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Question

A certain piece of copper is to be shaped into a conductor of minimum resistance. Its length and diameter should be respectively


A

l,d

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B

2l,d

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C

12l,2d

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D

2l,12d

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Solution

The correct option is C

12l,2d


Explanation for the correct option:

Step 1: Given

The length of the conductor is = l

The diameter of the conductor is = d

Step 2: Formula Used

The resistance of a conductor is given by R=ρlA, where ρ is resistivity of the conductor, l is length and A is area of the conductor. This can be further reduced to the expression

R=ρlπr2=ρlπd22=4ρlπd2

Where l is the length of the conductor and d is diameter

Step 3: Calculation

Find the resistance when length is l and diameter is d.

R=4ρlπd2

Find the resistance when length is 2l and diameter is d.

R=4ρ2lπd2=8ρlπd2

Finding the resistance when length is 12l and diameter is 2d.

R=4ρ12lπ2d2=2ρl4πd2=ρl2πd2

Finding the resistance when length is 2l and diameter is 12d.

R=4ρ2lπ12d2=8ρl14πd2=32ρlπd2

From the above values of resistance, it is clear that the least value is when length is 12l and diameter is 2d.

Hence, the correct option is option (C).


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