Question

Judge the equivalent resistance when the following are connected in parallel
(i) $1\Omega$ and ${10}^6\Omega$
(ii) $1\Omega ,{10}^3\Omega$ and ${10}^6\Omega .$

Answer 1

$\left(a\right)$The net resistance in parallel is given by

$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$

Here $R_1=1\Omega$ and $R_2={10}^6\Omega$

So,$\frac{1}{R}=\frac{R_1+R_2}{R_1{R}_2}$

$\Rightarrow R=\frac{R_1{R}_2}{R_1+R_2}$

$=\frac{1\times {10}^6}{{10}^6+1}$

$=\frac{1000000‬}{1000001}$

$=0.999\approx 1\Omega$

$\left(b\right)$The net resistance in parallel is given by

$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$

Here $R_1=1\Omega ,R_2={10}^3\Omega$ and $R_3={10}^6\Omega$

So,$\frac{1}{R}=\frac{1}{1}+\frac{1}{{10}^3}+\frac{1}{{10}^6}$

$=\frac{{10}^6+{10}^3+1}{1000000}$

$=\frac{1000000‬+1000+1}{1000000}$

$=\frac{1000000‬+1000+1}{1000000}$

$=\frac{1001001‬}{1000000}$

$\Rightarrow R=\frac{1000000}{1001001‬}=0.999\Omega \approx 1\Omega$