Nichrome wire of length 10 m is used to make a heating coil. This coil consumes power of 160 W when there is a potential difference of 40 V across it. Find the diameter of this wire. Resistivity of Nichrome is \( 10^{-6}\,\Omega\,\mathrm{m} \).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1.128 mm
The power fixes the resistance, \( R = V^2/P \); the resistance then gives the area through \( R = \rho l/A \), and the area gives the diameter.
\( \displaystyle R = \frac{V^{2}}{P} = \frac{(40\,\mathrm{V})^{2}}{160\,\mathrm{W}} = 10\,\Omega \)
\( \displaystyle A = \frac{\rho\,l}{R} = \frac{10^{-6}\,\Omega\,\mathrm{m} \times 10\,\mathrm{m}}{10\,\Omega} = 10^{-6}\,\mathrm{m^2} \)
\( \displaystyle A = \frac{\pi d^{2}}{4} \;\Rightarrow\; d = \sqrt{\frac{4A}{\pi}} = \sqrt{\frac{4\times10^{-6}}{3.1416}} = 1.128\times10^{-3}\,\mathrm{m} \)
\( d = 1.128\,\mathrm{mm} \)