Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/130

the west. The resultant force will make an angle 6 with the magnetic meridian, measured towards the west, and such that

$R=H\tan \theta .(4)$

(4)

Hence, to determine ${\frac {R}{H}}$ we proceed as follows :—

The direction of the magnetic north having been ascertained, a magnet, whose dimensions should not be too great, is suspended as in the former experiments, and the deflecting magnet $M$ is placed so that its centre is at a distance $r$ from that of the suspended magnet, in the same horizontal plane, and due magnetic east.

The axis of $M$ is carefully adjusted so as to be horizontal and in the direction of $r$.

The suspended magnet is observed before $M$ is brought near and also after it is placed in position. If is the observed deflexion, we have, if we use the approximate formula (1),

${\frac {M}{H}}={\frac {r^{3}}{2}}\tan \theta ;$

(5)

or, if we use the formula (3),

${\frac {1}{2}}{\frac {H}{M}}r^{3}\tan \theta =1+A_{1}{\frac {1}{r}}+A_{2}{\frac {1}{r^{2}}}+\&{\text{c.}}$

(6)

Here we must bear in mind that though the deflexion $\theta$ can be observed with great accuracy, the distance $r$ between the centres of the magnets is a quantity which cannot be precisely deter mined, unless both magnets are fixed and their centres defined by marks.

This difficulty is overcome thus :

The magnet $M$ is placed on a divided scale which extends east and west on both sides of the suspended magnet. The middle point between the ends of $M$ is reckoned the centre of the magnet. This point may be marked on the magnet and its position observed on the scale, or the positions of the ends may be observed and the arithmetic mean taken. Call this $s_{1}$, and let the line of the suspension fibre of the suspended magnet when produced cut the scale at $s_{0}$ , then $r_{1}=s_{1}-s_{0}$ where $s_{1}$ is known accurately and $s_{0}$ approximately. Let $\theta _{1}$ be the deflexion observed in this position of $M$.

Now reverse $M$, that is, place it on the scale with its ends reversed, then $r$ will be the same, but $M$ and $A_{1}$,$A_{3}$,$\&{\text{c.}}$ will have their signs changed, so that if $\theta _{2}$ is the deflexion,

$-{\frac {1}{2}}{\frac {H}{M}}{r_{1}}^{3}\tan \theta _{2}=1-A_{1}{\frac {1}{r_{1}}}+A_{2}{\frac {1}{{r_{1}}^{2}}}-\&{\text{c.}}$

(7)