Question

Judge the equivalent resistance when the following are connected in parallel:
A.\[1\Omega \] and \[{10^6}\Omega \]
B.\[1\Omega \], \[{10^3}\Omega \], and \[{10^6}\Omega \]

Answer 1

Hint: Resistance is defined as the measure of the limit on charge flow. The resistor can be connected in the simplest form as the series and parallel connections. When the resistance is connected in a parallel circuit then the total resistance is equal to the sum of the inverse of each individual resistance. We can see that the total resistance in a parallel circuit is always less than the smallest of the individuals connected in parallel.

Formula used:
\[\dfrac{1}{{{R_T}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + \dfrac{1}{{{R_3}}} + ........ + \dfrac{1}{{{R_n}}}\]
Here, \[{R_T}\]is the equivalent resistance of all the individual resistance that is connected in parallel.
\[{R_1}\],\[{R_2}\],\[{R_3}\] are the individual resistance connected in the parallel.

Complete step by step answer:
Given that we need to calculate the equivalent resistance of the following,
\[{R_1} = 1\Omega \], \[{R_2} = {10^6}\Omega \]
Therefore substituting this in the above-given formula,
\[\dfrac{1}{{{R_T}}} = \dfrac{1}{1} + \dfrac{1}{{{{10}^6}}}\]
$R = \dfrac{{{{10}^6}}}{{{{10}^6} + 1}} \sim \dfrac{{{{10}^6}}}{{{{10}^6}}}$
\[{R_T} = 1\Omega \]
\[{R_1} = 1\Omega \], \[{R_2} = {10^3}\Omega \], \[{R_3} = {10^6}\Omega \]
Therefore substituting this in the above-given formula we get,
\[\dfrac{1}{{{R_T}}} = \dfrac{1}{1} + \dfrac{1}{{{{10}^3}}} + \dfrac{1}{{{{10}^6}}}\]
\[\dfrac{1}{{{R_T}}} = \dfrac{{{{10}^6} + {{10}^3} + 1}}{{{{10}^6}}}\]
\[\dfrac{1}{{{R_T}}} = \dfrac{{10010001}}{{1000000}}\]
\[{R_T} = 0.99\Omega \]
Therefore the value of the equivalent resistance for the following are,
\[{R_1} = 1\Omega \], \[{R_2} = {10^6}\Omega \]: \[{R_T} = 1\Omega \]
\[{R_1} = 1\Omega \], \[{R_2} = {10^3}\Omega \], \[{R_3} = {10^6}\Omega \]: \[{R_T} = 0.99\Omega \]

Note:
If we connect two resistances in parallel have equal value then the total or equivalent resistance is equal to half the value of the one resistor. Then the value of the total resistance will be\[\dfrac{R}{2}\]. Similarly, if we connected three resistances in a parallel circuit and then the equal resistance will be equal to \[\dfrac{R}{3}\]. Parallel resistance will give us the value known as conductance. The value of conductance is equal to the inverse of the resistance. The unit of conductance is given as the Siemens with a symbol S. Each resistor that is connected in parallel will have the same voltage of the source applied to it.