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Equivalent Resistance

Learn equivalent resistance with examples, and short circuit definition. Practice equation to calculate combination of resistors in series and parallel circuits.

Series and Parallel Combination

Series Combination

  • Two resistors connected without other joint present between them are said to be in series.
  • Resistors R1 and R2 are in series combination as no other joint is present in figure.

  • In this figure since a joint is present so, this is not a series combination.

Current in Series Combination

  • Current  \(I\) will be same in R1 and R2 because current will not divide due to absence of any other joint.

  • Current  \(I\) will divide due to presence of joint.

Current in Parallel Combination

  • Since a joint is present between A and A' and similarly between B and B' due to this same current will not flow in R1 and R2 .
  • Such combination is known as parallel combination.

  • Consider a circuit as shown in figure.

  • This circuit is neither simple series nor simple parallel combination because it contains elements of both.

Illustration Questions

Choose the correct option regarding the following circuit.

A R1 and R2 in series

B R2 and R3 in series

C R1 and R3 in series

D None

×

This combination is neither in series nor in parallel.

\(\therefore\)  option D is correct

Choose the correct option regarding the following circuit.

image
A

R1 and R2 in series

.

B

R2 and R3 in series

C

R1 and R3 in series

D

None

Option D is Correct

Resistors in Series

  • In series connection current through each resistor is same.

  • Some amount of charge Q exists on resistor R1. The same charge Q must also enter in R2 otherwise the charge will accumulate between resistors.

  • The potential difference applied across the series combination of resistors divide between the resistors.

\(\Delta V=IR_1+IR_2\)

or,  \(\Delta V=IR_{eq}\)

where  \(R_{eq}=R_1+R_2\)

  • The equivalent resistance for more than two resistors connected in series is 

\(R_{eq}=R_1+R_2+R_3+...\)

Illustration Questions

Determine the equivalent resistance for the circuit. Given \(R_1=3\,\Omega\), \(R_2=5\,\Omega\), \(R_3=7\,\Omega\)

A \(15\,\Omega\)

B \(16\,\Omega\)

C \(17\,\Omega\)

D \(18\,\Omega\)

×

Equivalent resistance for circuit is given as 

\(R_{eq}=R_1+R_2+R_3\)

\(=3+5+7\)

\(R_{eq}=15\,\Omega\)

Determine the equivalent resistance for the circuit. Given \(R_1=3\,\Omega\), \(R_2=5\,\Omega\), \(R_3=7\,\Omega\)

image
A

\(15\,\Omega\)

.

B

\(16\,\Omega\)

C

\(17\,\Omega\)

D

\(18\,\Omega\)

Option A is Correct

Resistors in Parallel

 

  • Potential difference across each resistor is same.

  • Total current \((I)= I_1+I_2\)

\(I=\dfrac{\Delta V}{R_{eq}},\,I_1=\dfrac{\Delta V}{R_1},\,I_2=\dfrac{\Delta V}{R_2}\)

\(\dfrac{\Delta V}{R_{eq}}=\dfrac{\Delta V}{R_1}+\dfrac{\Delta V}{R_2}\)

\(\dfrac{1}{R_{eq}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}\)

  • For more than two resistance connected in parallel

\(\dfrac{1}{R_{eq}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+...\)

Illustration Questions

Determine the equivalent resistance. Given :  \(R_1=3\,\Omega\), \(R_2=5\,\Omega\), \(R_3=5\,\Omega\) 

A \(9\,\Omega\)

B \(8\,\Omega\)

C \(\dfrac{15}{11}\,\Omega\)

D \(\dfrac{11}{15}\,\Omega\)

×

Equivalent resistance for circuit is given as

\(\dfrac{1}{R_{eq}}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}\)

or,  \(\dfrac{1}{R_{eq}}=\dfrac{1}{3}+\dfrac{1}{5}+\dfrac{1}{5}\)

or,  \(\dfrac{1}{R_{eq}}=\dfrac{5+3+3}{15}\)

\({R_{eq}}=\dfrac{15}{11}\,\Omega\)

Determine the equivalent resistance. Given :  \(R_1=3\,\Omega\), \(R_2=5\,\Omega\), \(R_3=5\,\Omega\) 

image
A

\(9\,\Omega\)

.

