Question

A heater coil is to be constructed with a nichrome wire (ρ = 1.0 × 10⁻⁶ Ωm) that can operate at 500 W when connected to a 250 V supply. (a) What would be the resistance of the coil? (b) If the cross-sectional area of the wire is 0.5 mm², what length of the wire will be needed? (c) If the radius of each turn is 4.0 mm, how many turns will be there in the coil?

Answer 1

(a) Let R be the resistance of the coil.
The power P consumed by a coil of resistance R when connected across a supply V is given by
$$\begin{gathered} P=\frac{V^2}{R} \\ \Rightarrow R=\frac{V^2}{P} \\ \Rightarrow R=\frac{{\left(250\right)}^2}{500}=125\mathrm{\Omega } \end{gathered}$$

(b) We know:
$$\begin{gathered} R=\mathrm{\rho }\frac{l}{A} \\ \Rightarrow l=\frac{RA}{\mathrm{\rho }} \\ \Rightarrow l=\frac{125\times 0.5\times {10}^{-6}}{{10}^{-6}}=62.5\mathrm{m} \end{gathered}$$

(c) Let n be the number of turns in the coil. Then,
$$\begin{gathered} l=2\mathrm{\pi }rn \\ \mathit{\Rightarrow }n\mathit{=}\frac{\mathit{l}}{\mathit{2}\mathit{\pi }\mathit{r}} \\ \mathit{\Rightarrow }n\mathit{=}\frac{\mathit{62}\mathit{.}\mathit{5}}{\mathit{2}\mathit{\times }\mathit{3}\mathit{.}\mathit{14}\mathit{\times }\mathit{4}\mathit{\times }{\mathit{10}}^{\mathit{-}\mathit{3}}}\mathit{\approx }2500 \end{gathered}$$