Question

5
There
are m resistar each of resistance R
they connected in series & equvalent resistence
is X. Now they are conned in parallel the resistance
isY.What is the ratioof Xand Y.
​

Answer 1

'm' identical resistors of magnitude R are connected in :

1. Series Connection : The equivalent resistance is X.

$\sf R + R + R + \cdots + R = X \\ \\ \longrightarrow \sf mR = X ---------(1)$

2. Parallel Connection : The equivalent resistance is Y.

$\sf \dfrac{m}{R} + \dfrac{m}{R} + \cdots + \dfrac{1}{R} = Y \\ \\ \longrightarrow \sf \dfrac{m}{R} = Y --------(2)$

We have to find ratio of X and Y. From equations (1) and (2),

$\sf \dfrac{X}{Y} = \dfrac{mR}{ \dfrac{m}{R} } \\ \\ \implies \boxed{ \boxed{ \sf X : Y = {R}^{2} : 1}}$

Thus, the ratio of X and Y is R² : 1.

Answer 2

@ BeBrainliest is here to help you buddy. :)

Thanks for asking your doubts.

⌬ Keep asking .....

⌬ Keep Answering .....

⌬ Keep Contributing .....

$\Large{\underline{\underline{\textsf{\maltese\: {\red{Given :-}}}}}}$

✏ There are ‘m’ number of resistors each of resistance ‘R’.

✏ Equivalent resistance of the resistors when connected in series is X.

✏ Equivalent resistance of the resistors when connected in parallel is Y.

$\Large{\underline{\underline{\textsf{\maltese\: {\red{To Find :-}}}}}}$

✏ $\sf \dfrac{X}{Y} \; = \; ?$

$\Large{\underline{\underline{\textsf{\maltese\: {\red{Recalling the concepts :-}}}}}}$

✏ Equivalent resistance of resistors when connected in series.

✰ Rₙₑₜ = R₁ + R₂ + R₃ ..... Rₙ

✏ Equivalent resistance of resistors when connected in parallel.

✰ $\sf \dfrac{1}{R_{net}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} \; .... \; \dfrac{1}{R_n}$

$\Large{\underline{\underline{\textsf{\maltese\: {\red{Solution :-}}}}}}$

✏ Equivalent resistance of resistors when connected in series.

✰ Rₙₑₜ = R₁ + R₂ + R₃ ..... Rₙ

⇒ X = R + R + R ... ‘m’ times

⇒ X = Rm

✏ Equivalent resistance of resistors when connected in parallel.

✰ $\sf \dfrac{1}{R_{net}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} \; .... \; \dfrac{1}{R_n}$

⇒ Y = $\sf \dfrac{1}{R} + \dfrac{1}{R} + \dfrac{1}{R} \; .....\; m \: times$

⇒ Y = $\sf \dfrac{1}{R} \; * \; m$

⇒ Y = $\sf \dfrac{m}{R}$

$\sf \implies \dfrac{X}{Y} \; = \; \dfrac{Rm}{\dfrac{m}{R}}$

$\implies \sf \dfrac{X}{Y} = Rm \div \dfrac{m}{R}$

$\implies \sf \dfrac{X}{Y} = Rm \times \dfrac{R}{m}$

$\implies \sf \dfrac{X}{Y} = R\not{m} \times \dfrac{R}{\not{m}}$

$\implies \sf \dfrac{X}{Y} = R \times R$

$\implies \sf \dfrac{X}{Y} = R^2$

${\underline{ \boxed{\therefore \; \sf X : Y = R^2 : 1}}}$