Question

Which combination of statements is correct?
i) If the electrical resistivity of metal $X$ is lower than that of metal $Y$ , it means metal $X$ is better conductor of electricity than metal $Y$.
ii) If a wire of resistance $~8\Omega ~$ is doubled on itself, the resistance of the new wire will be $2\Omega$.
iii) The coils of electric toasters and irons are made of an alloy rather than a pure metal because resistivity of the alloy is higher than that of its constituent metals.
iv) If a conductor of length $65~cm~$ has a resistance of $4\Omega$, the same conductor of length $260~cm~$ will have the resistance of .
v) The maximum amount of current that can pass through a conductor is $440~A$.
A) (i), (iii), (v)
B) (i), (ii), (iii), (iv)
C) (iii), (iv), (v)
D) (iii) and (iv) only

Answer 1

In this question, keep in mind that resistivity is the property of material. It is only one of others factor who decides the resistance of a conductor resistance of a conductor also depends on its length and cross section area. Resistance is directly proportional to resistivity, and length of conductor while inversely proportional to area of cross section of conductor.

Formula used:
$R=\rho \dfrac{l}{A}$
Where $R$ resistance, $\rho$ is resistivity, $l$ is length, $A$ is cross section area of conductor.

Complete step by step solution:
If electrical resistivity of metal $X$ is lower than that of metal $Y$ , it doesn’t conclude that metal $X$ has less resistance than metal $Y$ we are not able to calculate their resistance without length and area of cross section of conductor, so, we can’t say metal $X$ is better conductor of electricity than metal $Y$ .
If a wire of resistance $~8\Omega ~$ is doubled on itself. Then its cross section area also changes but its volume remains the same. So,
$\Rightarrow \pi {{r}^{2}}l=\pi r_{2}^{2}{{l}_{2}}$
Where $r$ and $l$ are the radius of cross section and length before stretching, ${{r}_{2}}$ and ${{l}_{2}}$ are radius of cross section and length after stretching.
If, ${{l}_{2}}=2l$,
$\Rightarrow {{r}^{2}}l=r_{2}^{2}2l$
$\Rightarrow {{r}_{2}}=\dfrac{1}{\sqrt{2}}r$
And we know resistivity is material property so it doesn’t change.
So, $\Rightarrow \dfrac{R}{{{R}_{2}}}=\dfrac{\dfrac{l}{A}}{\dfrac{{{l}_{2}}}{{{A}_{2}}}}$
Where ${{R}_{2}}$is the resistance after stretching.
Put the values from above equations and solve for ${{R}_{2}}$

$\Rightarrow {{R}_{2}}=4R$
If $R=8$
Then, $\Rightarrow {{R}_{2}}=32\Omega$

The coils of electric toasters and irons are made of an alloy rather than a pure metal because the resistivity of the alloy is higher than that of its constituent metals. Because for more heating or heat generated due to resistance we need more resistance so it is true that we use alloy to maximize the resistance of iron.
We know that $R\propto l$
If a conductor of length $65~cm~$ has a resistance of $4\Omega$, then for conductor of $260cm$ is
$\Rightarrow \dfrac{4}{65}=\dfrac{{{R}_{2}}}{260}$
$\Rightarrow {{R}_{2}}=16\Omega$

The current on the conductor is dependent on resistance and terminal voltage across the conductor.
$V=IR$
It hasn’t any maximum value.

Hence, Option (D) is correct.

Note: Resistance is directly proportional to resistivity, and length of conductor while inversely proportional to area of cross section of conductor. Resistivity is material property. If we stretch a wire then its length increases but its cross section area decreases.