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Question: A copper wire of cross-sectional area \(3.4m{{m}^{2}}\) and length of the wire \(400m\), specific re...

A copper wire of cross-sectional area 3.4mm23.4m{{m}^{2}} and length of the wire 400m400m, specific resistivity of copper is 1.7×10−8Ωm1.7\times {{10}^{-8}}\Omega m. The resistance of the wire is
(A). 20Ω20\Omega
(B). 200Ω200\Omega
(C). 2Ω2\Omega
(D). 2kΩ2k\Omega

Explanation

Solution

An element opposes the flow of current through it f=due to its resistance. Resistance in circuit along with the potential difference at the ends of circuit determines the current flowing through the circuit. The resistance of a material depends on the type of material as well as its dimensions.

Formula used:
R=ρlAR=\dfrac{\rho l}{A}

Complete step-by-step solution:
The property of a material by virtue of which it opposes the flow of current through it is called its resistance. Its SI unit is Ω\Omega .

According to Ohm’s law, resistance is given by
R=VIR=\dfrac{V}{I}
Here, RR is the resistance
VV is the potential difference applied at the ends
II is current flowing between the ends

Resistance depends on the type of material as well as the dimensions. Its is also given by-
R=ρlAR=\dfrac{\rho l}{A} - (1)
Here, ρ\rho is the resistivity of the material
ll is the length
AA is the area of cross section

Given, a wire has specific resistivity 1.7×10−8Ωm1.7\times {{10}^{-8}}\Omega m, its length is 400m400m and area is 3.4mm23.4m{{m}^{2}}. Substituting given values in eq (1), we get,
R=1.7×10−8×4003.4×10−6 ⇒R=200×10−2Ω ∴R=2Ω \begin{aligned} & R=\dfrac{1.7\times {{10}^{-8}}\times 400}{3.4\times {{10}^{-6}}} \\\ & \Rightarrow R=200\times {{10}^{-2}}\Omega \\\ & \therefore R=2\Omega \\\ \end{aligned}
The resistance of the wire is 2Ω2\Omega .

Therefore, the resistance of the wire is 2Ω2\Omega . Hence, the correct option is (C).

Note:
The specific resistance or resistivity is defined as the resistance of an element of unit length and unit area of cross section. It is a constant quantity and is different for different materials. It depends on temperature. Copper is a metal, hence it is a very low resistance. The resistance in metals is lowest, in semiconductors, the resistance is more than conductors while in insulators, the resistance is the highest.