The ratio of the length of the two wires made of the same material is 1 : 3 and their volumes are in the ratio of 2 : 3. What is the ratio of their resistances?
Question
The ratio of the length of the two wires made of the same material is 1 : 3 and their volumes are in the ratio of 2 : 3. What is the ratio of their resistances?
Solution
The resistance of a wire is given by the formula R = ρL/A, where R is the resistance, ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
Given that the wires are made of the same material, their resistivities are the same. Therefore, the ratio of their resistances is directly proportional to the ratio of their lengths and inversely proportional to the ratio of their cross-sectional areas.
The ratio of the lengths of the wires is given as 1:3.
The volume of a wire is given by the formula V = AL, where V is the volume, A is the cross-sectional area, and L is the length of the wire. Therefore, the ratio of the volumes of the wires is equal to the ratio of the product of their cross-sectional areas and lengths.
Given that the ratio of the volumes of the wires is 2:3, and knowing that the ratio of their lengths is 1:3, we can find the ratio of their cross-sectional areas by dividing the ratio of their volumes by the ratio of their lengths. This gives us a ratio of cross-sectional areas of 2:1.
Therefore, the ratio of the resistances of the wires is (1/2) : (3/1) = 1:6.
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