Question

A copper wire of radius r and length l has a resistance R. A second copper wire with radius 2r and length l is taken and the two wires are joined in a parallel combination. The resultant resistance of the parallel combination of the two wires will be?

Answer 1

4R/5

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Answer 2

Given that,

A copper wire of radius r and length l has a resistance R.

To find,

The resultant resistance of the parallel combination of the two wires if a second copper wire with radius 2r and length l is taken.

Solution,

The resistance of a wire is given by :

For wire 1,

$R=\rho\dfrac{l}{A}\\\\R=\rho\dfrac{l}{\pi r^2}$

For wire 2,

$R'=\rho\dfrac{l}{A}\\\\R'=\rho\dfrac{l}{\pi (2r)^2}\\\\R'=\dfrac{1}{4}\rho\dfrac{l}{\pi r^2}\\\\R'=\dfrac{R}{4}$

The parallel combination of R and R/4 is :

$\dfrac{1}{R_{eq}}=\dfrac{1}{R}+\dfrac{1}{(R/4)}\\\\\dfrac{1}{R_{eq}}=\dfrac{1}{R}+\dfrac{1}{(R/4)}\\\\\dfrac{1}{R_{eq}}=\dfrac{5}{R}\\\\R_{eq}=\dfrac{R}{5}$

or

$R_{eq}=\dfrac{\rho l}{5\pi r^2}$

Learn more,

Parallel combination of resistor

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