Combination of Resistances

Combination of Resistances: Resistance measures the amount of opposition offered by a conducting material to the flow of current through it, and a resistor is an electrical component that decreases the electric current passing through a circuit.

Every conductor has some internal resistance that affects the current flow through the circuit. We can connect several resistors in two different types of circuit combinations to maintain the current and potential difference of the circuit to the required levels. To achieve this, we need to find the circuit’s equivalent resistance for a given combination of resistances.

What is a Combination of Resistances?

Sometimes to get a desired value of resistance in a circuit, we combine two or more resistances together. There are two types of combinations of resistances namely, the series combination of resistances and parallel combination of resistances. When the resistance of a combination is to be increased, we connect resistances in series, and when the resistance of the combination is to be decreased, we connect the resistances in parallel.

Equivalent Resistance

If we can replace the combination of two or more resistances connected between two points in an electric circuit by a single resistance such that both the voltage across two points and the current flowing between these two points remain unaltered, then that single resistance is the equivalent resistance of the combination of the resistances.

The equivalent resistance for a series combination of resistances is greater than all individual resistance connected in the circuit, whereas the equivalent resistance for a parallel combination of resistance is less than the smallest resistance connected in the circuit.

Series and Parallel Resistance

When two or more resistances are joined end-to-end so that the output current of the first resistance is the input current of the second resistance and the output current of the second resistance is the input current of the third resistance, and so on, then this circuit combination is known as the series combination of resistances.

In the case of a parallel combination of resistances, two or more resistances are connected in such a way that their terminals are connected to the same common point and the current at these common points or nodes are the same, but the current flowing through each branch may be different.

Series Combination of Resistances

In a series combination of resistances, there is only one path for the current to flow. So, the current flowing through all the individual resistances will be the same. But the voltage across all the resistances may be different. Now, suppose three bulbs of resistances namely ?1,?2R1,R2 and ?3R3 are joined together in a series connection. Let ?I be the current flowing through the circuit, ?1,?2V1,V2, and ?3V3 are the potential difference across each bulb.

Now consider the circuit diagram of resistances given for the bulbs.

Study About Ohm’s Law

The sum of potential differences across these resistances will be equal to the total potential difference (?)(V) across the ends of the circuit

?=?1+?2+?3 ….(1)V=V1+V2+V3 ….(1)

By applying Ohm’s law, we will get

?=???V=IRs

Where ??Rs is the equivalent resistance of the series combination of resistances.

Similarly, potential differences for three resistors will be given by

?1=??1V1=IR1
?2=??2V2=IR2
?3=??3V3=IR3

Putting these values of potential difference in equation (1),(1), we will get

???=??1+??2+??3IRs=IR1+IR2+IR3
⇒??=?1+?2+?3⇒Rs=R1+R2+R3

From here, we can see that the equivalent resistance is the sum of all the individual resistances connected in the circuit and it is greater than all the individual resistances.

Parallel Combination of Resistances

In a parallel combination of resistances, each branch may receive a different amount of current depending upon the value of the resistor connected to that particular branch. The potential difference across each branch, in the parallel circuit, is the same. Suppose three bulbs of resistances namely ?1,?2R1, R2, and ?3R3 are joined in parallel circuit combinations. Let (?)(V) be the potential difference across the circuit and ?1,?2, I1I2, and ?3I3 be the current flowing through each bulb as shown in the figure below.

Now consider the circuit diagram of resistances given for the bulbs.

The sum of current flowing through each branch is equal to the total current flowing through the circuit.

?=?1+?2+?3…..(2)I=I1+I2+I3…..(2)

By applying Ohm’s law, we will get

?=???I=VRp

Where ??Rp is the equivalent resistance of the parallel combination of resistances.

Similarly, the current flowing through each resistor is given by

?1=??1I1=VR1
?2=??2I2=VR2
?3=??3I3=VR3

Putting these values of current in equation (2),(2), we will get

1??=1?1+1?2+1?31Rp=1R1+1R2+1R3

From here, we can see that the equivalent resistance is lesser than each of the three resistances. Also, when resistances are joined in parallel, the area of cross-section effectively increases, so the overall value of resistance decreases.

Why is Parallel Arrangement Used in Domestic Wiring?

