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27 December, 20:11

The lengths and diameters of three wires are given below. Wire Length Diameter Wire 1 0.5 m 0.5 cm Wire 2 2 m 0.25 cm Wire 3 1 m 1 cm If they all have the same resistance, find the ratio of their resistivities.

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  1. 27 December, 20:17
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    The ratio reisistivity of the wires = 15.72 : 1 : 31.44

    Explanation:

    ρ = RA/L ... Equation 1

    Where ρ = resistivity of the wire, R = resistance, A = cross sectional area, L = length of the wire.

    Note: All the resistance are the same.

    For the wire 1,

    L = 0.5 m, A = πd²/4, d = 0.5 cm = 0.005 m, π = 3.143

    A = 3.143 (0.005) ²/4 = 0.0000196 m²

    ρ₁ = R (0.0000196) / 0.5

    ρ₁ = 0.0000393R

    For Wire 2,

    L = 2 m, d = 0.25 cm = 0.0025 m

    A = πd²/4 = 3.143 (0.0025) ²/4

    A = 0.0000049 m²

    ρ₂ = R (0.0000049) / 2

    ρ₂ = 0.0000025R

    For Wire 3

    L = 1 m, d = 1 cm = 0.01 m

    A = πd²/4 = 3.143 (0.01) ²/4

    A = 0.0000786 m²

    ρ₃ = R (0.0000786) / 1

    ρ₃ = 0.0000786R

    Therefore,

    ρ₁ : ρ₂ : ρ₃ = 0.0000393R : 0.0000025R : 0.0000786R

    ρ₁ : ρ₂ : ρ₃ = 3.93 : 0.25 : 7.86

    ρ₁ : ρ₂ : ρ₃ = 15.72 : 1 : 31.44

    Thus the ratio reisistivity of the wires = 15.72 : 1 : 31.44
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