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Question 26.20: A certain wire has a resistance R. What is the resistance of......

A certain wire has a resistance R. What is the resistance of a second wire, made of the same material, that is half as long and has half the diameter?

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The thickness (diameter) of the wire is denoted by D. We use R \propto L / A (Eq. 26-16) and note that A=\frac{1}{4} \pi D^2 \propto D^2 . The resistance of the second wire is given by

R=\rho \frac{L}{A},       (26-16)

R_2=R\left(\frac{A_1}{A_2}\right)\left(\frac{L_2}{L_1}\right)=R\left(\frac{D_1}{D_2}\right)^2\left(\frac{L_2}{L_1}\right)=R(2)^2\left(\frac{1}{2}\right)=2 R .

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