It is given that there is a wire of radius r, length l1 having a resistance R.
Now this wire is said to be extended to a new length l2 , which is n times l1 .
Since , there is no change in radius of the wire, let us assume that the volume before and after stretching are constant. V1=V2 ,
- V1 is volume before stretching
- V2 is volume after stretching.
Now we know that volume is a product of area and height.
In our case, height is the length of the wire A1 × l1 = A2 × l2 We know that , l2 = n×l1 .
Substituting this , we get ⇒A1 × l1 =A2 × n × l1 ⇒ A1/n = A2
We know that resistance R of a material is calculated by using the formula R=ρ l/A , where
- ρ is represented as resistivity of the material
- A is area and l is length of the wire.
Now , before extension , resistivity is given as , R1 = ρl1/A1
After extension, resistivity is given as, R2 =ρl2 /A2
Substituting for A2 in the equation for R2 , we get
⇒ R2 = ρl2 × n/A1
Now substituting l2= n ×l1 ,
we get ⇒ R2 = ρl1 × n2/A1 (The term ρl1A1 is equal to R1 and R1 value given in the question is R )
⇒ R2 = R × n2
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