Question

A metal wire is cut into several pieces.The length and radius of two pieces are in ratio 1:2.Compare the resistance a and the equivalent resistance when a and b are connected in parallel.

Answer 1

Answer: The resistance of a is 3 times of equivalent resistance

Explanation:

Given that,

The ratio of the length of two pieces

$\dfrac{l_{a}}{l_{b}} = \dfrac{1}{2}$

The ratio of the radius of two pieces

$\dfrac{r_{a}}{r_{b}} = \dfrac{1}{2}$

We know that, the resistance is

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

Where, l = length of the wire

A = cross sectional area

R = resistance

Now, the ratio of resistance

$\dfrac{R_{a}}{R_{b}} = \dfrac{\rho l_{a}}{\pi r_{a}^{2}}\times \dfrac{\pi r_{b}^{2}}{\rho l_{b}}$

$\dfrac{R_{a}}{R_{b}} = \dfrac{1}{2}\times 4$

$\dfrac{R_{a}}{R_{b}} = 2$...(I)

When a and b are connected in parallel

Then, the equivalent resistance

$R_{eq} = \dfrac{R_{a}R_{b}}{R_{a}+ R_{b}}$

Put the value of $R_{b}$ from equation (I)

$R_{eq} = \dfrac{R_{a}^{2}}{3R_{a}}$

$R_{eq} = \dfrac{R_{a}}{3}$

$R_{a} = 3 R_{eq}$

Hence, the resistance of a is 3 times of equivalent resistance