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Resistances in series and parallel-explanation, Practice problems, FAQs

Resistors find applications in everyday life; microwaves, heaters, toasters, electric stoves, remote controls. It is used to regulate the flow of current in the circuit. Suppose you take five resistors , and and connect them next to each other in such a way that only one terminal of is connected to one terminal of , one terminal of is connected to one terminal of so on and so forth. This type of connection is called “series arrangement”. You record the current flowing through them by connecting an ammeter. You disconnect the resistors, and then you connect them in such a way that both terminals of are connected to both terminals of are connected to that of and so on. This type of connection is called “parallel arrangement”. You record the current, and cross check it with the value obtained in the previous case. You notice that both values are different. Why? By Ohm’s law, the current . The value of “effective resistance” or “net resistance” is different in series and parallel arrangement. In this article, we will explore resistors in series and parallel, in detail.

Combination of resistors

Combination of resistors refers to multiple resistors connected to each other. They are of two types: series and parallel.

Resistors in series

When the resistors are connected with only one terminal touching one terminal of another resistor, i.e end to end, that is called series connection. When resistors are connected in series, the current flowing will be the same, but the voltage across each resistor may be different.

Below is the equivalent circuit.

Let be the total voltage that is applied across the series combination of resistors and Then

Let indicate current flowing through them.

Now are the respective voltages across and

Let be the equivalent resistance of the series combination.

Substituting these values in equation

If identical resistors are connected in series, then in the above equation, then the effective resistance of the combination becomes,

times

Resistors in parallel

When both terminals of one resistor are connected to both ends of the other resistor, then it is termed a parallel combination. When resistors are connected in parallel, the voltage across them would be the same, but the current flowing through each resistor may be different.

Below is the equivalent circuit.

Let be the total current flowing across the parallel combination of resistors and

Then

Let be the voltage across each resistor. Then,

Now are the respective currents across and

Let be the equivalent resistance of the parallel combination.

Substituting the above values in equation , we get,

When two resistors and are connected in parallel, then

When identical resistors are connected in parallel, then in the above equation,

I.e times.

Then the effective resistance of the combination becomes

Note:

The equivalent resistance in series combination is more than the equivalent resistance in parallel combination.

Video explanation

From 37:40 to 42:56

Practice problems

Q1. ) An infinite ladder network is made with resistances and as shown. The effective resistances between the terminals and is

(a) ∞ (b) (c) (d)

Solution) c The given network can be reduced to

The ladder is nothing but repetition. The effective resistance of the whole combination is nothing but

Since and are connected in parallel, then

(ignoring negative value of R)

Q2) Find the equivalent resistance between and in the following case

Solution) The dotted portion shows three resistors connected in series. Then , the effective resistance of the ladder

Now and are connected in parallel.

The given network can be further reduced to,

Now are connected in series.

and are in parallel.

Let be effective resistance of this. Then

Now is in series with and The equivalent resistance of this network will be

Q3) Calculate the effective resistance between and .

Solution) Let us name the nodes as and The resistor lying between points and are at the same potential, also those lying between and are at the same potential.

Resistors are connected in parallel.

The equivalent resistance between and is

Q4) Calculate the effective resistance between the points and shown in the figure.

Solution) The ratio of resistances in the left arm = ratio of resistances in the right arm

Given network of resistors is known as a balanced wheatstone bridge. The resistor can be removed.

The effective resistance of the upper arm

The effective resistance of the lower arm

and are now connected in parallel.

The effective resistance of the combination,

FAQs

Q1. Write the equation for effective resistance of two resistors and connected in series.

Ans)

Q2. Write the equation for effective resistance of two resistors and connected in parallel.

Ans)

Q3. The effective resistance in series is more than the effective resistance in parallel. Why?

Ans)

If the resistors are identical in value, then

Then and

Resistance in parallel is less compared to resistance in series.

Q4. Write the unit of resistance.

Ans) The unit of resistance is ohm

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