Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/388

 

Comparison of a Coefficient of Self-induction with a Coefficient of Mutual Induction.

Fig. 62. 756.] In the branch $AF$ of Wheatstone's Bridge let a coil be inserted, the coefficient of self-induction of which we wish to find. Let us call it $L$.

In the connecting wire between $A$ and the battery another coil is inserted. The coefficient of mutual induction between this coil and the coil in $AF$ is $M$. It may be measured by the method described in Art. 755.

If the current from $A$ to $F$ is $x$, and that from $A$ to $H$ is $y$, that from $Z$ to $A$, through $B$, will be $x+y$. The external electromotive force from $A$ to $F$ is

$A-F=Px+L{\frac {dx}{dt}}+M\left({\frac {dx}{dt}}+{\frac {dy}{dt}}\right)$.

(9)

The external electromotive force along $AH$ is

$A-H=Qy$.

(10)

If the galvanometer placed between $F$ and $H$ indicates no current, either transient or permanent, then by (9) and (10), since $H-F=0$,

$Px=Qy$;

(11)

and

$L{\frac {dx}{dt}}+M\left({\frac {dx}{dt}}+{\frac {dy}{dt}}\right)=0$,

(12)

whence

$L=-\left(1+{\frac {P}{Q}}\right)M$.

(13)

Since $L$ is always positive, $M$ must be negative, and therefore the current must flow in opposite directions through the coils placed in $P$ and in $B$. In making the experiment we may either begin by adjusting the resistances so that

$PS=QR$,

(14)

which is the condition that there may be no permanent current, and then adjust the distance between the coils till the galvanometer ceases to indicate a transient current on making and breaking the battery connexion; or, if this distance is not capable of adjustment, we may get rid of the transient current by altering the resistances $Q$ and $S$ in such a way that the ratio of $Q$ to $S$ remains constant.

If this double adjustment is found too troublesome, we may adopt