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Question: The resistivity of a steel wire is \(1 \times {10^{ - 7}}\Omega - m\) . The resistance of a steel wi...

The resistivity of a steel wire is 1×10−7Ω−m1 \times {10^{ - 7}}\Omega - m . The resistance of a steel wire of a particular length and thickness is 1Ω1\Omega . If the length and diameter of the wire both are doubled, then the resistivity in Ω−m\Omega - m will be
A. 1×10−71 \times {10^{ - 7}}
B. 2×10−72 \times {10^{ - 7}}
C. 4×10−74 \times {10^{ - 7}}
D. 8×10−78 \times {10^{ - 7}}ṣ$

Explanation

Solution

Resistivity is a proportionality constant and it is represented by ρ\rho , it depends on the nature of material of the conductor whereas resistance is the opposition offered by a conductor to the flow of electrons through it and depends on the dimensions of the conductor.

Complete step by step answer:
Resistance of a conductor is directly proportional to its length and inversely proportional to its area of cross-section.
R=ρlAR = \rho \dfrac{l}{A}
Where ρ\rho is a proportionality constant called resistivity or specific resistance of the conductor.
Now we know that, R=mlne2τR = \dfrac{{ml}}{{n{e^2}\tau }}
Where m=m = the mass of electron, l=l = length of conductor, n=n = number of free electrons per unit volume, e=e = charge on an electron and τ=\tau = average relaxation time
On comparing the above two equations, we get
ρlA=mlne2τA   ⟹  ρ=mne2τ  \rho \dfrac{l}{A} = \dfrac{{ml}}{{n{e^2}\tau A}} \\\ \implies \rho = \dfrac{m}{{n{e^2}\tau }} \\\
We see that the resistivity is a characteristic property and doesn’t depend on the dimensions of the conductor.
Thus, the resistivity of wire will not change by doubling the length of the wire and the diameter.

So, the correct answer is “Option A”.

Additional Information:
Resistance is the total opposition offered by a conductor to the flow of current through it. It is the ratio of potential difference applied across the conductor to the current flowing through it.
R=VIR = \dfrac{V}{I}
Where V is the applied potential difference and I is the current passing through the conductor.
Due to the potential difference applied across the conductor, free electrons drift towards the positive end of the conductor but on their way these electrons collide with the fixed positive ions. These collisions become the cause of resistance.

Note:
Resistivity depends on the nature of the material of the conductor, so it can be changed by changing the temperature. On increasing the temperature of the conductor, its resistivity increases while on decreasing the temperature resistivity also decreases. Some materials become superconductors when their temperature falls below a certain temperature called critical temperature. Thus, resistivity doesn’t change on changing the dimensions of the conductor but can be changed on changing the temperature.