Q147 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Physics
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.
- Correct answerA$0.036 \Omega$
- B$0.09 \Omega$
- C$1.2 \Omega$
- D$3.1 \Omega$
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,
$$
\begin{aligned}
l_2= & 3 \mathrm{~cm}=3 \times 10^{-2} \mathrm{~m}, r_2=0.5 \mathrm{~mm}=5 \times 10^{-4} \mathrm{~m} \\
\therefore R_2 & =\rho \frac{l_2}{A_2}=\rho \frac{l_2}{\pi r_2^2} \\
& =9.42 \times 10^{-7} \times \frac{3 \times 10^{-2}}{3.14 \times\left(5 \times 10^{-4}\right)^2} \\
& =0.36 \times 10^{-1} \Omega \\
& =0.036 \Omega
\end{aligned}
$$
Hence, the correct answer is option: 'A'.