Answer to Question #190244 in Civil and Environmental Engineering for John Prats

Question #190244

An aluminum wire of diameter D and length L has a resistance of 1.5 Ω. Find the length of another aluminum wire at the same temperature if its diameter is 2D and its resistance is 6 Ω.

Expert's answer

Wire of the first case , Aluminum has the following information .

Length = L

Resistance = 1.5 "\\Omega"

Diameter = D

Area A= "\\pi r^2= \\pi (\\frac{D}{2})^2= \\pi \\frac{D^2}{4}"

R= "\\rho \\frac{L}{A}" , "\\rho= \\frac{AR}{L}"

"\\rho" = "\\frac{\\pi \\times 0.25D^2\\times 1.5}{L} = \\frac{0.375D^2}{L} \\pi"

"\\rho" = Resistivity is constant for all size of one type of material at a particular temperature .

Therefore the second aluminum also has the same resistivity as="\\frac{0.375D^2}{L} \\pi"

the second aluminum piece has the following information,

R= 6 "\\Omega" Diameter = 2D ,

Area = "\\pi r^2= \\pi (\\frac{2D}{2})^2 = \\pi D^2"

Length of the wire = "\\frac{AR}{\\rho}" = "\\pi(\\frac{D^2 \\times 6\\times L}{\\pi \\times 0.375\\times D^2})= 16 L"

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