Answer:
The total resistance of these three resistors connected in parallel is [tex]1.7143\Omega[/tex]
Step-by-step explanation:
The attached image has the circuit for finding the total resistance. The circuit is composed by a voltage source and three resistors connected in parallel: [tex]R_1=6\Omega [/tex], [tex]R_2=3\Omega [/tex] and [tex]R_3=12\Omega [/tex].
First step: to find the source current
The current that the source provides is the sum of the current that each resistor consumes. Keep in mind that the voltage is the same for the three resistors ([tex]R_1[/tex], [tex]R_2[/tex] and [tex]R_3[/tex]).
[tex]I_{R_1}=\frac{V_S}{R_1}[/tex]
[tex]I_{R_2}=\frac{V_S}{R_2}[/tex]
[tex]I_{R_3}=\frac{V_S}{R_3}[/tex]
The total current is:
[tex]I_S=I_{R_1}+I_{R_2}+I_{R_3}=\frac{V_S}{R_1}+\frac{V_S}{R_2}+\frac{V_S}{R_3}=\frac{R_2\cdot R_3 \cdot V_S+R_1\cdot R_3 \cdot V_S+R_1\cdot R_2 \cdot V_S}{R_1\cdot R_2\cdot R_3}[/tex]
[tex]I_S=V_S\cdot \frac{R_2\cdot R_3+R_1\cdot R_3+R_1\cdot R_2}{R_1\cdot R_2\cdot R_3}[/tex]
The total resistance ([tex]R_T[/tex]) is the source voltage divided by the source current:
[tex]R_T=\frac{V_S}{I_S}[/tex]
Now, replace [tex]I_S[/tex] by the previous expression and the total resistance would be:
[tex]R_T=\frac{V_S}{V_S\cdot \frac{R_2\cdot R_3+R_1\cdot R_3+R_1\cdot R_2}{R_1\cdot R_2\cdot R_3}}[/tex]
Simplify the expression and you must get:
[tex]R_T=\frac{R_1\cdot R_2\cdot R_3}{R_2\cdot R_3+R_1\cdot R_3+R_1\cdot R_2}[/tex]
The last step is to replace the values of the resistors:
[tex]R_T=\frac{(6\Omega )\cdot (3\Omega)\cdot (12\Omega)}{(3\Omega)\cdot (12\Omega)+(6\Omega)\cdot (12\Omega)+(6\Omega)\cdot (3\Omega)}=\frac{12}{7}\Omega=1.7143\Omega [/tex]
Thus, the total resistance of these three resistors connected in parallel is [tex]1.7143\Omega[/tex]
