Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/265

 Rh electromagnetic phenomena depend, is not the same thing as $$\mathfrak{K}$$, the current of conduction, but that the time-variation of $$\mathfrak{D}$$, the electric displacement, must be taken into account in estimating the total movement of electricity, so that we must write, or, in terms of the components, {{numb form | $$ \left. \begin{align} u = p + \frac{df}{dt}, \\ v = q + \frac{dg}{dt}, \\ w = r + \frac{dh}{dt}. \end{align} \right\} $$|(H*)}}

611.] Since both $$\mathfrak{K}$$ and $$\mathfrak{D}$$ depend on the electromotive force $$\mathfrak{E}$$, we may express the true current & in terms of the electromotive force, thus or, in the case in which C and K are constants, {{numb form | $$ \left. \begin{align} u = CP + \frac{1}{4\pi} K \frac{dP}{dt}, \\ v = CQ + \frac{1}{4\pi} K \frac{dQ}{dt}, \\ w = CR + \frac{1}{4\pi} K \frac{dR}{dt}. \end{align} \right\} $$|(I*)}}

612.] The volume-density of the free electricity at any point is found from the components of electric displacement by the equation

613.] The surface-density of electricity is where l, m, n are the direction-cosines of the normal drawn from the surface into the medium in which f, g, h are the components of the displacement, and l', m', n' are those of the normal drawn from the surface into the medium in which they are f', g', h'.

614.] When the magnetization of the medium is entirely induced by the magnetic force acting on it, we may write the equation of induced magnetization, where μ is the coefficient of magnetic permeability, which may be considered a scalar quantity, or a linear and vector function operating on $$\mathfrak{H}$$, according as the medium is isotropic or not.