components of the density of the current at any point of the conducting circuit, and if $S$ is the transverse section of the circuit, then we may write and we may also write the volumeand we now find where the integration is to be extended to every part of space where there are electric currents.

635.] Let us now substitute for $u$, $v$, $w$ their values as given by the equations of electric currents (E), Art. 607, in terms of the components $\alpha$, $\beta$, $\gamma$ of the magnetic force. We then have where the integration is extended over a portion of space including all the currents.

If we integrate this by parts, and remember that, at a great distance $r$ from the system, $\alpha$, $\beta$, and $\gamma$ are of the order of magnitude $r^{-3}$ , we find that when the integration is extended throughout all space, the expression is reduced to

By the equations (A), Art. 591, of magnetic induction, we may substitute for the quantities in small brackets the components of magnetic induction $a$, $b$, $c$, so that the kinetic energy may be written where the integration is to be extended throughout every part of space in which the magnetic force and magnetic induction have values differing from zero.

The quantity within brackets in this expression is the product of the magnetic induction into the resolved part of the magnetic force in its own direction.

In the language of quaternions this may be written more simply,where ${\mathfrak {B}}$ is the magnetic induction, whose components are $a$, $b$, $c$, and ${\mathfrak {H}}$ is the magnetic force, whose components are $\alpha$, $\beta$, $\gamma$.

636.] The electrokinetic energy of the system may therefore be expressed either as an integral to be taken where there are electric currents, or as an integral to be taken over every part of the field in which magnetic force exists. The first integral, however, is the natural expression of the theory which supposes the currents to act upon each other directly at a distance, while the second is appropriate to the theory which endeavours to explain the action between the currents by means of some intermediate action in the space between them. As in this treatise we have adopted the latter method of investigation, we naturally adopt the second expression as giving the most significant form to the kinetic energy.

According to our hypothesis, we assume the kinetic energy to exist wherever there is magnetic force, that is, in general, in every part of the field. The amount of this energy per unit of volume is $-{\frac {1}{8\pi }}S{\mathfrak {B}}{\mathfrak {H}}$, and this energy exists in the form of some kind of motion of the matter in every portion of space.

When we come to consider Faraday's discovery of the effect of magnetism on polarized light, we shall point out reasons for believing that wherever there are lines of magnetic force, there is a rotatory motion of matter round those lines. See Art. 821.

Magnetic and Electrokinetic Energy compared.

637.] We found in Art. 423 that the mutual potential energy of two magnetic shells, of strengths $\phi$ and $\phi ^{\prime }$ and bounded by the closed curves $s$ and $s^{\prime }$ respectively, is where $\epsilon$ is the angle between the directions of $ds$ and $ds^{\prime }$, and $r$ is the distance between them. We also found in Art. 521 that the mutual energy of two circuits $s$ and $s^{\prime }$, in which currents $i$ and $i^{\prime }$ flow, is

If $i$, $i^{\prime }$ are equal to $\phi$, $\phi ^{\prime }$ respectively, the mechanical action between the magnetic shells is equal to that between the corresponding electric circuits, and in the same direction. In the case of the magnetic shells, the force tends to diminish, their mutual potential energy, in the case of the circuits it tends to increase their mutual energy, because this energy is kinetic.

It is impossible, by any arrangement of magnetized matter, to produce a system corresponding in all respects to an electric circuit, for the potential of the magnetic system is single valued at every point of space, whereas that of the electric system is many- valued.

But it is always possible, by a proper arrangement of infinitely small electric circuits, to produce a system corresponding in all respects to any magnetic system, provided the line of integration which we follow in calculating the potential is prevented from passing through any of these small circuits. This will be more fully explained in Art. 833.

The action of magnets at a distance is perfectly identical with that of electric currents. We therefore endeavour to trace both to the same cause, and since we cannot explain electric currents by means of magnets, we must adopt the other alternative, and explain magnets by means of molecular electric currents.

638.] In our investigation of magnetic phenomena, in Part III of this treatise, we made no attempt to account for magnetic action at a distance, but treated this action as a fundamental fact of experience. We therefore assumed that the energy of a magnetic system is potential energy, and that this energy is diminished when the parts of the system yield to the magnetic forces which act on them.

