The given data:
- Wire of Radius = r
- Length = l1
- Resistance = R
This wire is now expanded to a new length l2, that is n times l1.
Because there is zero change in the wire’s radius, then it can be considered that the volume before and after stretching is constant, that is, V1 = V2.
Here,
- V1 = Volume before stretching
- V2 = Volume after stretching
It is already known that Volume is the product of area and height.
In this case, height can be considered the length of the wire, A1 × l1 = A2 × l2. We are aware that, l2 = n × l1.
After replacing, we can get,
⇒ A1 × l1 = A2 × n × l1
⇒ A1/n = A2
It is already known that resistance R of a material is estimated by means of the formula:
| R = ρ l/A |
Where,
- ρ = Resistivity of the material
- A = Area
- l = length of the wire
Before the extension, the resistivity was expressed as R1 = ρl1/A1
Now, after the extension, resistivity can be expressed as R2 =ρl2 /A2
Upon value substitution for A2 in the equation for R2, we can obtain:
⇒ R2 = ρl2 × n/A1
Now after replacing, we get: l2= n ×l1,
⇒ R2 = ρl1 × n2/A1
Therefore, the term ρl1A1 = R1 and R1 (value given in the question is R )
⇒ R2 = R × n2
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