$A$ cable consists of a copper wire of radius $9 \, mm$ having a resistance of $5 \, \Omega$. If this cable is replaced by $6$ copper wires each of radius $3 \, mm$,what will be the new resistance of the cable in $\Omega$?

$A$ cell sends a current through a resistance $R_1$ for time $t$. Now,the same cell sends a current through a resistance $R_2$ for the same time $t$. If the heat produced in both cases is the same,then the internal resistance of the cell is:

Given below are two statements:
Statement $I$: $A$ uniform wire of resistance $80\,\Omega$ is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be $5\,\Omega$.
Statement $II$: Two resistances $2\,R$ and $3\,R$ are connected in parallel in an electric circuit. The value of thermal energy developed in $3\,R$ and $2\,R$ will be in the ratio $3:2$.
In the light of the above statements,choose the most appropriate answer from the options given below.

$A$ battery of $emf$ $E$ is being charged from a charger such that the positive terminal of the battery is connected to terminal $A$ of the charger and the negative terminal of the battery is connected to terminal $B$ of the charger. The internal resistance of the battery is $r$.

Which of the following graphs represents the variation of power loss in the external load with external resistance $R$?

One junction of a certain thermoelectric couple is at a fixed temperature $T_r$ and the other junction is at temperature $T$. The thermo electromotive force for this is expressed by $E = K(T - T_r) \left[ T_0 - \frac{1}{2}(T + T_r) \right]$. At temperature $T = \frac{1}{2}T_0$,the thermoelectric power is