The resistance of a uniform metal wire of circular cross section with radius r a

Question

The resistance of a uniform metal wire of circular cross section with radius r and length L is measured to be equal to R. A second wire of the same material which is measured to have the same resistance R is four times as long as the first wire. What is the ratio of the radius of the second wire to that of the first wire?

The resistance of a uniform metal wire of circular cross section with radius r and length L is measured to be equal to R. A second wire of the same material which is measured to have the same resistance R is four times as long as the first wire. What is the ratio of the radius of the second wire to that of the first wire?

1:4

1:4

Explanation / Answer

Note that

R = rho L / A

They have the same material, so rho1 = rho2. R1 = R2 as well. Thus,

R1/R2 = [rho1 L1 A2] / [rho2 L2 A1]

R1/R2 = (rho1/rho2) (L1/L2) / (A1/A2)

As R1/R2 = 1 = rho1/rho2,

1 = 1 (L1/L2) / (A1/A2)

As L1/L2 = 1/4,

1 = 1 (1/4) / (A1/A2)

Thus,

A1/A2 = 1/4

As A = pi r^2, then

(pi r1^2) / (pi r2^2) = 1/4

(r1/r2)^2 = 1/4

r1/r2 = 1/2

r2/r1 = 2/1

Thus, the ratio is 2:1 [ANSWER]

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