Question

The equivalent resistance of n resistors each of same resistance when connected in series is R.
If the same resistances are connected in parallel, the equivalent resistance will be

Answer 1

Explanation:

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Answer 2

Given :

▪ n identical resistors are connected first in series and then in parallel.

▪ Equivalent resistance of series = R

To Find :

▪ Equivalent resistance of parallel connection.

Formula :

✴ Eq. of series connection :

$\bigstar\:\underline{\boxed{\bf{\red{R_s=R_1+R_2+...+R_n}}}}$

✴ Eq. of parallel connection :

$\bigstar\:\underline{\boxed{\bf{\blue{\dfrac{1}{R_p}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...+\dfrac{1}{R_n}}}}}$

Calculation :

↗ Let, resistance of each resistor be X.

✴ For series connection :

$\implies\sf\:R_s=R_1+R_2+...+R_n\\ \\ \implies\sf\:R=X+X+...\:n\:times\\ \\ \implies\sf\:R=nX\\ \\ \implies\bf\:X=\dfrac{R}{n}$

✴ For parallel connection :

$\implies\sf\:\dfrac{1}{R_p}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...+\dfrac{1}{R_n}\\ \\ \implies\sf\:\dfrac{1}{R'}=\dfrac{1}{X}+\dfrac{1}{X}+...\:n\:times\\ \\ \implies\sf\:\dfrac{1}{R'}=\dfrac{n}{X}\\ \\ \implies\sf\:\dfrac{1}{R'}=\dfrac{n^2}{R}\\ \\ \implies\underline{\boxed{\bf{\green{R'=\dfrac{R}{n^2}}}}}\:\orange{\bigstar}$