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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.

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.

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

$$R_{eq}=R_1+R_2+R_3+...$$

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 $$R_1=3\,\Omega$$, $$R_2=5\,\Omega$$, $$R_3=7\,\Omega$$

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

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

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 :  $$R_1=3\,\Omega$$, $$R_2=5\,\Omega$$, $$R_3=5\,\Omega$$

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

$$R_{eq}=R'+R''$$

$$R_{eq}=\dfrac{R}{2}+\dfrac{R}{2}$$

$$R_{eq}=R$$

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

$$\dfrac{1}{R'}=\dfrac{1}{10}+\dfrac{1}{10}$$

$$\dfrac{1}{R'}=\dfrac{2}{10}$$

$$R'=5\,\Omega$$

R' and $$5\,\Omega$$ are in series

$$R_{eq}=5+5$$

$$=10\,\Omega$$

Determine $$R_{eq}$$ of the circuit shown.

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.

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 $$10\Omega$$.

A

5 A

.

B

10 A

C

2.5 A

D

zero

Option D is Correct

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

$$\dfrac{1}{R'}=\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{2}{4}$$

$$R'=2\,\Omega$$

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

$$\dfrac{1}{R''}=\dfrac{1}{6}+\dfrac{1}{6}=\dfrac{2}{6}$$

$$R''=3\,\Omega$$

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

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

$$R'''=\dfrac{6}{5}\,\Omega$$

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

$$\dfrac{1}{R''''}=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{2}{2}$$

$$R''''=1\,\Omega$$

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

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

Find the equivalent resistance of the following circuit.

A

$$6\,\Omega$$

.

B

$$4\,\Omega$$

C

$$2.2\,\Omega$$

D

$$3\,\Omega$$

Option C is Correct