Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/113

As X increases, the number of molecules belonging to the cone about B continually diminishes, and when X becomes equal to D all the molecules have been wrenched out of their former positions of equilibrium, and have been forced into the fringe of the cone round A, so that when X becomes greater than D all the molecules form part of the cone round A or of its fringe.

When the force X is removed, then in the case in which X is less than L everything returns to its primitive state. When X is between L and D then there is a cone round A whose angle

$AOP=\theta _{0}+\beta _{0},\,$

and another cone round B whose angle

$BOQ=\theta _{0}-\beta _{0}.\,$

Within these cones the axes of the molecules are distributed uniformly. But all the molecules, the original direction of whose axes lay outside of both these cones, have been wrenched from their primitive positions and form a fringe round the cone about A.

If X is greater than D, then the cone round B is completely dispersed, and all the molecules which formed it are converted into the fringe round A, and are inclined at the angle θ₀ + β₀.

445.] Treating this case in the same way as before, we find for the intensity of the temporary magnetization during the action of the force X, which is supposed to act on iron which has never before been magnetized,

When X is less than L,

$I={\frac {2}{3}}M{\frac {X}{D}}.$

When X is equal to L,

$I={\frac {2}{3}}M{\frac {L}{D}}.$

When X is between L and D

$I=M\left\{{\frac {2}{3}}{\frac {X}{D}}+\left(1-{\frac {L^{2}}{X^{2}}}\right)\left[{\sqrt {1-{\frac {L^{2}}{D^{2}}}}}-{\frac {2}{3}}{\sqrt {{\frac {X^{2}}{D^{2}}}-{\frac {L^{2}}{D^{2}}}}}\right]\right\}.$

When X is equal to D,

$I=M\left\{{\frac {2}{3}}+{\frac {1}{3}}\left(1-{\frac {L^{2}}{X^{2}}}\right)^{\frac {3}{2}}\right\}.$

When X is greater than D,

$I=M\left\{{\frac {1}{3}}{\frac {X}{D}}+{\frac {1}{2}}-{\frac {1}{6}}{\frac {D}{X}}+{\frac {(D^{2}-L^{2})^{\frac {3}{2}}}{6X^{2}D}}-{\frac {\sqrt {X^{2}-L^{2}}}{6X^{2}D}}(2X^{2}-3XD+L^{2})\right\}.$

When X is infinite, I = M.

When X is less than L the magnetization follows the former law, and is proportional to the magnetizing force. As soon as X exceeds L the magnetization assumes a more rapid rate of increase