User:Henceforth1212/sandbox

Maxwell's equations are derived using Faraday's induction experiment. The equations that form Maxwell's electric curl equation are derived using an internal electric field (F, G, H) formed within the conduction wire.

"PART III. – GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD......

Electromagnetic Momentum (F, G, H).

(57) Let F, G, H represent the components of electromagnetic momentum at any point of the field, due to any system of magnets or currents." (Maxwell1, Part III).

"If the circuit be the boundary of the elementary area dydz, then its electromagnetic momentum is

(dH/dy − dG/dz)dydz..........................23

and this is the number of lines of magnetic force which pass through the area dydz." (Maxwell1, Part III).

......................................................

According to Simpson, the "elementary area dydz" represents the area formed by a wire loop and H and G represents an internal electric field (F, G, H) that forms the induction current by an external magnetic flux (α, β, γ).

"the elementary area dydz" Maxwell now very helpfully (at last!) takes a simplified case (Fig. 4.11) in which the circuit is a rectangle perpendicular to the x-direction. Recall that F, G, and H are the electromagnetic momenta per unit length of the circuit. As we travel around the loop in a counter clockwise direction to determine the total electromagnetic momentum............Our integration of the electromagnetic momentum around the loop thus becomes at the same time a measurement of the number of lines of magnetic force through it." (Simpson, p. 323-324).

....................................................................

Maxwell's electric curl equation is derived using Faraday's induction effect represented with the magnetic flux ((α, β, γ),

"Magnetic Force (α, β, γ).

(59) Let α, β , γ represent the force acting on a unit magnetic pole placed at the given point resolved in the directions of x , y , and z." (Maxwell, Part III).

"(61) Expressing the electric momentum of small circuits perpendicular to the three axes in this notation, we obtain the following

Equations of Magnetic Force. 7

3

5

6

6

6

________________________________________________________________________________________________

Maxwell's magnetic curl equation is derived using the electric displacement,

"Electrical Displacements (f, g, h).

(55) Electrical displacement consists in the opposite electrification of the sides of a molecule or particle of a body which may or may not be accompanied with transmission through the body. Let the quantity of electricity which would appear on the faces dy. dz of an element dx, dy , dz cut from the body be f. dy. dz, then f is the component of electric displacement parallel to x. We shall use f, g, h to denote the electric displacements parallel to x, y , z respectively.

The variations of the electrical displacement must be added to the currents p, q , r to get the total motion of electricity, which we may call p′ , q′ , r′, so that

p′ = p + df/ dt.............................................34

q′ = q + dg/dt...............................................35

r′ = r + dh/dt"  (Maxwell, Part III).........................36

.........................................................................

"Hence if there is no electric current,

dγ/dy − dβ/dz = 0..................23

but if there is a current p′

.

.

.

.

We may call these the Equations of Currents." (Maxwell, Part III).

_________________________________________________________________________________________

"1. The induction of electric currents by the increase or diminution of neighbouring currents according to the changes in the lines of force passing through the circuit.

2. The distribution of magnetic intensity according to the variations of a magnetic potential.

3. The induction (or influence) of statical electricity through dielectrics." (Maxwell, Part III)