Question

There are two wires of same material. Their redii and lengths are both in the ratio 1:2. If the extensions produced are equal, what is the ratio of loads?

Answer 1

Steps:
1) We know that the Resistance of a same wire is proportional to (l/A) .
where L = Length of wire,
A = Area of cross section
And, Area is proportional to d^2 ,
where 'd' is diameter of wire.


=> Resistance is proportional to


2) We have,
Ratio of their length = 1:2
Ratio of diameters = 2:3
=> Ratio of Area of cross- section = 2^2:3^2 = 4:9
=> Ratio of resistance :

9:16

3) Since, current in series circuit is same.
=> Ratio of Potential difference across wires :
R(1) : R(2) = 9:16

Answer 2

Given, There are two wires of same material. so, Young's modulus of both wires are same. e.g., $Y_1=Y_2$

extensions produced are equal.
so, $\Delta L_1=\Delta L_2$

and their radii and lengths are both in the ratio 1 : 2.
e.g., $\frac{r_1}{r_2}=\frac{1}{2}$

$\frac{L_1}{L_2}=\frac{1}{2}$

We have to find the ratio of their loads.

e.g., $\frac{F_1}{F_2}=\frac{Y_1A_1\Delta L_1 L_2}{Y_2A_2\Delta L_2 L_1}$

= $\frac{πr_1^2}{πr_2^2}\frac{L_2}{L_1} = [tex]\frac{1}{4}\frac{2}{1}$

= $\frac{1}{2}$