Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/101

from the ends. When κ is infinite the free magnetism at any point of the cylinder is simply proportional to its distance from the middle point, the distribution being- similar to that of free electricity on a conductor in a field of uniform force.

440.] In all substances except iron, nickel, and cobalt, the coefficient of magnetization is so small that the induced magnetization of the body produces only a very slight alteration of the forces in the magnetic field. We may therefore assume, as a first approximation, that the actual magnetic force within the body is the same as if the body had not been there. The superficial magnetization of the body is therefore, as a first approximation, $-\kappa {\frac {dV}{d\nu }}$, where ${\frac {dV}{d\nu }}$ is the rate of increase of the magnetic potential due to the external magnet along a normal to the surface drawn inwards. If we now calculate the potential due to this superficial distribution, we may use it in proceeding to a second approximation.

To find the mechanical energy due to the distribution of magnetism on this first approximation we must find the surface-integral

$E=\iint {\kappa V{\frac {dV}{d\nu }}dS}$

taken over the whole surface of the body. Now we have shewn in Art. 100 that this is equal to the volume-integral

$E=-\iiint {\kappa \left({\frac {\overline {dV}}{dx}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dy}}{\Bigg |}^{2}+{\frac {\overline {dV}}{dz}}{\Bigg |}^{2}\right)dxdydz},$

taken through the whole space occupied by the body, or, if R is the resultant magnetic force,

$E=-\iiint {\kappa R^{2}dxdydz}.$

Now since the work done by the magnetic force on the body during a displacement δx is Xδx where X is the mechanical force in the direction of x, and since

$\int {X\delta x}+E={\text{constant}},$

$X=-{\frac {dE}{dx}}={\frac {d}{dx}}\iiint {\kappa R^{2}dxdydz}=\iiint {\kappa {\frac {d\cdot R^{2}}{dx}}R^{2}dxdydz}$

which shews that the force acting on the body is as if every part of it tended to move from places where R² is less to places where it is greater with a force which on every unit of volume is

$\kappa {\frac {d\cdot R^{2}}{dx}}.$