Question

If the length of a wire is halved and its cross-sectional area is doubled, then what would be the resistance of the wire?

(Given, initially the resistance of the wire is R)

Answer 1

The formula of resistance is:

R = \(\frac{ρ\,\times\,l}{A}\)

Where, ρ resistivity 

I = Length of the conductor 

A = Area of the conductor 

If length of the conductor is halved then, I₁ = \(\frac12\)

And, the cross sectional area is doubled, A₁ =2A

∴ R₁ = \(\frac{ρ\,\times\,l_1}{A_1}\)

R_{1 =} \(\frac{ρ\,\times\,\frac{1}2}{2A}\)

R_{1 =}  \(\frac14\) x \(\frac{ρ\,\times\,1}{A}\)

Now,  \(\frac{ρ\,\times\,1}{A}\) = R

∴ R₁ = \(\frac{R}4\)

Thus, we can conclude that the new resistance of the wire when the area of wire is doublead and length is haved become 1/4 the initial resistance.