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A wire of length $L$ and resistance $R$ is stretched so that length is doubled and area of cross-section is halved. How will (i) resistance change (ii) resistivity change?

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Last updated date: 20th Jun 2024
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Answer
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Hint: Use the relation between the resistance, length, area and resistivity of a material.

Formula used: $R=\dfrac{\rho L}{A}$
$R$ is the resistance
$L$ is the length
$A$ is the area of cross-section
$\rho$ is the resistivity of the material

Complete step by step solution:
If
$L'=2L$
$A'=\dfrac{A}{2}$
Then,
$R'=\dfrac{\rho L'}{A'}=\dfrac{\rho (2L)}{(A'/2)}=\dfrac{4\rho L}{A}=4R$
The answer is
(i) The resistance increases by 4 times if the length is doubled and the area is halved.

(ii) The resistivity remains unchanged. Resistivity is an intrinsic property of a material and it is not scalable.

Additional information: Superconductivity is a property of certain materials where they show zero resistance when cooled below a critical temperature. If current is passed through a loop of superconducting wire then the current can persist indefinitely even in the absence of any power source. It was discovered by Kamerlingh Onnes. It can be explained using quantum mechanics. Magnetic field lines are completely expelled from the interior of a superconductor when it transitions into the superconducting state. This is known as the Meissner effect. Some examples of superconductors include aluminium, niobium, magnesium diboride and iron pnictides. Superconductors are used to make extremely powerful electromagnets to accelerate charged particles to great speeds. They are also used in magnetic resonance imaging (MRI).

Note: Resistivity is the fundamental property of a material that describes how strongly it resists or conducts electric current. A low resistivity implies that current can easily flow through it. Conductors have low resistivity and insulators have high resistivity. Semiconductors have intermediate resistivity but resistivity can be decreased by suitable doping.