Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/320

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288

PARALLEL CURRENTS.

[684.

and outside the tube,

\beta =2{\frac {C}{r}}

,

(13)

the same as when the current flows through a solid wire.

684.] The magnetic induction at any point is $b=\mu \beta$ , and since, by equation (2),

⁠	$b$	${}=-{\frac {dH}{dr}}$ ,	(14)
	$H$	${}=-\int \mu \beta \,dr$ .	(15)

The value of $H$ outside the tube is

A-2\mu _{0}C\log r

,

(16)

where $\mu _{0}$ is the value of $\mu$ , in the space outside the tube, and $A$ is a constant, the value of which depends on the position of the return current. In the substance of the tube,

H=A-2\mu _{0}C\log a_{1}+{\frac {\mu C}{a_{1}^{2}-a_{2}^{2}}}\left(a_{1}^{2}-r^{2}+2a_{2}^{2}\log {\frac {r}{a_{1}}}\right)

.

(17)

In the space within the tube $H$ is constant, and

H=A-2\mu _{0}C\log a_{1}+\mu C\left(1+{\frac {2a_{2}^{2}}{a_{1}^{2}-a_{2}^{2}}}\log {\frac {a_{2}}{a_{1}}}\right)

.

(18)

685.] Let the circuit be completed by a return current, flowing in a tube or wire parallel to the first, the axes of the two currents being at a distance $b$ . To determine the kinetic energy of the system we have to calculate the integral

T={\frac {1}{2}}\iiint Hw\,dx\,dy\,dz

.

(19)

If we confine our attention to that part of the system which lies between two planes perpendicular to the axes of the conductors, and distant $l$ from each other, the expression becomes

T={\frac {1}{2}}l\iint Hw\,dx\,dy

.

(20)

If we distinguish by an accent the quantities belonging to the return current, we may write this

{\frac {2T}{l}}=\iint Hw^{\prime }\,dx^{\prime }\,dy^{\prime }+\iint H^{\prime }w\,dx\,dy+\iint Hw\,dx\,dy+\iint H^{\prime }w^{\prime }\,dx^{\prime }\,dy^{\prime }

.

(21)

Since the action of the current on any point outside the tube is the same as if the same current had been concentrated at the axis of the tube, the mean value of $H$ for the section of the return current is $A-2\mu _{0}C\log b$ , and the mean value of $H^{\prime }$ for the section of the positive current is $A^{\prime }-2\mu _{0}C^{\prime }\log b$ .