A metal wire of $1 \mathrm{~cm}$ length and $1 \mathrm{~mm}$ radius has a resistance of $3 \times 10^{-3}…

A metal wire of $1 \mathrm{~cm}$ length and $1 \mathrm{~mm}$ radius has a resistance of $3 \times 10^{-3} \Omega$. If a wire of the same metal of length $3 \mathrm{~cm}$ and radius $0.5 \mathrm{~mm}$ is drawn, the resistance of the wire.
  1. $0.036 \Omega$
  2. $0.09 \Omega$
  3. $1.2 \Omega$
  4. $3.1 \Omega$

Solution

length of metal wire, $l_1=1 \mathrm{~cm}=10^{-2} \mathrm{~m}$ $ \begin{aligned} & r_1=1 \mathrm{~mm}=10^{-3} \mathrm{~m} \\ & R_1=3 \times 10^{-3} \Omega \end{aligned} $ Resistivity of wire is given as $ \begin{aligned} \rho & =R \frac{A}{l} . \\ & =3 \times 10^{-3} \times \frac{\pi r_1^2}{l_1}=3 \times 10^{-3} \times \frac{3.14 \times\left(10^{-3}\right)^2}{10^{-2}} \\ & =9.42 \times 10^{-7} \Omega-\mathrm{m} \end{aligned} $ Again for second case, $ \begin{aligned} l_2= & 3 \mathrm{~cm}=3 \times 10^{-2} \mathrm{~m}, r_2=0.5 \mathrm{~mm}=5 \times 10^{-4} \mathrm{~m} \\ \therefore R_2 & =\rho \frac{l_2}{A_2}=\rho \frac{l_2}{\pi r_2^2} \\ & =9.42 \times 10^{-7} \times \frac{3 \times 10^{-2}}{3.14 \times\left(5 \times 10^{-4}\right)^2} \\ & =0.36 \times 10^{-1} \Omega \\ & =0.036 \Omega \end{aligned} $

Asked in: AP EAMCET 2022 (06 Jul Shift 2)

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