Page:Motion of Electrification through a Dielectric.djvu/13

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516

ELECTRICAL PAPERS.

possibility, in connexion with the structure of the ether, that is not in question.^[1]

A Charged Straight Line moving in its own Line.

16. Let us now derive from (29), or from (27), the results in some cases of distributed electrification, in steady rectilinear motion. The integrations to be effected being all of an elementary character, it is not necessary to give the working.

First, let a straight line AB be charged to linear density q, and be in motion at speed u in its own line from left to right. Then at P we shall have

$\scriptstyle {A=qu\ \log \left({\frac {r_{1}}{r_{2}}}\cdot {\frac {\mu _{1}+\left(1-\nu _{1}^{2}u^{2}/v^{2}\right)^{\frac {1}{2}}}{\mu _{2}+\left(1-\nu _{2}^{2}u^{2}/v^{2}\right)^{\frac {1}{2}}}}\right),\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(39)}}}$

from which H=-dA/dh gives

$\scriptstyle {H=qu\left(1-{\frac {u^{2}}{v^{2}}}\right)\left[{\frac {\nu _{1}}{r_{1}\left(1-\nu _{1}^{2}u^{2}/v^{2}\right)+r_{1}\mu _{1}\left(1-\nu _{1}^{2}u^{2}/v^{2}\right)^{\frac {1}{2}}}}-\mathrm {same} \ f^{n}\ \mathrm {of} \ r_{2},\mu _{2},\nu _{2}\right],~{\text{(40)}}}$

where μ=cos θ, ν = sin θ.

When P is vertically over B, and A is at an infinite distance, we shall find

$\scriptstyle {H=qu/h,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(41)}}}$

which is one half the value due to an infinitely long (both ways) straight current of strength qu. The notable thing is the independence of the ratio u/v.

↑ [The difficulty about the above method and solution (29) is that it is not explicit enough when u>v, and does not indicate the limits of application. It gives a real solution for the hinder cone, a real solution for the forward cone, and an unreal solution in the rest of space, but we have no instruction to reject the part for the forward cone and the unreal part, nor have we any means of testing that the remainder, confined to the hinder cone, is the proper solution, viz., by the test of divergence, to give the right amount of electrification. The integral displacement comes to -∞. Now this may require to be supplemented by +∞+q on the boundary of the cone, but we have no way of testing it.

But certain considerations led me to the conclusion that the problem of u>v was really quite as definite a one as that of u<v, and that a correct method of a general character (independent of the magnitude of u) would show this explicitly. I therefore (in 1890) attacked the problem from a different point of view, employing the method of resistance-operators (or an equivalent method). Form the complete differential equation D=φu, connecting the displacement D associated with a moving point-charge with its velocity u, which is any function of the time t. Here φ is a differential operator, a function of p or d/dt. The solution of this equation gives D explicitly in terms of u, whether steady or variable, and its structure indicates the limits of application.

Taking u=constant, we obtain the result (29) when u<v. But when u>v, the formula tells us to exclude all space except the hinder cone, and that in it, the solution is not (29), but double as much. That is, double the right member of the first of (29) when u>v. The boundary of the cone is also a displacement sheet. The displacement is to the charge in the cone, and from the charge on its surface. Being so near the end of the second volume, I regret that there is no space here for the mathematical investigation, which cannot be given in a few words, and must be reserved.]

[1] [The difficulty about the above method and solution (29) is that it is not explicit enough when u>v, and does not indicate the limits of application. It gives a real solution for the hinder cone, a real solution for the forward cone, and an unreal solution in the rest of space, but we have no instruction to reject the part for the forward cone and the unreal part, nor have we any means of testing that the remainder, confined to the hinder cone, is the proper solution, viz., by the test of divergence, to give the right amount of electrification. The integral displacement comes to -∞. Now this may require to be supplemented by +∞+q on the boundary of the cone, but we have no way of testing it.

But certain considerations led me to the conclusion that the problem of u>v was really quite as definite a one as that of u<v, and that a correct method of a general character (independent of the magnitude of u) would show this explicitly. I therefore (in 1890) attacked the problem from a different point of view, employing the method of resistance-operators (or an equivalent method). Form the complete differential equation D=φu, connecting the displacement D associated with a moving point-charge with its velocity u, which is any function of the time t. Here φ is a differential operator, a function of p or d/dt. The solution of this equation gives D explicitly in terms of u, whether steady or variable, and its structure indicates the limits of application.

Taking u=constant, we obtain the result (29) when u<v. But when u>v, the formula tells us to exclude all space except the hinder cone, and that in it, the solution is not (29), but double as much. That is, double the right member of the first of (29) when u>v. The boundary of the cone is also a displacement sheet. The displacement is to the charge in the cone, and from the charge on its surface. Being so near the end of the second volume, I regret that there is no space here for the mathematical investigation, which cannot be given in a few words, and must be reserved.]

[1]