$A$ wire of length $10 \ m$ and resistance $30 \ \Omega$ is connected to a battery of $emf$ $2.5 \ V$ and internal resistance $5 \ \Omega$ through an external resistance $R$. If the potential gradient along the wire is $50 \ \mu V/mm$,then $R = $ ................. $\Omega$.

The potentiometer wire is $5 \ m$ long and a potential difference of $4 \ V$ is maintained between the ends. The e.m.f. of the cell which balances against a length of $200 \ cm$ of the potentiometer wire is: (in $V$)

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:

In the arrangement shown in the figure,when the switch $S_2$ is open,the galvanometer shows no deflection for $l = L/2$. When the switch $S_2$ is closed,the galvanometer shows no deflection for $l = 5L/12$. The internal resistance $(r)$ of the $6\, V$ cell and the $emf$ $E$ of the other battery are respectively:

$A$ $6 \, V$ battery of negligible internal resistance is connected across a uniform wire $AB$ of length $1 \, m$. The positive terminal of another battery of emf $4 \, V$ and internal resistance $1 \, \Omega$ is joined to point $A$ as shown in the figure. The ammeter shows zero deflection when the jockey touches the wire at point $C$. The length $AC$ is equal to:

In a potentiometer arrangement,$E_1$ is the cell establishing the current in the primary circuit,$E_2$ is the cell to be measured,$AB$ is the potentiometer wire,and $G$ is a galvanometer. Which of the following are the essential conditions for a balance point to be obtained?