Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/258

representing the element of the circuit and the magnetic induction, and the multiplication is to be understood in the Hamiltonian sense.

603.] If the conductor is to be treated not as a line but as a body, we must express the force on the element of length, and the current through the complete section, in terms of symbols denoting the force per unit of volume, and the current per unit of area.

Let X, Y, Z now represent the components of the force referred to unit of volume, and u, v, w those of the current referred to unit of area. Then, if S represents the section of the conductor, which we shall suppose small, the volume of the element ds will be S ds, and $u={\frac {i_{2}}{S}}{\frac {dx}{ds}}$. Hence, equation (7) will become

${\frac {XSds}{ds}}=S(vc-wb),$

(10)

or

$\left.{\begin{aligned}X&=vc-wb,\\Y&=wa-uc,\\Z&=ub-va.\end{aligned}}\right\}{\begin{matrix}{\text{(Equations of}}\\{\text{Electromagnetic}}\\{\text{Force.)}}\end{matrix}}$

(C)

Here X, Y, Z are the components of the electromagnetic force on an element of a conductor divided by the volume of that element; u, v, w are the components of the electric current through the element referred to unit of area, and a, b, c are the components of the magnetic induction at the element, which are also referred to unit of area.

If the vector ${\mathfrak {F}}$ represents in magnitude and direction the force acting on unit of volume of the conductor, and if ${\mathfrak {C}}$ represents the electric current flowing through it,

${\mathfrak {F}}=V.{\mathfrak {C\,B}}.$

(11)