Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/90

action of the magnetized shell on a point within it is always opposed to that of the external magnetic force whether the shell be paramagnetic or diamagnetic. The actual value of the resultant potential within the shell is

$(c_{i}+A_{1})S_{i}r^{i},\,$

or

$(1+4\pi \kappa )(2i+1)^{3}N_{i}C_{i}S_{i}r.\,$

(13)

432.] When κ is a large number, as it is in the case of soft iron, then, unless the shell is very thin, the magnetic force within it is but a small fraction of the external force.

In this way Sir W. Thomson has rendered his marine galvanometer independent of external magnetic force by enclosing it in a tube of soft iron.

433.] The case of greatest practical importance is that in which i = 1. In this case

$N_{1}={\frac {1}{9(1+4\pi \kappa )+2(4\pi \kappa )^{2}\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{3}\right)}},$

(14)

${\begin{aligned}A_{1}&=-2(4\pi \kappa )^{2}\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{3}\right)N_{1}C_{1},\\A_{2}&=-4\pi \kappa \left[3+8\pi \kappa \left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{3}\right)\right]N_{1}C_{1},\\B_{2}&=12\pi \kappa a_{1}^{3}N_{1}C_{1},\\B_{3}&=4\pi \kappa (3+8\pi \kappa )(a_{2}^{3}-a_{1}^{3})N_{1}C_{1}.\end{aligned}}$

(15)

The magnetic force within the hollow shell is in this case uniform and equal to

$C_{1}+A_{1}={\frac {9(1+4\pi \kappa )}{9(1+4\pi \kappa )+2(4\pi \kappa )^{2}\left(1-\left({\frac {a_{1}}{a_{2}}}\right)^{3}\right)}}C_{1}.$

(16)

If we wish to determine κ by measuring the magnetic force within a hollow shell and comparing it with the external magnetic force, the best value of the thickness of the shell may be found from the equation

$1-{\frac {a_{1}^{3}}{a_{2}^{3}}}={\frac {9}{2}}{\frac {1+4\pi \kappa }{(4\pi \kappa )^{2}}}.$

(17)

The magnetic force inside the shell is then half of its value outside.

Since, in the case of iron, κ is a number between 20 and 30, the thickness of the shell ought to be about the hundredth part of its radius. This method is applicable only when the value of κ is large. When it is very small the value of A_l becomes insensible, since it depends on the square of κ.