Question

A metal wire is cut into several pieces The length and radius of two pieces are in the ratio 1:2, compare the resistance of A and the equivalent resistance when A and B are connected in parallel

Answer 1

Answer:

$\frac{R_{A}}{R_{eq}} = \frac{3}{1}$

Step-by-step explanation:

Hey Mate,

Given,

Two pieces of metal are A and B.

The Ratio of length of pieces A and B is 1 : 2 and ratio of radius of pieces A and B is 1 : 2.

= $l_{a} / A_{b} = 1/2$  and  $r_{a} / r_{b} = 1/2$

we know that $R = p\frac{l}{a}$

where R is resistance, l is length of wire and a is cross sectional area of wire.

so, $R_{A} = \frac{l_{A}}{\pi r^{2}_{A} }$ and $R_{B} = \frac{l_{B}}{\pi r^{2}_{B} }$

We get,

$\frac{R_{A}}{R_{B}}=\frac{l_{A} r^2_{B}}{l_{B}r^2_{A}}$

$= \frac{1*2^2}{2*1^2} = 2$

when both pieces are connected in parallel.

$R_{eq} = \frac{R_{A}R_{B}}{R_{A}+ R_{B}} =\frac{1}{3} R_{A}$

So,

$\frac{R_{A}}{R_{eq}} = \frac{3}{1}$