Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/147

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463.]

joule's suspension.

115

The magnetization of the magnet is then reversed so that the end $B$ dips, the magnetic moment is ascertained, and eight observations are taken in this state, and the sixteen observations combined to determine the true dip.

462.] It is found that in spite of the utmost care the dip, as thus deduced from observations made with one dip circle, differs perceptibly from that deduced from observations with another dip circle at the same place. Mr. Broun has pointed out the effect due to ellipticity of the bearings of the axle, and how to correct it by taking observations with the magnet magnetized to different strengths.

The principle of this method may be stated thus. We shall suppose that the error of any one observation is a small quantity not exceeding a degree. We shall also suppose that some unknown but regular force acts upon the magnet, disturbing it from its true position.

If $L$ is the moment of this force, $\theta _{0}$ the true dip, and $\theta$ the observed dip, then

L=MI\sin(\theta -\theta _{0})

,

(17)

{}=MI(\theta -\theta _{0})

,

(18)

since $\theta -\theta _{0}$ is small.

It is evident that the greater $M$ becomes the nearer does the needle approach its proper position. Now let the operation of taking the dip be performed twice, first with the magnetization equal to $M_{1}$ , the greatest that the needle is capable of, and next with the magnetization equal to $M_{2}$ , a much smaller value but sufficient to make the readings distinct and the error still moderate. Let $\theta _{1}$ and $\theta _{2}$ be the dips deduced from these two sets of observations, and let $L$ be the mean value of the unknown disturbing force for the eight positions of each determination, which we shall suppose the same for both determinations. Then

L=M_{1}I(\theta _{1}-\theta _{0})=M_{2}I(\theta _{2}-\theta _{0})

.

(19)

Hence

\theta _{0}={\frac {M_{1}\theta _{1}-M_{2}\theta _{2}}{M_{1}-M_{2}}}

,

L=M_{1}M_{2}I{\frac {\theta _{1}-\theta _{2}}{M_{2}-M_{1}}}

.

(20)

If we find that several experiments give nearly equal values for $L$ , then we may consider that $\theta _{0}$ must be very nearly the true value of the dip.

463.] Dr. Joule has recently constructed a new dip-circle, in which the axis of the needle, instead of rolling on horizontal agate planes, is slung on two filaments of silk or spider s thread, the ends

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