B

\(8\,\Omega\)

C

\(\dfrac{15}{11}\,\Omega\)

D

\(\dfrac{11}{15}\,\Omega\)

Option C is Correct

Combination of Series and Parallel Combination with 3 or 4 Resistors

  • Consider combination of four resistors connected as shown in figure.

 

Step - 1

For loop (I) R and R in parallel

\(\dfrac{1}{R'}=\dfrac{1}{R}+\dfrac{1}{R}\)

\(R'=\dfrac{R}{2}\)

Step - 2

For loop (II) R and R in parallel

\(\dfrac{1}{R''}=\dfrac{1}{R}+\dfrac{1}{R}\)

\(R''=\dfrac{R}{2}\)

Step - 3

R' and R'' in series

\(R_{eq}=R'+R''\)

\(R_{eq}=\dfrac{R}{2}+\dfrac{R}{2}\)

\(R_{eq}=R\)

Illustration Questions

Determine \(R_{eq}\) of the circuit shown.

A \(5\,\Omega\)

B \(15\,\Omega\)

C \(20\,\Omega\)

D \(10\,\Omega\)

×

\(10\,\Omega\) and \(10\,\Omega\) resistor are in parallel

\(\dfrac{1}{R'}=\dfrac{1}{10}+\dfrac{1}{10}\)

\(\dfrac{1}{R'}=\dfrac{2}{10}\)

\(R'=5\,\Omega\)

image

R' and \(5\,\Omega\) are in series 

\(R_{eq}=5+5\)

\(=10\,\Omega\)

image

Determine \(R_{eq}\) of the circuit shown.

image
A

\(5\,\Omega\)

.

B

\(15\,\Omega\)

C

\(20\,\Omega\)

D

\(10\,\Omega\)

Option D is Correct

Short Circuit

  • When two points of a circuit are connected together by a conducting wire, they are said to be short circuited.
  • The connecting wire is assumed to have zero resistance.

  • Voltage difference between terminal B and D is zero.

                                 \(V_{BD}=0\)

  • Hence, there will be no current in resistances R2 and R3.

Note- The shorted components are not damaged, they will function normally when short circuit is removed.

Illustration Questions

Calculate the value of current in resistance \(10\Omega\).

A 5 A

B 10 A

C 2.5 A

D zero

×

Since B and C are short circuited, the voltage different between B and C is zero.

Hence, no current will flow in resistance \(10\,\Omega\) .

Calculate the value of current in resistance \(10\Omega\).

image
A

5 A

.

B

10 A

C

2.5 A

D

zero

Option D is Correct

Illustration Questions

Find the equivalent resistance of the following circuit.

A \(6\,\Omega\)

B \(4\,\Omega\)

C \(2.2\,\Omega\)

D \(3\,\Omega\)

×

(1) \(4\,\Omega\) and \(4\,\Omega\) resistors are connected in parallel 

\(\dfrac{1}{R'}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{2}{4}\)

\(R'=2\,\Omega\)

image

(2) \(6\,\Omega\) and \(6\,\Omega\) resistors are connected in parallel

\(\dfrac{1}{R''}=\dfrac{1}{6}+\dfrac{1}{6}=\dfrac{2}{6}\)

\(R''=3\,\Omega\)

image

(3) R' and R'' are connected in parallel

\(\dfrac{1}{R'''}=\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{3+2}{6}=\dfrac{5}{6}\)

\(R'''=\dfrac{6}{5}\,\Omega\)

image

(4) \(2\,\Omega\) and \(2\,\Omega\) resistors are connected in parallel 

\(\dfrac{1}{R''''}=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{2}{2}\)

\(R''''=1\,\Omega\)

image

(5) R''' and R'''' are connected in series

\(R_{eq}=R'''+R''''=\dfrac{6}{5}+1=\dfrac{11}{5}=2.2\,\Omega\)

image

Find the equivalent resistance of the following circuit.

image
A

\(6\,\Omega\)

.

B

\(4\,\Omega\)

C

\(2.2\,\Omega\)

D

\(3\,\Omega\)

Option C is Correct

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