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A conducting wire of uniform diameter was made from a piece of copper. If l and d are taken as the symbols of length and diameter of the wire respectively then for which l and d given below the resistance of the wire would be minimum?
1. l, d
2. 2l, d
3. l/2, 2d
4. 2l , d/2

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Correct Answer - Option 3 : l/2, 2d

Concept:

  • Resistivity (ρ): The property of a conductor that opposes the flow of electric current through them and independent of the shape and size of the materials but depends on the nature and temperature of the materials is called resistivity.
    • The unit for resistivity is the ohm-meter (Ω-m).
    • The resistivity of a material depends on its nature and the temperature of the conductor.
    • The resistivity of a material doesn't depend on its shape and size (length and area).
  • Resistance: The property of any conductor that opposes the flow of electric current through it and depends on the shape and size of the materials, temperature, and nature of the materials is called resistance.
    • It is denoted by R and the SI unit is the ohm (Ω).

The resistance is given by:

\(R = \rho\frac{l}{A}\)

where ρ is resistivity, L is the length and A is the area of the cross-section. 

Explanation:

The resistance of the copper wire ⇒\(R = \rho\frac{l}{A}\)

where ρ is the resistivity of the material of wire and is constant.

Area of wire A = πd2/4

So, \(R = k\frac{l}{d^2}\)  where, k = ρπ/4  = constant.

1) \(R_1 = k\frac{l}{d^2}\)

2) \(R_2 = k\frac{2l}{d^2}\)

3) \(R_3 = k\frac{l/2}{(2d)^2} = k \frac{l}{8d}\)

4) \(R_4 = k\frac{2l}{(d/2)^2} = k\frac{8l}{d} \)

So, for minimum resistance, the length and diameter will be l/2, 2d respectively.

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