The electrical resistance of an iron wire is $R$. If both its length and radius are doubled,then:

  • A
    The resistance will be halved and the resistivity will remain unchanged.
  • B
    The resistance will be halved and the resistivity will be doubled.
  • C
    Both resistance and resistivity will remain unchanged.
  • D
    The resistance will be doubled and the resistivity will be halved.

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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 $D$.
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