In the case of a series arrangement, there is only one path for the current to flow, so the failure of any one device or component of the series arrangement will stop the flow of current through all other devices and components.

The circuit will be open in this case. But in the case of a parallel arrangement, each branch receives a different amount of current. The failure of one device in the parallel arrangement will not affect the flow of current through other branches. Also, the potential difference across each device or component will be the same.

Summary

Resistances can be combined in series as well as in parallel combinations. The effective resistance of a series combination of resistance is the sum of individual resistances. In a series circuit, the current remains the same through all resistances. For a parallel circuit, the effective resistance is smaller than all resistances. The current gets divided in a parallel combination of resistance and the voltage across all branches remains the same. This makes the parallel combination more effective in domestic electrical wiring.

Solved Examples

Q.1. Three resistors of values 2Ω,4Ω2Ω,4Ω, and 6Ω6Ω are connected in parallel. Find the equivalent resistance of the combination of resistors.
Ans: The equivalent resistance, in the case of parallel connection, is equal to the inverse of the sum of the reciprocals of all resistances.
Let ??Rp be the equivalent resistance of the parallel combination, then
1??=1?1+1?2+1?31Rp=1R1+1R2+1R3
⇒1??=12+14+16=1112⇒1Rp=12+14+16=1112
⇒??=1211Ω⇒Rp=1211Ω

Q.2. Find the equivalent resistance between A and B in the circuit diagram given below.

Ans: Here resistances 4Ω4Ω and 8Ω8Ω are in parallel and the resultant is in series with 2Ω2Ω and 4Ω.Ω.
For a parallel combination of resistances, the equivalent resistance is given by
1??=1?1+1?2+1?3…+1??1Rp=1R1+1R2+1R3…+1Rn
Here the equivalent resistance of the parallel combination is given by
1??=14+18=381Rp=14+18=38
??=83=2.67ΩRp=83=2.67Ω
Now, the resultant circuit will be

For a series combination of resistances, will get equivalent resistance(??)(Rs) by using the following formula:
??=?1+?2+?3Rs=R1+R2+R3
??=2+4+2.67=8.67ΩRs=2+4+2.67=8.67Ω
So, the equivalent resistance between point AA, and BB is 8.67Ω8.67Ω.

FAQs

Q.1. Why resistance of several resistances joined in parallel is lesser than the least resistance?
Ans: Suppose three resistors namely ?1,?2R1,R2 and ?3R3 are connected in parallel then the equivalent resistance of this combination will be given by 1??=1?1+1?2+1??1Rp=1R1+1R2+1Rs From this relation, it is clear that 1??>1?11Rp>1R1. Therefore, ??<?1Rp<R1. Similarly, ??<?2Rp<R2 and ??<?3Rp<R3. So, the equivalent resistance is lesser than the least resistance connected in a parallel circuit. When resistances are joined in parallel, the area of cross-section effectively increases, so resistance decreases.

Q.2. What are the laws of the combination of resistances?
Ans: The laws of combination of resistances are as follows:
(a) When two or more resistances are connected in a series combination, then the equivalent resistance of the series combination will be equal to the sum of the individual resistances.
(b) When two or more resistances are connected in parallel combination, then the equivalent resistance of the parallel combination will be equal to the inverse of the sum of the reciprocals of all resistances.

Q.3. What are the disadvantages of a series combination?
Ans: When components are connected in series, then the overall resistance of the circuit increases. This results in the loss of energy. If one component connected in the series circuit gets damaged, then all other components will also stop working because the circuit will be incomplete.

Q.4. What do you know about the combination of resistances?
Ans: There are two methods to combine two more resistances. These methods include the series combination of resistances where all resistances are connected end-to-end and the other method is the parallel combination of resistances where all resistances are connected to common terminals along which the current remains unchanged.
In the series combination of resistances, the current remains the same, and the potential difference across each resistance changes, whereas in the parallel combination of resistances, the voltage across each branch is the same, but the current gets divided in each branch depending upon the value of the resistance.

Q.5. Which combination of resistances is more effective?
Ans: When we consider the electrical connectivity of household appliances, then a parallel combination of resistance is more effective because it reduces unwanted energy loss.
When any appliance connected in a parallel circuit gets damaged, then the other devices will not get affected. They all will work properly.