Question

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

Answer 1

Let two pieces of metal are A and B.
a/c to question,
ratio of length of pieces A and B is 1 : 2 and ratio of radius of pieces A and B is 1 : 2.
e.g., $\frac{l_A}{A_B}=\frac{1}{2}$
and $\frac{r_A}{r_B}=\frac{1}{2}$

we know, $R=\rho\frac{l}{a}$
where R is resistance, l is length of wire and a is cross sectional area of wire.
so, $R_A=\rho\frac{l_A}{\pi r_A^2}$
and $R_B=\rho\frac{l_B}{\pi r_B^2}$

here, we get, $\frac{R_A}{R_B}=\frac{l_A}{l_B}\frac{r_B^2}{r_A^2}=\frac{1}{2}\frac{2^2}{1^2}=2$

when both pieces are connected in parallel.
then, $R_{eq}=\frac{R_AR_B}{R_A+R_B}$
=$\frac{R_AR_A/2}{R_A+R_A/2}=\frac{1}{3}R_A$

hence, $\frac{R_A}{R_{eq}}=\frac{3}{1}$