Question

Nichrome wire, often used for heating elements, has resistivity of 1.0 × 10-6 Ω ∙ m at room temperature. What length of No. 30 wire (of diameter 0.250 mm) is needed to wind a resistor that has 50 ohms at room temperature?

Answer 1

Answer:

Length = 2.453 m

Explanation:

Given:

Resistivity of the wire (ρ) = 1 × 10⁻⁶ Ω-m

Diameter of the wire (d) = 0.250 mm = 0.250 × 10⁻³ m

Resistance of the wire (R) = 50 Ω

Length of the wire (L) = ?

The area of cross section is given as:

$A=\frac{1}{4}\pi d^2\\\\A=\frac{1}{4}\times\ 3.14\times (0.250\times 10^{-3})^2\\\\A=0.785\times 6.25\times 10^{-8}\\\\A=4.906\times 10^{-8}\ m^2$

We know that, for a constant temperature, the resistance of a wire is directly proportional to its length and inversely proportional to its area of cross section. The constant of proportionality is called the resistivity of the wire. Therefore,

$R=\rho \frac{L}{A}$

Expressing the above in terms of length 'L', we get:

$L=\frac{RA}{\rho}$

Plug in the given values and solve for 'L'. This gives,

$L=\frac{50\times 4.906\times 10^{-8}}{1\times 10^{-6}}\ m\\\\L=\frac{2.453}{1}=2.453\ m$

Therefore, length of No. 30 wire (of diameter 0.250 mm) is 2.453 m.