Question

Resistance n,each of r ohm,when collected in parallel give an equivalent resistance of R ohm.If these resistance were connected in series ,the combination would have a resistance in ohm,equal to

Answer 1

Answer:

Initially , all the n resistors had been connected in parallel combination. The net resistance is R

We can say that :

$\dfrac{1}{R} = \dfrac{1}{r} + \dfrac{1}{r} + ..... \: n \: times$

$= > \dfrac{1}{R} = n \times \bigg \{\dfrac{1}{r} \bigg \}$

$= > R = \dfrac{r}{n}$

Now all the resistors have been connected in series , and we need to find out the equivalent Resistance in terms of R :

$R \: eq. = r + r + r + ....n \: times$

$= > R \: eq. = n \times (r)$

$= > R \: eq. = n \times(n \times R)$

$= > R \: eq. = {n}^{2} R$

So final answer :

$\boxed{ \red{ \bold{ \huge{ \sf{R \: eq. = {n}^{2} R }}}}}$

Answer 2

$\large{\underline{\underline{\mathfrak{Answer :}}}}$

• Equivalent Resistance is Req = n²R

$\rule{200}{0.5}$

$\underline{\underline{\mathfrak{Step-By-Step-Explanation :}}}$

Given :

• n Resistance of r ohm are connected in series gives equivalent resistance R ohm.

$\rule{200}{1}$

To Find :

• Resistance in series

$\rule{200}{1}$

Solution :

First, we have to add all the n resistances in parallel which gives result as R ohm .

$\implies {\sf{\dfrac{1}{R} \: = \: \dfrac{1}{r} \: + \: \dfrac{1}{r} \: + \: ....... \: + \: n}} \\ \\ \implies {\sf{\dfrac{1}{R} \: = \: n \: \times \: \dfrac{1}{r}}} \\ \\ \implies {\sf{R \: = \: \dfrac{r}{n}}} \\ \\ \implies {\sf{r \: = \: nr \: -----(1)}}$

$\rule{100}{1}$

Now, add all the resistances in series to give Equivalent resistance.

$\implies {\sf{R_{eq} \: = \: r \: + \: r \: + \: ....... \: + \: n}} \\ \\ \implies {\sf{R_{eq} \: = \: n \: \times \: r}} \\ \\ \footnotesize{\underline{\sf{\dag \: \: \: \: \: \: \: Put \: value \: of \: r \: from \: (1) \: \: \: \: \: \: \: }}} \\ \\ \implies {\sf{R_{eq} \: = \: n \: \times \: nR}} \\ \\ \implies {\sf{R_{eq} \: = \: n^2R}} \\ \\ \underline{\sf{\therefore \: Equivalent \: Resistance \: is \: n^2 R}}$