$A$ wire of length $10 \ cm$ is connected to a cell of $emf$ $2 \ V$ and negligible internal resistance. The resistance of the wire is $3 \ \Omega$. The value of the resistance required to obtain a potential gradient of $1 \ mV/cm$ is ................ $\Omega$.

$A$ resistance of $R \; \Omega$ draws current from a potentiometer. The potentiometer has a total resistance $R_{0} \; \Omega$ (Figure). $A$ voltage $V$ is supplied to the potentiometer. Derive an expression for the voltage across $R$ when the sliding contact is in the middle of the potentiometer.

The reading of the voltmeter in the following circuit is ................ $V$.

In a potentiometer experiment,when three cells $A, B$ and $C$ are connected in series,the balancing length is found to be $420 \ cm$. If cells $A$ and $B$ are connected in series,the balancing length is $220 \ cm$ and for cells $B$ and $C$ connected in series,the balancing length is $320 \ cm$. The emf of cells $A, B$ and $C$ are respectively in the ratio of:

The given figure represents an arrangement of a potentiometer for the calculation of the internal resistance $(r)$ of an unknown battery $(E)$. The balance length is $70.0 \, cm$ with the key open and $60.0 \, cm$ with the key closed. $R$ is $132.40 \, \Omega$. The internal resistance $(r)$ of the unknown cell will be ....... $\Omega$ (Given $E_o > E$):-

Two cells having unknown e.m.f.s $E_{1}$ and $E_{2}$ $(E_{1} > E_{2})$ are connected in a potentiometer circuit so as to assist each other. The null point is obtained at $490 \ cm$ from the higher potential end. When cell $E_{2}$ is connected so as to oppose cell $E_{1}$,the null point is obtained at $90 \ cm$ from the same end. The ratio of the e.m.f.s of the two cells $(\frac{E_{1}}{E_{2}})$ is: