Question

There are m resistors each of resistance R. First they all are connected in series and equivalent resistance is X. Now they are connected in parallel and equivalent resistance is Y. What is the ratio of X and Y?

Answer 1

M resistors in series. so net resistance=X=MR then M resistors in parallel so net resistance =Y=R\M.so relation between X & Y is X\Y=M^2:1

Answer 2

Aɴꜱᴡᴇʀ

$\huge{\sf{x : y = m^2 : 1}}$

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Gɪᴠᴇɴ

• m resistors of resistance R are first connected in series and then connected in parallel.

• resistance of series connection = x

• resistance of parallel connection = y

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ᴛᴏ ꜰɪɴᴅ

The ratio between X and Y

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Sᴛᴇᴘꜱ

Here we can use the formula

$\boxed{ \fbox{\sf{\red{R_{eq}=R_1+R_2+R_3+.....}}}}$

• Formula of equation resistance in parallel connection is given by >>

$\boxed{ \fbox{\sf{\pink{\dfrac{1}{R_{eq}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+.....}}}}$

Series Connection

$\begin{lgathered}\dashrightarrow\sf\:R_{s}=R+R+..... m \:times\\ \\ \dashrightarrow\:{\sf{\orange{\large{x=mR}}}}\end{lgathered}$

Parallel Connection

$\begin{lgathered}\leadsto\sf\:\dfrac{1}{R_p}=\dfrac{1}{R}+\dfrac{1}{R}+.....m \: times\\ \\ \dashrightarrow\sf\:\dfrac{1}{y}=\dfrac{m}{R}\\ \\ \dashrightarrow\:{\sf{\orange{\large{y=\dfrac{R}{m}}}}}\end{lgathered}$

Ratio of X and Y is

$\begin{lgathered}\mapsto\sf\:\dfrac{x}{y}=\dfrac{mR}{\frac{R}{m}}\\ \\ \mapsto\:\boxed{ \fbox{\tt{\pink{\large{x : y = m^2 : 1}}}}}\end{lgathered}</p><p>$

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$\huge{\mathfrak{\purple{hope\; it \;helps}}}$