A copper wire and an iron wire, each having an area of cross section $A$ and lengths $L_1$ and $L_2$ are joined end to end. The copper end is maintained at a potential $V_1$ and the iron end at lower potential $V_2$. If ${\sigma }_1$ and ${\sigma }_2$ are the conductivities of copper and iron respectively, the potential of the junction will be

In a conductor, if the number of conduction electrons per unit volume is $8.5\times {10}^{28}{\mathrm{m}}^{-3}$ and mean free time is $25f\mathrm{s}$ (femtosecond), its approximate resistivity is: $\left({\mathrm{m}}_{\mathrm{e}}=9.1\times {10}^{-31}\mathrm{k}\mathrm{g}\right)$

Model a torch battery of length $l$ to be made up of a thin cylindrical bar of radius $a$ and a concentric thin cylindrical shell of radius $b$ filled in between with an electrolyte of resistivity $\rho$ (see figure). If the battery is connected to a resistance of value $R$, the maximum Joule heating in $R$ will take place for :

If the last band on the carbon resistor is absent, then the tolerance is