If, however, we regard magnets as deriving their properties from electric currents circulating within their molecules, their energy is kinetic, and the force between them is such that it tends to move them in a direction such that if the strengths of the currents were maintained constant the kinetic energy would increase.

This mode of explaining magnetism requires us also to abandon the method followed in Part III, in which we regarded the magnet as a continuous and homogeneous body, the minutest part of which has magnetic properties of the same kind as the whole.

We must now regard a magnet as containing a finite, though very great, number of electric circuits, so that it has essentially a molecular, as distinguished from a continuous structure.

If we suppose our mathematical machinery to be so coarse that our line of integration cannot thread a molecular circuit, and that an immense number of magnetic molecules are contained in our element of volume, we shall still arrive at results similar to those of Part III, but if we suppose our machinery of a finer order, and capable of investigating all that goes on in the interior of the molecules, we must give up the old theory of magnetism, and adopt that of Ampère, which admits of no magnets except those which consist of electric currents.

We must also regard both magnetic and electromagnetic energy as kinetic energy, and we must attribute to it the proper sign, as given in Art. 635.

In what follows, though we may occasionally, as in Art. 639, &c., attempt to carry out the old theory of magnetism, we shall find that we obtain a perfectly consistent system only when we abandon that theory and adopt Ampère's theory of molecular currents, as in Art. 644.

The energy of the field therefore consists of two parts only, the electrostatic or potential energyand the electromagnetic or kinetic energy

ON THE FORCES WHICH ACT ON AN ELEMENT OF A BODY PLACED IN THE ELECTROMAGNETIC FIELD.

Forces acting on a Magnetic Element.

639.] The potential energy of the element $dx\,dy\,dz$ of a body magnetized with an intensity whose components are $A$, $B$, $C$, and placed in a field of magnetic force whose components are $\alpha$, $\beta$, $\gamma$, is Hence, if the force urging the element to move without rotation in the direction of $x$ is $X_{1}\,dx\,dy\,dz$,and if the moment of the couple tending to turn the element about the axis of $x$ from $y$ towards $z$ is $L\,dx\,dy\,dz$,

The forces and the moments corresponding to the axes of $y$ and $z$ may be written down by making the proper substitutions.

640.] If the magnetized body carries an electric current, of which the components are $u$, $v$, $w$, then, by equations C, Art. 603, there will be an additional electromagnetic force whose components are $X_{2}$, $Y_{2}$, $Z_{2}$, of which $X_{2}$ is Hence, the total force, $X$, arising from the magnetism of the molecule, as well as the current passing through it, is

The quantities $a$, $b$, $c$ are the components of magnetic induction, and are related to $\alpha$, $\beta$, $\gamma$, the components of magnetic force, by the equations given in Art. 400,

$a=\alpha +4\pi A$, $b=\beta +4\pi B$, $c=\gamma +4\pi C$.

(5)

The components of the current, $u$, $v$, $w$, can be expressed in terms of $\alpha$, $\beta$, $\gamma$ by the equations of Art. 607,

$4\pi u={\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}$, $4\pi v={\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}$, $4\pi w={\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}$.

(6)

Hence

$X$	${}={\frac {1}{4\pi }}\left\{(a-\alpha ){\frac {d\alpha }{dx}}+(b-\beta ){\frac {d\beta }{dx}}+(c-\gamma ){\frac {d\gamma }{dx}}+b\left({\frac {d\alpha }{dy}}-{\frac {d\beta }{dx}}\right)+c\left({\frac {d\alpha }{dx}}-{\frac {d\gamma }{dx}}\right)\right\}$,
	${}={\frac {1}{4\pi }}\left\{a{\frac {d\alpha }{dx}}+b{\frac {d\alpha }{dy}}+c{\frac {d\alpha }{dz}}-{\frac {1}{2}}{\frac {d}{dx}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right\}$.

