57.

A certain mechanical force is exerted on the substance of the conductor by the electromagnetic field considered. We shall calculate the component of this force normal to the surface. It is partly of electric, partly of magnetic, origin. Let us first consider the former, ${𝖥}_e$. Since the electric charge existing on the surface of the conductor is in an electric field, a mechanical force equal to the product of the charge and the field-strength is exerted on it. Since, however, the field-strength is discontinuous, having the value $-2\sin \theta f$ on the side of the vacuum and $0$ on the side of the conductor, from a well-known law of electrostatics the magnitude of the mechanical force ${𝖥}_e$ acting on an element of surface $d\sigma$ of the conductor is obtained by multiplying the electric charge of the element of area calculated in [eqn:(60)] (60) by the arithmetic mean of the electric field-strength on the two sides. Hence ${𝖥}_e=\frac{\sin \theta }{2\pi }fd\sigma (-\sin \theta f)=-\frac{{\sin }^2\theta }{2\pi }{f}^{2}d\sigma .$ This force acts in the direction toward the vacuum and therefore exerts a tension.