Question

The radii of two wire of same material and same length are in the ratio

2:3. What is the ratio of their i) Resistances ii) Resistivity.

Answer 1

Answer:

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Answer 2

Given:

$\frac{r_1}{r_2} = \frac{2}{3}\\L_1=L_2 = L (length)\\R_1 , R_2 resistances\\\rho _1 , \rho_2 resistivity$

where $r_1 , r_2$ are radii of two wires

To find :

$\frac{R_1}{R_2}$ and $\frac{\rho_1} {\rho_2}$

Solution:

• The resistance of a wire can be expressed as $R = \rho \times \frac {L}{A}$

​   where,

rho is the resistivity of the wire,

L is the length of the wire,

A is the area of cross section of the wire, that is, square of the radius.

In this case, two wires of the same material and same length have radii r1 and r2 respectively are taken. So, the resistance will differ only in the square of the radius.

i)    For resistances , we have ;

$\frac{R_1}{R_2} = \frac{r_2^2}{r_1^2}$   from the above equation , since L is same for both.

Substituting values of $\frac{r_2}{r_1}$ , we get :

$\frac{R_1}{R_2} = \frac{3^2}{2^2} = \frac{9}{4}$

ii)      Similarly for resistivity , we get :

$\frac{\rho_1}{\rho_2} = \frac{2^2}{3^2} = \frac{4}{9}$