Question

If the resistance of wire A is four times the resistance of wire B, find the ratio of their cross sectional area and the ratio of the radii og their wires

Answer 1

length is same must be clarified

Answer 2

Answer:

The ratio of the cross sectional area and radius are  1:2 and 1:√2.

Explanation:

Given that,

The resistance of wire A is four times the resistance of wire B,

$R_{A}=4R_{B}$

Resistance of wire A = R

Resistance of wire B = R'

The formula of resistivity of the wire

$R =\dfrac{\rho l}{A}$

The volume of the wire is

$R = \dfrac{\rho l A}{A^2}$

Where, R = resistance

$\rho$=Resistivity

A = area of cross section

So, The resistance of the wire is inversely proportional to the square of area of cross section.

$R\propto\dfrac{1}{A^2}$

The ratio of the wire A and B

$\dfrac{R_{A}}{R_{B}}=\dfrac{A_{B}^2}{A_{A}^2}$

$\dfrac{4}{1}=\dfrac{A_{B}^2}{A_{A}^2}$

The ratio of their cross sectional area of the wires

$\dfrac{A_{A}}{A_{B}}=\dfrac{1}{2}$

The ratio of their radius of the wires

$\dfrac{\pi r_{A}^2}{\pi r_{B}^2}=\dfrac{1}{2}$

$\dfrac{r_{A}}{r_{B}}=\dfrac{1}{\sqrt{2}}$

Hence, The ratio of the cross sectional area and radius are  1:2 and 1:√2.