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Part D

For the combination of resistors shown, find the equivalent resistance between points A and B.

(Figure 6)

Express your answer in Ohms.

Req=
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Answer #1
Concepts and reason

The concepts used to solve this problem are effective resistance of resistors connected in series and parallel.

First, calculate the effective resistance of the resistors in top and bottom branch connected in parallel. Now, the effective resistances of the parallel resistors are in series with the rest of the resistors in their respective branches. Then, calculate the effective resistance of the series combinations of resistors in top and bottom branch.

Now, the equivalent resistance of the top and bottom branch will be parallel to each other. Finally, use the formula to calculate the resistance for the resistors connected in parallel to calculate the equivalent resistance of the resistors in top and bottom branch between point A and B.

Fundamentals

The expression for the equivalent resistance of the resistor connected in series is,

Req=R1+R2{R_{{\rm{eq}}}} = {R_1} + {R_2}

Here, Req{R_{{\rm{eq}}}} is the equivalent resistance of the resistors connected in series, R1,R2{R_1},{R_2} are the resistors connected in series.

The expression for the equivalent resistance of the resistor connected in parallel is,

1Req=1R3+1R4+1R5\frac{1}{{{{R'}_{{\rm{eq}}}}}} = \frac{1}{{{R_3}}} + \frac{1}{{{R_4}}} + \frac{1}{{{R_5}}}

Here, Req{R'_{{\rm{eq}}}} is the equivalent resistance of the resistors connected in parallel, R3,R4,R5{R_3},{R_4},{R_5} are the resistors connected in parallel.

(D)

The given combination of resistors can be divided into top and bottom branch of resistors.

The top branch resistors are shown in the figure below:

R
Ry

The expression to calculate the equivalent resistance of the resistor connected in parallel to the top branch is,

1Reqt=1R3+1R4+1R5\frac{1}{{{{R'}_{{\rm{eqt}}}}}} = \frac{1}{{{R_3}}} + \frac{1}{{{R_4}}} + \frac{1}{{{R_5}}}

Here, Reqt{R'_{{\rm{eqt}}}} is the equivalent resistance for three resistors connected in parallel to the top branch in the circuit and R3,R4,R5{R_3},{R_4},{R_5} are the resistors connected in parallel combination.

Substitute 4Ω4\,\Omega for R3{R_3} , 6Ω6\,\Omega for R4{R_4} , and 12Ω12\,\Omega for R5{R_5} to find Reqt{R'_{{\rm{eqt}}}} .

1Reqt=14Ω+16Ω+112Ω=12ΩReqt=2Ω\begin{array}{c}\\\frac{1}{{{{R'}_{{\rm{eqt}}}}}} = \frac{1}{{4\,\Omega }} + \frac{1}{{6\,\Omega }} + \frac{1}{{12\,\Omega }}\\\\ = \frac{1}{{2\,\Omega }}\\\\{{R'}_{{\rm{eqt}}}} = 2\,\Omega \\\end{array}

The expression for the equivalent resistance of the resistor connected in series at the top branch is, Rtop=R1+R2+Reqt{R_{{\rm{top}}}} = {R_1} + {R_2} + {R'_{{\rm{eqt}}}}

Substitute 1Ω1\,\Omega for R1{R_1} , 3Ω3\,\Omega for R2{R_2} , 2Ω2\,\Omega for Reqt{R'_{{\rm{eqt}}}} .

Rtop=1Ω+3Ω+2Ω=6Ω\begin{array}{c}\\{R_{{\rm{top}}}} = 1\,\Omega + 3\,\Omega + 2\,\Omega \\\\ = 6\,\Omega \\\end{array}

The bottom branch resistors are shown in the figure below:

Rs

The expression for the equivalent resistance of the resistor connected in parallel at the bottom branch is,

1Reqb=1R6+1R7\frac{1}{{{{R'}_{{\rm{eqb}}}}}} = \frac{1}{{{R_6}}} + \frac{1}{{{R_7}}}

Here, Reqb{R'_{{\rm{eqb}}}} is the equivalent resistance for two resistors at the bottom branch of the circuit and R6,R7{R_6},{R_7} are the resistors connected in parallel combination.

Substitute 5Ω5\,\Omega for R6{R_6} and 20Ω20\,\Omega for R7{R_7} to find Reqb{R'_{{\rm{eqb}}}} .

1Reqb=1R6+1R71Reqb=15Ω+120Ω=14ΩReqb=4Ω\begin{array}{c}\\\frac{1}{{{{R'}_{{\rm{eqb}}}}}} = \frac{1}{{{R_6}}} + \frac{1}{{{R_7}}}\\\\\frac{1}{{{{R'}_{{\rm{eqb}}}}}} = \frac{1}{{5\,\Omega }} + \frac{1}{{20\,\Omega }}\\\\ = \frac{1}{{4\,\Omega }}\\\\{{R'}_{{\rm{eqb}}}} = 4\,\Omega \\\end{array}

The expression for the equivalent resistance of the resistor connected in series at the bottom branch is, Rbottom=Reqb+R8{R_{{\rm{bottom}}}} = {R'_{{\rm{eqb}}}} + {R_8}

Substitute 4Ω4\,\Omega for Reqb{R'_{{\rm{eqb}}}} and 2Ω2\,\Omega for R8{R_8} in the above expression.

Rbottom=4Ω+2Ω=6Ω\begin{array}{c}\\{R_{{\rm{bottom}}}} = 4\,\Omega + 2\,\Omega \\\\ = 6\,\Omega \\\end{array}

Now, the equivalent resistance between point A and B is,

Rop
В
A
Rbottom

The expression to calculate the equivalent resistance for parallel combination of resistors between point A and B is,

1REQ=1Rtop+1Rbottom\frac{1}{{{R_{EQ}}}} = \frac{1}{{{R_{{\rm{top}}}}}} + \frac{1}{{{R_{{\rm{bottom}}}}}}

Substitute 6Ω6\,\Omega for Rtop{R_{{\rm{top}}}} and 6Ω6\,\Omega for Rbottom{R_{{\rm{bottom}}}} in the above expression.

1REQ=16Ω+16Ω=(6Ω)(6Ω)(6Ω+6Ω)=3Ω\begin{array}{c}\\\frac{1}{{{R_{EQ}}}} = \frac{1}{{6\,\Omega }} + \frac{1}{{6\,\Omega }}\\\\ = \frac{{\left( {6\,\Omega } \right)\left( {6\,\Omega } \right)}}{{\left( {6\,\Omega + 6\,\Omega } \right)}}\\\\ = 3\,\Omega \\\end{array}

Ans: Part D

Thus, the equivalent resistance between points A and B is 3Ω3{\rm{ }}\,\Omega .

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