ELECTRO-DYNAMICS.

By PERCIVAL FROST, M.A.

AMPERE, in his theory of Electro-Magnetics, claims the credit of having thrown over the hypotheses of philosophers, which contradicted Newton's law that Action and Reaction were opposite as well as equal; and his formula for the action of elements of currents upon one another, combined with his hypothesis that every magnet acts as a solenoid, meets all the cases of mutual action of voltaic currents and magnets which are at rest. But the main point, upon which he rests his assertion, although very distinctly stated by him, is passed over cursorily by most writers who reproduce his arguments, and seems to be scarcely appreciated by others. I have endeavoured in this paper to make in a simple manner the calculations which are sufficient to establish the mode of action of currents upon currents; and to shew the identity of the action of magnetic fluids, distributed according to Ampere's supposition, with the actions of currents upon the pole of a solenoid or magnet.

I have also pointed out an important error made or quoted by Wiedeman in his work on Magnetism, and taken that opportunity of illustrating the different methods of calculating the action of currents of a finite size upon magnets.

Total action of one small plane closed current upon another. 1. Let the plane of the circuit acted on be that of xy, the origin being a point 0 in the middle of the circuit (fig. 20).

Let the axis of y be parallel to the plane of the acting circuit, the normal to which is inclined at the angle a to Ox. ,,, the coordinates of a point C in the middle of the acting circuit.

,,, 0, those of an element ds at P, in the circuit 0. A, A' the areas of the circuits C and O

neglecting squares of small quantities;

then

fxdy,=X, fy,de, -,,

ds, being measured in the direction of the current,

also

fdx=0, fdy, =0, fxdx=0, fydy,=0.

the integrals being taken over the whole circuit.

The formula for the action of a current on an element resolved parallel to the axis of z, p being the perpendicular from O on the plane of λ, is

x

- x1) dy, — (y—y,) dx ̧

d =
dp

2.3

.

If dZ be the action on ds,, parallel to Oz, neglecting squares of small quantities

dZ= şiir ψιλ dp

therefore, for the action on the complete circuit '

+3

dp to

= ελλ'

dp

2

=

Σλλ'

dp dx dz

when is the potential of the area of the circuit & with respect to the point 0, for if C be the centre of gravity, and squares of small quantities be neglected, and §, n, be the coordinates of dA with reference to axes through C

2. It can easily be shown by way of testing these results, that the reversed action of these forces gives the action of the dronit at 0 on that at C, in a similar form.

This is obvious for the single forces, by the results quoted in the last number of the Journal, since the components depend upon an expression of the form

d2 dpdp

and the reversed couples about the axes are

L1 = − L - Yz + Zy,
M1 = − M− Zx + Xz,

N-N-Xy+ Yx;

L

therefore if L', M', N' be the components of the reversed couple round the axes of x', y', z', chosen with respect to O in the same manner as those of x, y, z are with respect to C, L, cosa - N1 N, sina

Whole action of an infinite solenoid upon a small plane

current.

3. Let the same axes be taken as before, let λ be the area of each of the elementary currents of the solenoid, and use the same notation as in Cor. 5 of the last number.

The action of the solenoid upon the current is compounded

of a single force

λλ' 1 Jü T 9

(1),

in the direction perpendicular to the plane of the circuit, and

whose axis is perpendicular to the plane containing the line joining the pole with the centre of the circuit, and the projection of this line on the plane of the circuit.

The action of the circuit on the solenoid consists of the reversed forces (1) and (2), for the couple vanishes, being compounded of the reversed couple (3), and the couple introduced by the transference of (1) to this pole, the moment of which is the same as that of (3).

4. Gauss' construction for the action of a small bar magnet on a particle of free magnetism, is represented by the expressions for the forces (1) and (2), for, let C (fig. 21) be the pole of the solenoid, O the small circuit, OD normal to it, CD perpendicular to OC, and OE=OD.

If OC represent the force (2), OCEO will repre

3z

sent force (1), and CE will represent the resultant action of O in direction and magnitude.

On the action and reaction of an element of a current and the pole of a solenoid.

5. From the above investigation of the actions of currents, it follows,

That it is correct to state, that the action of the pole of a solenoid upon an element of a current is a single force acting on the element perpendicular to the plane containing the element and the pole, according to Biot's law.

That it is incorrect to state that the action of the element on the pole is a single force acting at the pole, according to that law, since by Newton's law of Action and Reaction, that action must also be a force acting at the element.

That the correctness of the results obtained in many calculations, as in the case of a straight current, arises from a composition of two errors, the couples which are neglected in the part of the circuit considered, being counteracted by those in other portions of the closed current which are not considered in the calculations.