Question Number 41231 by tanmay.chaudhury50@gmail.com last updated on 03/Aug/18
Commented by tanmay.chaudhury50@gmail.com last updated on 03/Aug/18
$${find}\:{equivalant}\:{resistance}\:{between}\:{point}\:{A}\:{andB} \\ $$
Answered by ajfour last updated on 03/Aug/18
$${V}_{{A}} −{ir}−{IR}={V}_{{B}} \\ $$$$\Rightarrow\:\:{V}_{{A}} −{V}_{{B}} =\:{ir}+{IR}\:\:\:\:...\left({i}\right) \\ $$$${left}\:{loop}\:{eq}. \\ $$$${V}_{{A}} −{ir}−\left({i}−{I}\right){r}+{IR}\:=\:{V}_{{A}} \\ $$$$\Rightarrow\:\:{I}\left({R}+{r}\right)\:=\:\mathrm{2}{ir} \\ $$$${or}\:\:\:\:\frac{{I}}{{i}}\:=\:\frac{\mathrm{2}{r}}{{R}+{r}}\:\:\:\:\:\:\:....\left({ii}\right) \\ $$$${R}_{{eq}} \:=\:\frac{{V}_{{A}} −{V}_{{B}} }{{i}+{I}}\:\:\:\:\:\:\:....\left({iii}\right) \\ $$$${using}\:\left({i}\right)\&\:\left({ii}\right)\:{in}\:\:\left({iii}\right) \\ $$$$\:\:{R}_{{eq}} \:=\:\frac{{ir}+{IR}}{{i}+{I}}\:=\:\frac{{r}+\frac{{I}}{{i}}{R}}{\mathrm{1}+\frac{{I}}{{i}}} \\ $$$$\:\:\:\:{R}_{{eq}} \:=\:\frac{{r}+\frac{\mathrm{2}{rR}}{{R}+{r}}}{\mathrm{1}+\frac{\mathrm{2}{r}}{{R}+{r}}}\:=\:\frac{{r}\left({r}+\mathrm{3}{R}\right)}{{R}+\mathrm{3}{r}}\:. \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 04/Aug/18
$${thank}\:{you}... \\ $$