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Chapter 9: Problem 56

A heater is being designed that uses a coil of 14-gauge nichrome wire to generate \(300 \mathrm{W}\) using a voltage of \(V=110 \mathrm{V} .\) How long should the engineer make the wire?

Short Answer

Expert verified
The engineer should make the 14-gauge nichrome wire approximately 76.16 meters long to generate 300W using a voltage of 110V.

Step by step solution

01

Write down the given values and formulas

Given values: Power (P) = 300 W Voltage (V) = 110 V Wire gauge = 14-gauge Formulas needed: 1. Ohm's law: V = IR 2. Power formula: P = IV 3. Resistance formula for a wire: R = 蟻(L/A) 4. Area formula for a cylindrical wire: A = 蟺(D/2)^2 Here, I is the current, R is the resistance, 蟻 is the resistivity of the material, L is the length of the wire, A is the cross-sectional area of the wire, and D is the wire diameter.
02

Calculate current using the Power formula

Rearrange the Power formula for I: I = P / V Plugging in the given values: I = 300 W / 110 V I 鈮� 2.73 A
03

Calculate wire resistance using Ohm's Law

Rearrange Ohm's Law for Resistance: R = V / I Plugging in the given voltage and calculated current: R 鈮� 110 V / 2.73 A R 鈮� 40.29 惟
04

Find the resistivity and diameter for 14-gauge nichrome wire

Lookup the resistivity 蟻 of nichrome, which is approximately \( 1.10 \times 10^{-6} \Omega \mathrm{m} \). Lookup the diameter (D) of 14-gauge wire, which is approximately \( 1.628 \times 10^{-3} m \).
05

Calculate the cross-sectional area (A) of the wire

Use the Area formula for cylindrical wire: A = 蟺(D/2)^2 Plug in the diameter D: A = 蟺\((1.628 \times 10^{-3} m / 2)^2\) A 鈮� \( 2.080 \times 10^{-6} m^2 \)
06

Calculate the wire length (L) using the resistance formula

Rearrange the Resistance formula for wire length: L = R(A / 蟻) Plug in the values of R, A, and 蟻: L 鈮� \( 40.29 惟 \) (\( 2.080 \times 10^{-6} m^2 \) / \( 1.10 \times 10^{-6} 惟 \mathrm{m} \)) L 鈮� 76.16 m The engineer should make the nichrome wire approximately 76.16 meters long.

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