(7)

Multiplying this equation, (8), by $\alpha$, and dividing by $4\pi$, we may add the result to (7), and we findwhere $X$ is the force referred to unit of volume in the direction of $x$, and $L$ is the moment of the forces about this axis.

On the Explanation of these Forces by the Hypothesis of a Medium in a State of Stress.

641.] Let us denote a stress of any kind referred to unit of area by a symbol of the form $P_{hk}$ , where the first suffix, ${}_{h}$, indicates that the normal to the surface on which the stress is supposed to act is parallel to the axis of $h$, and the second suffix, ${}_{k}$, indicates that the direction of the stress with which the part of the body on the positive side of the surface acts on the part on the negative side is parallel to the axis of $k$.

The directions of $h$ and $k$ may be the same, in which case the stress is a normal stress. They may be oblique to each other, in which case the stress is an oblique stress, or they may be perpendicular to each other, in which case the stress is a tangential stress.

The condition that the stresses shall not produce any tendency to rotation in the elementary portions of the body is

In the case of a magnetized body, however, there is such a tendency to rotation, and therefore this condition, which holds in the ordinary theory of stress, is not fulfilled.

Let us consider the effect of the stresses on the six sides of the elementary portion of the body $dx\,dy\,dz$, taking the origin of coordinates at its centre of gravity.

On the positive face $dy\,dz$, for which the value of $x$ is ${\frac {1}{2}}\,dx$, the forces are—

Parallel to $x$, ⁠ $\left(P_{xx}+{\frac {1}{2}}{\frac {dP_{xx}}{dx}}\,dx\right)\,dy\,dz=X_{+x}$, $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ Parallel to $y$, $\left(P_{xy}+{\frac {1}{2}}{\frac {dP_{xy}}{dx}}\,dx\right)\,dy\,dz=Y_{+x}$, Parallel to $z$, $\left(P_{xz}+{\frac {1}{2}}{\frac {dP_{xz}}{dx}}\,dx\right)\,dy\,dz=Z_{+x}$.

(12)

The forces acting on the opposite side, $-X_{-x}$, $-Y_{-x}$ and $-Z_{-x}$, may be found from these by changing the sign of $dx$. We may express in the same way the systems of three forces acting on each of the other faces of the element, the direction of the force being indicated by the capital letter, and the face on which it acts by the suffix.

If $X\,dx\,dy\,dz$ is the whole force parallel to $x$ acting on the element,

whence

$X\,dx\,dy\,dz$	${}=X_{+x}+X_{+y}+X_{+z}+X_{-x}+X_{-y}+X_{-z}$,
	${}=\left({\frac {dP_{xx}}{dx}}+{\frac {dP_{yx}}{dx}}+{\frac {dP_{zx}}{dx}}\right)\,dx\,dy\,dz$,
$X$	${}={\frac {d}{dx}}P_{xx}+{\frac {d}{dy}}P_{yx}+{\frac {d}{dz}}P_{zx}$.

(13)

If $L\,dx\,dy\,dz$ is the moment of the forces about the axis of $x$ tending to turn the element from $y$ to $z$,

whence

$L\,dx\,dy\,dz$	${}={\frac {1}{2}}\,dy(Z_{+y}-Z_{-y})-{\frac {1}{2}}\,dz(Y_{+z}-Y_{-z})$,
	${}=(P_{yz}-P_{zy})\,dx\,dy\,dz$,
$L$	${}=P_{yz}-P_{zy}$.

(14)

Comparing the values of $X$ and $L$ given by equations (9) and (11) with those given by (13) and (14), we find that, if we make

$P_{xx}={\frac {1}{4\pi }}\left(a\alpha -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$,		$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
$P_{yy}={\frac {1}{4\pi }}\left(b\beta -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$,
$P_{zz}={\frac {1}{4\pi }}\left(c\gamma -{\frac {1}{2}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\right)$.
$P_{yz}={\frac {1}{4\pi }}b\gamma$,	$P_{zy}={\frac {1}{4\pi }}c\beta$,
$P_{zx}={\frac {1}{4\pi }}c\alpha$,	$P_{xz}={\frac {1}{4\pi }}a\gamma$,
$P_{xy}={\frac {1}{4\pi }}a\beta$,	$P_{yz}={\frac {1}{4\pi }}b\alpha$,

(15)

the force arising from a system of stress of which these are the components will be statically equivalent, in its effects on each element of the body, with the forces arising from the magnetization and electric currents.

