Question

Two cylindrical wires of the same material have their lengths in the ratio of 4 : 9. What

should be the ratio of their radii so that their resistances are in the ratio of 4 : 1?

step by step answer, no spams.....easy explanation needed​

Answer 1

Answer:

I think the answer is 1:3....

Answer 2

Answer:

Two cylindrical wires of the same material have their lengths in the ratio of 4: 9 Their resistances are in the ratio of 4: 1  ratio of their radii will be $1:3$

Explanation:

• Resistance (R) is an electrical quantity that measures the opposition to the flow of current.

• Resistance depends on the length (L) and area (A) of the material in which the current is flowing.  

       $R= \frac{\rho L}{A} }$

        $\rho$ is a constant and area is given by $\pi r^{2}$

• That is,

        $R=\frac{\rho L}{\pi r^{2} }$     or        $R \propto\frac{L}{r^2}$

• Let $\text L_1$  and $\text L_2$ are  the lengths and $\text r_1$ and $\text r_2$ are the radii of two materials.

         $\therefore \frac{R_{1}}{R_{2}}=\frac{L_{1}}{L_{2}} \times \frac{r_{2}^{2}}{r_{1}^{2}}$     ....(1)

     

In the question, it is given that,

$L_1:L_2 = 4:9$     or       $\frac{L_1}{L_2} =4:9$

$R_1:R_2 = 4:1$        or     $\frac{R_1}{R_2} = \frac{4}{1}$

Using equation(1),

$\frac{4}{1}=\frac{4}{9} \times \frac{r_2^{2}}{r_1^{2}}$

$\frac{r_1^{2} }{r_2^2} = \frac{1}{9}$

$\therefore \frac{r_1}{r_2} =\frac{1}{3}$

• The ratio of their radii will be $1:3$