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Question: A circular conductor is made of a uniform wire of resistance \(2\times {10^{−3}}\) ohm per metre and...

A circular conductor is made of a uniform wire of resistance 2×10−32\times {10^{−3}} ohm per metre and the diameter of this circular conductor is 2 metres. Then the resistance measured between the ends of the diameter is (in ohms)
A. π×10−3\pi \times {10^{ - 3}}
B. 2π×10−32\pi \times {10^{ - 3}}
C. 4π×10−34\pi \times {10^{ - 3}}
D. 4×10−34 \times {10^{ - 3}}

Explanation

Solution

Let us first understand the concept of resistance and resistivity following which we can substitute the values in formula as shown below. So basically resistance is the property of the conductor which opposes the flow of electric current. It is the ratio of the voltage applied to the electric current flowing through it whereas resistivity is defined as the resistance offered by the material per unit length for unit cross-section.

Complete step by step answer:
Resistance is defined as R=ρLAR = \dfrac{{\rho L}}{A}
where
\rho = \text{resistivity} \\\ \Rightarrow L = \text{length} \\\ \Rightarrow A = \text{cross-sectional area} \\\
The resistance per unit length is given to be
R′=2×10−3R' = 2 \times {10^{ - 3}}Ohm per metre.
So the length can be calculated as
L=2πr ⇒L=πDL = 2\pi r \\\ \Rightarrow L = \pi D
where DD is given a diameter of 2 m. Hence L= 2π metres.
Therefore,
Resistance=2×10−3×2π ⇒Resistance=4π×10−3 \text{Resistance} = 2 \times {10^{ - 3}} \times 2\pi \\\ \Rightarrow \text{Resistance}= 4\pi \times {10^{ - 3}} \\\
Now when the resistance is measured between ends of any diameter, it can be seen as a parallel connection of two resistances R/2 each.
Solving we get
1R=2R+2R ⇒R=R4 ∴R=π×10−3 ohms\dfrac{1}{R} = \dfrac{2}{R} + \dfrac{2}{R} \\\ \Rightarrow R = \dfrac{R}{4} \\\ \therefore R =\pi \times {10^{ - 3}}\,ohms
So the resistance between two endpoints of a diameter will be R/4 i.e. π×10−3ohms\pi \times {10^{ - 3}}ohms.

Hence, option A is correct.

Note: The electrical resistance of a wire is greater for a longer wire and less for a wire of larger cross sectional area. As the temperature increases, the resistance of pure metals increases while the resistance of insulators decreases. Resistivity of conductors < Resistivity of alloys < Resistivity of insulators.