642.] The nature of the stress of which these are the components may be easily fouud, by making the axis of $x$ bisect the angle between the directions of the magnetic force and the magnetic induction, and taking the axis of $y$ in the plane of these directions, and measured towards the side of the magnetic force.

If we put ${\mathfrak {H}}$ for the numerical value of the magnetic force, ${\mathfrak {B}}$ for that of the magnetic induction, and $2\epsilon$ for the angle between their directions,

$\alpha ={\mathfrak {H}}\cos \epsilon$,	⁠	$\beta ={}$	${\mathfrak {H}}\sin \epsilon$,	⁠	$\gamma =0$,
$a={\mathfrak {B}}\cos \epsilon$,		$b=-{}$	${\mathfrak {B}}\sin \epsilon$,		$c=0$;

(16)

$P_{xx}={\frac {1}{4\pi }}($	${\mathfrak {BH}}\cos ^{2}\epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{yy}={\frac {1}{4\pi }}(-{}$	${\mathfrak {BH}}\sin ^{2}\epsilon -{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{zz}={\frac {1}{4\pi }}(-{\frac {1}{2}}{\mathfrak {H}}^{2})$,
$P_{yz}=P_{zx}=P_{zy}=P_{xz}=0$,
$P_{xy}={\frac {1}{4\pi }}{\mathfrak {BH}}\cos \epsilon \sin \epsilon$,
$P_{yx}=-{\frac {1}{4\pi }}{\mathfrak {BH}}\cos \epsilon \sin \epsilon$

(17)

Hence, the state of stress may be considered as compounded of—

(1) A pressure equal in all directions ${}={\frac {1}{8\pi }}{\mathfrak {H}}^{2}$.

(2) A tension along the line bisecting the angle between the directions of the magnetic force and the magnetic induction

(3) A pressure along the line bisecting the exterior angle between these directions ${}={\frac {1}{4\pi }}{\mathfrak {BH}}\sin ^{2}\epsilon$.

(4) A couple tending to turn every element of the substance in the plane of the two directions from the direction of magnetic induction to the direction of magnetic force ${}={\frac {1}{4\pi }}{\mathfrak {BH}}\sin 2\epsilon$.

When the magnetic induction is in the same direction as the magnetic force, as it always is in fluids and non-magnetized solids, then $\epsilon =0$, and making the axis of $x$ coincide with the direction of the magnetic force,

and the tangential stresses disappear.

The stress in this case is therefore a hydrostatic pressure ${\frac {1}{8\pi }}{\mathfrak {H}}^{2}$, combined with a longitudinal tension ${\frac {1}{4\pi }}{\mathfrak {BH}}$ along the lines of force.

643.] When there is no magnetization, ${\mathfrak {B}}={\mathfrak {H}}$, and the stress is still further simplified, being a tension along the lines of force equal to ${\frac {1}{8\pi }}{\mathfrak {H}}^{2}$, combined with a pressure in all directions at right angles to the lines of force, numerically equal also to ${\frac {1}{8\pi }}{\mathfrak {H^{2}}}$. The components of stress in this important case are

$P_{xx}$	${}={\frac {1}{8\pi }}(\alpha ^{2}-\beta ^{2}-\gamma ^{2})$,
$P_{yy}$	${}={\frac {1}{8\pi }}(\beta ^{2}-\gamma ^{2}-\alpha ^{2})$,
$P_{zz}$	${}={\frac {1}{8\pi }}(\gamma ^{2}-\alpha ^{2}-\beta ^{2})$,
$P_{yz}=P_{zy}$	${}={\frac {1}{4\pi }}\beta \gamma$,
$P_{zx}=P_{xz}$	${}={\frac {1}{4\pi }}\gamma \alpha$,
$P_{xy}=P_{yx}$	${}={\frac {1}{4\pi }}\alpha \beta$.

