Two wires are made from the same material. One wire has a resistance of 0.10 ω. The other wire is twice as long as the first wire and has a radius that is half as much. What is the resistance of the second wire?

Explanation:

Given that,

First wire has a resistance of

R1 = 0.1 Ω

We are told that

Second wire is twice as long as the first wire.

Then,

Let the first wire has length

L1 = x

Then, second wire will have

L2 = 2x

Also, the radius of the second wire is half the radius of the first wire

Let the first wire has radius

r1 = y

Then, it area is

A1 = πr1² = πy²

Then, the second wire has radius

r2 = ½y

It area is also

A2 = πr2² = π(½y)² = ½πy²

Since the wire is made of the same material, then, they will have the same resistivity ρ

Then, we want to find the resistance of the second wire

Using

R = ρL/A

Where

R = resistance

ρ = sensitivity

L = length

A = area

Then, make resistivity subject of formula since it is a constant

RA = ρL

Then, ρ = RA/L

Then,

R1 • A1 / L1 = R2 • A2 / L2

Substituting each value

0.1 × πy² / x = R2 × ¼πy² / 2x

Cross multiply

0.1 × πy² × 2x = R2 ×¼π y² × x

Then,

R2 = 0.1 × πy² × 2x / ¼πy² × x

R2 = 0.1 × 2 / ¼

R2 = 0.8Ω

The resistance of the second wire is 0.8Ω

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