Respuesta :
[tex] {\text{Consider four resistors, each of which has a resistance }}r\,{\text{ohm}}{\text{.}} \hfill \\ {\text{Case I:}} \hfill \\
{\text{Let pairs of two resistors connected in parallel, while each pair is in series}}{\text{.}} \hfill \\
{\text{Let }}R'\,{\text{ohm be the resultant of each pair of resistors combined in series}} \hfill \\ {\text{and }}R\,{\text{ohm be the resultant of whole combinations}}{\text{.}} \hfill \\R' = r + r \hfill \\ \Rightarrow R' = 2r \hfill \\ [/tex][tex] \therefore \frac{1}{R} = \frac{1}{{2r}} + \frac{1}{{2r}} \hfill \\
\Rightarrow \frac{1}{R} = \frac{2}{{2r}} \hfill \\
\Rightarrow \frac{1}{R} = \frac{1}{r} \hfill \\
\Rightarrow R = r \hfill \\
{\text{Case II:}} \hfill \\
{\text{Let pairs of two resistors connected in series, while each pair is in parallel}}{\text{.}} \hfill \\
{\text{Let }}R'\,{\text{ohm be the resultant of each pair of resistors combined in parallel }} \hfill \\ [/tex][tex] {\text{and }}R\,{\text{ohm be the resultant of whole combinations}}{\text{.}} \hfill \\
\frac{1}{{R'}} = \frac{1}{r} + \frac{1}{r} = \frac{2}{r} \hfill \\
\Rightarrow R' = \frac{r}{2} \hfill \\
\therefore R = \frac{r}{2} + \frac{r}{2} \hfill \\
\Rightarrow R = 2 \times \frac{r}{2} \hfill \\
\therefore R = r \hfill \\
{\text{Hence, there is two ways using which when four resistors, each of which has a resistance }}r\,{\text{ohm combined,}} \hfill \\ [/tex][tex] {\text{their resultant is again comes out to be }}r\,{\text{ohm.}}\hfill \\ {\text{Thus, there are more than one way to achieve this goal}}{\text{.}}\hfill \\ [/tex]
There are more than one way to connect the four resistor of resistance r to get the equivalent resistance of the combination equal to r.
What is series and parallel connection of resistor?
There are two types of connection of resistor-
- In the series connection of resistor, the resistor are connected end to end in the circuit. The equivalent resistance for series connection of resistor is calculated with the following formula.
[tex]R_{eq}=R_1+R_2+....R_n[/tex]
- In the Parallel connection of resistor, the resistor are connected parallely in the circuit and the terminals of all resistor is connected to same nodes. The equivalent resistance for parallel connection of resistor is calculated with the following formula.
[tex]R_{eq}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+....\dfrac{1}{R_n}[/tex]
For the four resistors, each of which has a resistance r. Let they are connect in parallel combination with pair of two in series combination.
Then the resultant resistance is,
[tex]\dfrac{1}{R_{eq}}=\dfrac{1}{r+r}+\dfrac{1}{r+r}\\\dfrac{1}{R_{eq}}=\dfrac{1}{2r}+\dfrac{1}{2r}\\\dfrac{1}{R_{eq}}=\dfrac{2}{2r}\\\dfrac{1}{R_{eq}}=\dfrac{1}{r}\\{R_{eq}}=r[/tex]
The resultant resitance is equal to r. To get this result we can connect two resistor in series and every pair in parallel combination.
There is more than one way to connect the four resistor of resistance r to get the equivalent resistance of the combination equal to r.
Learn more about the series and parallel connection of resistor here;
https://brainly.com/question/466269
