Question

$\Huge{\mathscr{\fcolorbox {navy}{pink}{\color {brown}{Question:-}}}}$

Prove that in a parallel connection the combine resistance is the sum of the reciprocals of all resistance.

$\boxed {\mathcal{\blue {\dfrac {1}{R}=\dfrac {1}{R_1}+\dfrac{1}{R_2}+\dfrac {1}{R}{3}}}$​

Answer 1

Explanation:

P.F.A the proof below:

Hope you got that.

Thank You.

Answer 2

$\large \pmb{\sf{To \: Prove :}}$

In a parallel connection the combine resistance is the sum of the reciprocals of all resistance.

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$\large{\pmb{\sf{Simplified :}}}$

$\boxed {\pmb{\green {\dfrac {1}{R}=\dfrac {1}{R_1}+\dfrac{1}{R_2}+\dfrac {1}{R}_{3}}}}$

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$\large\pmb{\sf{Proof :}}$

${: \mapsto{\footnotesize {\text{In a parallel circuit, Voltage (V) remains constant}}}}$

$: \mapsto \footnotesize \text{But the current gets distributed :}$

$\boxed{ \footnotesize{ \color{maroon}{\rm{Total \: current = current \: in \: R_1 + current \: in \: R_2 + current \: in \: R_3}}}}$

$\sf{Current \: in \: R_1} \begin{cases} \sf{V = I_1 × R_1 } \\ \sf{ \frac{V}{R_1} = I_1} \end{cases}$

$\sf{Current \: in \: R_2} \begin{cases} \sf{V = I_2 × R_2 } \\ \sf{ \frac{V}{R_2} = I_2} \end{cases}$

$\sf{Current \: in \: R_3} \begin{cases} \sf{V = I_3 × R_3} \\ \sf{ \frac{V}{R_3} = I_3} \end{cases}$

${ : \mapsto\textsf {\footnotesize{Total Voltage = Total current × Total resistance }}}$

${ : \mapsto{\sf{ V = Total \: current × Total \: resistance }}}$

$\boxed{ : \mapsto{\sf { Total \: current = \frac{V}{ Total \: resistance} }}}$

$\rm{Now, Total \: Current = I_1 + I_2 + I_3}$

$\sf{\frac{V}{ Total \: resistance} = \frac{V}{R_1} +\frac{V}{R_2} +\frac{V}{R_3} }$

$\sf{ : \leadsto\frac{ \cancel{V}}{ Total \: resistance} = \cancel{V} \left( \frac{1}{R_1} +\frac{1}{R_2} +\frac{1}{R_3} \right)}$

$\sf{ : \leadsto\frac{ 1}{ Total \: resistance} = \frac{1}{R_1} +\frac{1}{R_2} +\frac{1}{R_3} }$

$: \leadsto\boxed {\pmb{\green {\dfrac {1}{R}=\dfrac {1}{R_1}+\dfrac{1}{R_2}+\dfrac {1}{R}_{3}}}}$

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$\large\pmb{\sf{Note : }}$

$\rm {: \mapsto Swipe \: towards \: left\: to \: view \: full \: equation }$

$\rm {: \mapsto Diagram \: in \: the\: attachment }$

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