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Wize University Physics Textbook (Master) > DC Circuits

Resistors in Series and Parallel

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Resistors in Series and Parallel


Resistors in Series

Resistors are said to be in series if they are wired one after the other along the same piece of wire.




The equivalent resistance is the sum of the individual resistances:

 Req=R1+R2+...+Rn \boxed{\ R_{eq}=R_1+R_2+...+R_n \ }


The current is the same through all resistors:
 I1=I2=...=In \boxed{ \ I_1=I_2=...=I_n \ }


The voltage drops across each resistor add up to the total voltage:

 Vsource=V1+V2+...+Vn \boxed{\ V_{source}=V_1+V_2+...+V_n \ }








NOTE: You might see various notations used for the equivalent resistance: it can also be called net, effective, total etc.
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Resistors in Parallel

Resistors are said to be in parallel if they are wired as separate branches, so that the current splits up and recombines at the junctions.



The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances:

 1Req=1R1+1R2+...+1Rn \boxed{\ \dfrac{1}{R_{eq}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...+\dfrac{1}{R_n} \ }


The voltage drops are the same through all resistors:
 V1=V2=...=Vn \boxed{ \ V_1=V_2=...=V_n \ }


The currents across each resistor add up to the total current:

 Isource=I1+I2+...+In \boxed{\ I_{source}=I_1+I_2+...+I_n \ }




Exam Tip
You might see a combination of series and parallel resistors in a circuit. If possible, put them together one by one to find the equivalent resistance of the whole circuit.

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Example: Equivalent Resistance


All resistors in the circuit below have the same resistance RR. Find the equivalent resistance in terms of RR.



Let's first combine the two resistors R3R_3 and R4R_4, which are in parallel:

1R34=1R3+1R4\dfrac{1}{R_{34}}=\dfrac{1}{R_3}+\dfrac{1}{R_4}

=1R+1R=\dfrac{1}{R}+\dfrac{1}{R}

=2R=\dfrac{2}{R}

Taking the reciprocal of both sides we get:

R34=R2R_{34}=\dfrac{R}{2}

Now the combined R34R_{34} is in series together with R1R_1 and R2R_2:

Req=R1+R2+R34R_{eq}=R_1+R_2+R_{34}

=R+R+R2=R+R+\dfrac{R}{2}

=5R2=\dfrac{5R}{2}
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Practice: Combination of Series and Parallel


a) Find an algebraic expression for the equivalent resistance of the circuit below in terms of R1R_1, R2R_2 and R3R_3.

b) Find the current and voltage through each resistor. Use R1=7.5R_1=7.5 Ω, R2=8R_2=8 Ω, R3=4R_3=4 Ω and V=12V=12 V.