The length of a given cylindrical wire is increased by $100 \%$. Due to the consequent decrease in diameter, the change in the resistance of the wire will be .................. $\%$

The resistances of a carbon filament lamp and a tungsten filament lamp are measured individually when the lamps are lit and compared with their respective resistances when cold. Which one of the following statements will be true?

The resistance of a wire is $5\, \Omega$ at $50\, ^\circ C$ and $6\, \Omega$ at $100\, ^\circ C$. The resistance of the wire at $0\, ^\circ C$ will be .............. $\Omega$.

Consider a block of conducting material of resistivity $\rho$ shown in the figure. Current $I$ enters at $A$ and leaves from $D$. We apply the superposition principle to find the voltage $\Delta V$ developed between $B$ and $C$. The calculation is done in the following steps:
$(i)$ Take current $I$ entering from $A$ and assume it to spread over a hemispherical surface in the block.
$(ii)$ Calculate field $E(r)$ at distance $r$ from $A$ by using Ohm's law $E = \rho j$,where $j$ is the current per unit area at $r$.
$(iii)$ From the $r$ dependence of $E(r)$,obtain the potential $V(r)$ at $r$.
$(iv)$ Repeat $(i)$,$(ii)$ and $(iii)$ for current $I$ leaving $D$ and superpose results for $A$ and $D$.
For current entering at $A$,the electric field at a distance $r$ from $A$ is

$A$ material '$B$' has twice the specific resistance of '$A$'. $A$ circular wire made of '$B$' has twice the diameter of a wire made of '$A$'. Then,for the two wires to have the same resistance,the ratio $\frac{l_B}{l_A}$ of their respective lengths must be

$A$ and $B$ are two square plates of the same metal and same thickness,but the length of $B$ is twice that of $A$. The ratio of the resistances of $A$ and $B$ is