Resistance of a System of Resistors | Science Class 10 PDF Download

What is a Resistor?

A passive electrical component with two terminals that are used for either limiting or regulating the flow of electric current in electrical circuits.

• A conducting material (e.g., a wire) of a particular resistance meant for use in a circuit is called a resistor. 
• A resistor is sometimes simply referred to as resistance. It is represented by the symbol. 
• Two or more resistors can be connected in series, in parallel or in a manner that is a combination of these two.

Resistor

Resistors in Series

Two or more resistors are said to be connected in series if the current flowing through one also flows through the rest.

The total potential difference across the combination of resistors connected in series is equal to the sum of the potential differences across the individual resistors.

V = V₁ + V₂ + V₃

Resistors in Series 

Equivalent Resistance in Series Connection

Figure (a) shows three resistors of resistances R₁, R₂ and R₃ connected in series. The cell connected across the combination maintains a potential difference V across the combination. The current through the cell is i. The same current i flows through each resistor.

Let us replace the combination of resistors with a single resistor R_eq such that the current does not change, i.e., it remains i. This resistance is called the equivalent resistance of the combination, and its value is given by Ohm's law as R_eq = V / i

Thus V = i R_eq. 

The potential differences V₁ , V₂ and V₃ across the resistors R₁ , R₂ and R₃ respectively are given by
Ohm's law as : V₁ = iR₁ , V₂ = iR₂ , V₃ = iR₃
Since the resistors are in series, V = V₁ + V₂ + V₃
Substituting the values of the potential differences in the above equation,
iR_eq = iR₁ + iR₂ + iR₃
or iR_eq = i(R₁ +R₂ +R₃)
or R_eq = R₁ + R₂ + R₃
Similarly, for n resistors connected in series,
Equivalent resistance of resistors in series : R_eq = R₁ + R₂ + R₃ + .... R_n

Resistors in Parallel

The total current flowing into the combination is equal to the sum of the currents passing through the individual resistors.
i = i₁ + i₂ + i₃
If resistors are connected in such a way that the same potential difference gets applied to each of them, they are said to be connected in parallel.

Resistors in Parallel

Equivalent Resistance in Parallel Connection

Figure (a) shows three resistors of resistances R₁,  R₂ and R₃ connected in parallel across points A and B. The cell connected across these two points maintains a potential difference V across each resistor. The current through the cell is i. It gets divided at A into three parts i₁, i₂ and i₃, which flow through R₁, R₂ and R₃ respectively.

Let us replace the combination of resistors with an equivalent resistor R_eq such that the current i in the circuit does not change (Fig). The equivalent resistance is given by Ohm's law as R_eq = V/i.

Thus,

The currents i₁ , i₂ and i₃ through the resistors R₁, R₂ and R₃ respectively are given by Ohm's law as

Since the resistors are in parallel,

i = i₁ + i₂ + i₃

Substituting the values of the currents in the above equation,

Similarly, if there are n resistors connected in parallel, their equivalent resistance R_eq is given by

Equivalent resistance of resistors in parallel:

 For two resistances R₁ and R₂ connected in parallel,

The equivalent resistance in a parallel connection is less than each of the resistances.

When a resistance is joined parallel to a comparatively smaller resistance, the equivalent resistance is very close to the value of the smaller resistance.

Note: If a resistor connected in series with others is removed or fails, the current through each resistor becomes zero. On the other hand, if a resistor connected in parallel with others fails or is removed, the current continues to flow through the other resistors.

Distribution of Current in Two Resistors in Parallel

i = i₁ + i₂ .....(i)
Applying Ohm's law to the resistor R₁           
V_A - V_B =R₁i₁· .....(ii)
And applying Ohm's law to the resistor R₂
V_A - V_B = R₂i₂ .... (iii)
From (ii) and (iii), R₁i₁ = R₂i₂ or  
Substituting for i₂ in (i), we have

or  
Similarly,  
Thus,  
The current through each branch in a parallel combination of resistors is inversely proportional to its resistance.

Devices in Series and Parallel

You must have seen tiny bulbs strung together for decorating buildings during festivals like Diwali, and occasions like marriages, etc. 

• These bulbs are connected in series, and the mains voltage is applied to the combination. The potential difference (V) of the mains gets divided across the bulbs (V = V₁ + V₂ + V₃ + ... ). 
• So, a small potential difference exists across each bulb, close to that required to make the bulb work. However, the same current flows through all the bulbs. So, if one bulb goes bad, the current through it stops, and this stops the current through the rest of the bulbs as well. 
• To make the chain of lights work, we have to find and replace the defective bulb. This problem does not occur with the lights in our house. That is because in houses, lights, fans, etc., are connected in parallel. 
• In a parallel connection, the same mains voltage gets applied to each device, but the current through each is different. If one of them goes bad, the current in the other branches of the parallel connection does not stop. 
• Another advantage of parallel connection is that, unlike series connection, each device can draw a different current, as per its requirement.