The force arising from these stresses on an element of the medium referred to unit of volume is

$X$	${}={\frac {d}{dx}}p_{xx}+{\frac {d}{dy}}p_{yx}+{\frac {d}{dz}}p_{zx}$,
	${}={\frac {1}{4\pi }}\left\{\alpha {\frac {d\alpha }{dx}}-\beta {\frac {d\beta }{dx}}-\gamma {\frac {d\gamma }{dx}}\right\}+{\frac {1}{4\pi }}\left\{\alpha {\frac {d\beta }{dy}}+\beta {\frac {d\alpha }{dy}}\right\}+{\frac {1}{4\pi }}\left\{\alpha {\frac {d\gamma }{dz}}-\gamma {\frac {d\alpha }{dz}}\right\}$,
	${}={\frac {1}{4\pi }}\alpha \left({\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}\right)+{\frac {1}{4\pi }}\gamma \left({\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right)-{\frac {1}{4\pi }}\beta \left({\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right)$.

Now

${\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}=4\pi m$, ${\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}=4\pi v$, ${\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}=4\pi w$,

where $m$ is the density of austral magnetic matter referred to unit of volume, and $v$ and $w$ are the components of electric currents referred to unit of area perpendicular to $y$ and $z$ respectively. Hence,

Similarly

$X=\alpha m+v\gamma -w\beta$ $Y=\beta m+w\alpha -u\gamma$, $Z=\gamma m+u\beta +v\alpha$.

(Equations of Electromagnetic Force.)

(20)

644.] If we adopt the theories of Ampère and Weber as to the nature of magnetic and diamagnetic bodies, and assume that magnetic and diamagnetic polarity are due to molecular electric currents, we get rid of imaginary magnetic matter, and find that everywhere $m=0$,and so that the equations of electromagnetic force become,

$X=v\gamma -w\beta$, $Y=w\alpha -u\gamma$, $Z=u\beta -v\alpha$.

(22)

These are the components of the mechanical force referred to unit of volume of the substance. The components of the magnetic force are $\alpha$, $\beta$, $\gamma$, and those of the electric current are $u$, $v$, $w$. These equations are identical with those already established. (Equations (C), Art, 603.)

645.] In explaining the electromagnetic force by means of a state of stress in a medium, we are only following out the conception of Faraday^[1], that the lines of magnetic force tend to shorten themselves, and that they repel each other when placed side by side. All that we have done is to express the value of the tension along the lines, and the pressure at right angles to them, in mathematical language, and to prove that the state of stress thus assumed to exist in the medium will actually produce the observed forces on the conductors which carry electric currents.

We have asserted nothing as yet with respect to the mode in which this state of stress is originated and maintained in the medium. We have merely shewn that it is possible to conceive the mutual action of electric currents to depend on a particular kind of stress in the surrounding medium, instead of being a direct and immediate action at a distance.

Any further explanation of the state of stress, by means of the motion of the medium or otherwise, must be regarded as a separate and independent part of the theory, which may stand or fall without affecting our present position. See Art. 832.

In the first part of this treatise, Art. 108, we shewed that the observed electrostatic forces may be conceived as operating through the intervention of a state of stress in the surrounding medium. We have now done the same for the electromagnetic forces, and it remains to be seen whether the conception of a medium capable of supporting these states of stress is consistent with other known phenomena, or whether we shall have to put it aside as s unfruitful.

In a field in which electrostatic as well as electromagnetic action is taking place, we must suppose the electrostatic stress described in Part I to be superposed on the electromagnetic stress which we have been considering.

646.] If we suppose the total terrestrial magnetic force to be 10 British units (grain, foot, second), as it is nearly in Britain, then the tension perpendicular to the lines of force is 0.128 grains weight per square foot. The greatest magnetic tension produced by Joule^[2] by means of electromagnets was about 140 pounds weight on the square inch.

1. Exp. Res., 3286, 3267, 3268.
2. Sturgeon's Annals of Electricity, vol. v. p. 187 (1840); or Philosophical Magazine, Dec., 1851.