A wire of length L and diameter D has a resistance R. If BOTH the length and the diameter of the same wire are decreased to half of t

Question

A wire of length L and diameter D has a resistance R.
If BOTH the length and the diameter of the same wire are decreased to half of their original values, the resistivity of the new wire:

A. doubles.
B. drops to half its original value.
C. quadruples
D. drops to one quarter its original value.
E. remains the same.

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Thu Thủy 4 years 2021-08-28T10:22:22+00:00 1 Answers 12 views 0

Answers ( )

  1. Thunguyet
    0
    2021-08-28T10:23:37+00:00
    Reply

    There seems to be a typo in the question. The question asks to find the resistivity, instead of resistance. Resistivity is a property of the wire and does not change when its physical dimensions change.

    Answer:

    (A) doubles

    Explanation:

    The resistivity of a wire is defined by

    \rho=\dfrac{RA}{l}

    R is the resistance, A is the cross-sectional area and l is the length of the wire.

    R = \dfrac{\rho l}{A}

    It follows that

    R\propto\dfrac{l}{A}

    Assuming a circular cross section, the area is given by

    A =\pi \dfrac{D^2}{4}

    where D is the diameter.

    Then

    R\propto\dfrac{l}{D^2}

    R=k\dfrac{l}{D^2}

    k is an arbitrary constant

    When l and D are halved, the new resistance, R1 is

    R_1=k\dfrac{0. 5l}{(0.5D)^2} =2k\dfrac{l}{D^2} = 2R

    Hence, the resistance is doubled.

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