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2 January, 08:18

A wire of resistivity ρ must be replaced in a circuit by a wire of the same material but four times as long. If, however, the total resistance is to remain as before, the diameter of the new wire mustA - be four times the original diameter. B - be one-fourth the original diameter. C - be the same as the original diameter. D - be one-half the original diameter. E - be two times the original diameter.

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  1. 2 January, 08:25
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    E. two times the original diameter

    Explanation:

    Resistance of a wire is:

    R = ρ L/A

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

    For a round wire with diameter d:

    R = ρ L / (¼ π d²)

    The two wires must have the same resistance, so:

    ρ₁ L₁ / (¼ π d₁²) = ρ₂ L₂ / (¼ π d₂²)

    The wires are made of the same material, so ρ₁ = ρ₂:

    L₁ / (¼ π d₁²) = L₂ / (¼ π d₂²)

    The new length is four times the old, so 4 L₁ = L₂:

    L₁ / (¼ π d₁²) = 4 L₁ / (¼ π d₂²)

    1 / (¼ π d₁²) = 4 / (¼ π d₂²)

    Solving:

    1 / (d₁²) = 4 / (d₂²)

    (d₂²) / (d₁²) = 4

    (d₂ / d₁) ² = 4

    d₂ / d₁ = 2

    So the new wire must have a diameter twice as large as the old wire.
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