58.

We shall now calculate the mechanical force of magnetic origin ${𝖥}_m$. In the interior of the conducting substance there are certain conduction currents, whose intensity and direction are determined by the vector $𝖨$ of the current density $𝖨=\frac{c}{4\pi }\mathrm{\backslash curl}𝖧.\mathrm{\backslash Label}[eqn](61)\text{upshape (61)}$ A mechanical force acts on every element of space $d\tau$ of the conductor through which a conduction current flows, and is given by the vector product $\frac{d\tau }{c}[𝖨\times 𝖧].\mathrm{\backslash Label}[eqn](62)\text{upshape (62)}$

Hence the component of this force normal to the surface of the conductor $x=0$ is equal to $\frac{d\tau }{c}({𝖨}_y{𝖧}_z-{𝖨}_z{𝖧}_y).$ On substituting the values of ${𝖨}_y$ and ${𝖨}_z$ from [eqn:(61)] (61) we obtain $\frac{d\tau }{4\pi }[{𝖧}_z(\frac{\partial {𝖧}_x}{\partial z}-\frac{\partial {𝖧}_z}{\partial x})-{𝖧}_y(\frac{\partial {𝖧}_y}{\partial x}-\frac{\partial {𝖧}_x}{\partial y})].$ In this expression the differential coefficients with respect to $y$ and $z$ are negligibly small in comparison to those with respect to $x$, according to the remark at the end of [sect:56.] Sec. 56; hence the expression reduces to $-\frac{d\tau }{4\pi }({𝖧}_y\frac{\partial {𝖧}_y}{\partial x}+{𝖧}_z\frac{\partial {𝖧}_z}{\partial x}).$ Let us now consider a cylinder cut out of the conductor perpendicular to the surface with the cross-section $d\sigma$, and extending from $x=0$ to $x=\infty$. The entire mechanical force of magnetic origin acting on this cylinder in the direction of the $x$-axis, since $d\tau =d\sigma x$, is given by ${𝖥}_m=-\frac{d\sigma }{4\pi }{\int }_0^{\infty }dx({𝖧}_y\frac{\partial {𝖧}_y}{\partial x}+{𝖧}_z\frac{\partial {𝖧}_z}{\partial x}).$ On integration, since $𝖧$ vanishes for $x=\infty$, we obtain ${𝖥}_m=\frac{d\sigma }{8\pi }{({𝖧}_y^2+{𝖧}_z^2)}_{x=0}$ or by equation [eqn:(59)] (59) ${𝖥}_m=\frac{d\sigma }{2\pi }({\cos }^2\theta ·g^2+f^2).$

By adding ${𝖥}_e$ and ${𝖥}_m$ the total mechanical force acting on the cylinder in question in the direction of the $x$-axis is found to be $𝖥=\frac{d\sigma }{2\pi }{\cos }^2\theta (f^2+g^2).\mathrm{\backslash Label}[eqn](63)\text{upshape (63)}$ This force exerts on the surface of the conductor a pressure, which acts in a direction normal to the surface toward the interior and is

called "Maxwell's radiation pressure." The existence and the magnitude of the radiation pressure as predicted by the theory was first found by delicate measurements with the radiometer by P. Lebedew.P. Lebedew, Annalen d. Phys. 6, p. 433, 1901. See also E. F. Nichols and G. F. Hull, Annalen d. Phys. 12, p. 225, 1903.