Two resistors $R_1$ and $R_2$ arranged in parallel combination in an electrical closed circuit are made of the same material and of same thickness. If the length of $R_2$ is twice the length of $R_1$, then the total resistance $R$ satisfies
$3R = 2R_1$
$3R = 2R_2$
$2R = 3R_1$
$2R = 3R_2$
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC NDA-1 – 2022
Let the length of $R_1$ be $L_1$. Then $R_1 = \rho L_1 / A$.
The length of $R_2$ is $L_2 = 2L_1$. Then $R_2 = \rho L_2 / A = \rho (2L_1) / A = 2 (\rho L_1 / A) = 2R_1$.
Now, the resistors are in parallel, so $1/R = 1/R_1 + 1/R_2$.
Substitute $R_2 = 2R_1$:
$1/R = 1/R_1 + 1/(2R_1)$
$1/R = (2 + 1) / (2R_1)$
$1/R = 3 / (2R_1)$
$R = (2R_1) / 3$
Multiplying both sides by 3 gives $3R = 2R_1$.