Two conductors are made of the same metal. Find the ratio of the resistance of the first conductor to the resistance of the second for the following ratios of conductor sizes: l1 = l2, d1 = 4d2
To determine the resistance of a wire, we use the formula for the resistance of a conductor.
R = ρ * L / S, where ρ is the specific resistance of the conductor, L is the length of the conductor, S is the cross-section of the conductor.
Let us express the cross-section of the conductor in terms of its diameter.
S1 = π * d1 ^ 2/4;
S2 = π * d2 ^ 2/4;
Then S1 / S2 = (π * d1 ^ 2/4) / (π * d2 ^ 2/4) = d1 ^ 2 / d2 ^ 2;
The resistance of the first conductor is:
R1 = ρ1 * L1 / S1;
R2 = ρ2 * L2 / S2;
R1 / R2 = (ρ1 * L1 / S1) / (ρ2 * L2 / S2) = S2 / S1 = d2 ^ 2 / d1 ^ 2;
By condition d1 = 4 * d2.
R1 / R2 = (d2 ^ 2/16 * d2 ^ 2 = 1/16.
Answer: The ratio is 1/16.
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