System of Resistors
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail for 10th Science : Chapter 4 : Electricity : System of Resistors
SYSTEM OF RESISTORS
So far, you have learnt how the resistance of a conductor affects the current through a circuit. You have also studied the case of the simple electric circuit containing a single resistor. Now in practice, you may encounter a complicated circuit, which uses a combination of many resistors. This combination of resistors is known as ‘system of resistors’ or ‘grouping of resistors’. Resistors can be connected in various combinations. The two basic methods of joining resistors together are:
a) Resistors connected in series, and b) Resistors connected in parallel.
In the following sections, you shall compute the effective resistance when many resistors having different resistance values are connected in series and in parallel.
1. Resistors in series
A series circuit connects the components one after the other to form a ‘single loop’. A series circuit has only one loop through which current can pass. If the circuit is interrupted at any point in the loop, no current can pass through the circuit and hence no electric appliances connected in the circuit will work. Series circuits are commonly used in devices such as flashlights. Thus, if resistors are connected end to end, so that the same current passes through each of them, then they are said to be connected in series.
Figure 4.6 Series connection of resistors
Let, three resistances R1, R2 and R3 be connected in series (Figure 4.6). Let the current flowing through them be I. According to Ohm’s Law, the potential differences V1, V2 and V3 across R1, R2 and R3 respectively, are given by:
V1 = I R1 (4.7)
V2 = I R2 ( 4.8)
V3 = I R3 (4.9)
The sum of the potential differences across the ends of each resistor is given by:
V = V1 + V2 + V3
Using equations (4.7), (4.8) and (4.9), we get
V = I R1 + I R2 + I R3 (4.10)
The effective resistor is a single resistor, which can replace the resistors effectively, so as to allow the same current through the electric circuit. Let, the effective resistance of the series-combination of the resistors, be RS. Then,
V = I RS (4.11)
Combining equations (4.10) and (4.11), you get,
I RS = I R1 + I R2 + I R3
$$R_S = R_1 + R_2 + R_3 \quad (4.12)$$
Thus, you can understand that when a number of resistors are connected in series, their equivalent resistance or effective resistance is equal to the sum of the individual resistances. When ‘n’ resistors of equal resistance R are connected in series, the equivalent resistance is ‘n R’.
i.e., RS = n R
The equivalent resistance in a series combination is greater than the highest of the individual resistances.
Solved Problem
Three resistors of resistances 5 ohm, 3 ohm and 2 ohm are connected in series with 10 V battery. Calculate their effective resistance and the current flowing through the circuit.
Solution:
R1 = 5 Ω, R2 = 3 Ω, R3 = 2 Ω, V = 10 V
Rs = R1 + R2 + R3, Rs = 5 + 3 + 2 = 10, hence
Rs = 10 Ω
The current, I = V/Rs = 10/10 = 1 A
2. Resistances in Parallel
A parallel circuit has two or more loops through which current can pass. If the circuit is disconnected in one of the loops, the current can still pass through the other loop(s). The wiring in a house consists of parallel circuits.
Figure 4.7 Parallel connections of resistors
Consider that three resistors R1, R2 and R3 are connected across two common points A and B. The potential difference across each resistance is the same and equal to the potential difference between A and B. This is measured using the voltmeter. The current I arriving at A divides into three branches I1, I2 and I3 passing through R1, R2 and R3 respectively.
According to the Ohm’s law, you have,
$$ I_1 = \frac{V}{R_1} \quad (4.13) $$
$$ I_2 = \frac{V}{R_2} \quad (4.14) $$
$$ I_3 = \frac{V}{R_3} \quad (4.15) $$
The total current through the circuit is given by
I = I1 + I2 + I3
Using equations (4.13), (4.14) and (4.15), you get
$$ I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} \quad (4.16) $$
Let the effective resistance of the parallel combination of resistors be RP. Then,
$$ I = \frac{V}{R_P} \quad (4.17) $$
Combining equations (4.16) and (4.17), you have
$$ \frac{V}{R_P} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} $$
$$ \frac{1}{R_P} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \quad (4.18) $$
Thus, when a number of resistors are connected in parallel, the sum of the reciprocals of the individual resistances is equal to the reciprocal of the effective or equivalent resistance. When ‘n’ resistors of equal resistances R are connected in parallel, the equivalent resistance is R/n.
$$ R_P = \frac{R}{n} $$
The equivalent resistance in a parallel combination is less than the lowest of the individual resistances.
3. Series Connection of Parallel Resistors
If you consider the connection of a set of parallel resistors that are connected in series, you get a series – parallel circuit. Let R1 and R2 be connected in parallel to give an effective resistance of RP1. Similarly, let R3 and R4 be connected in parallel to give an effective resistance of RP2. Then, both of these parallel segments are connected in series (Figure 4.8).
Figure 4.8 Series-parallel combination of resistors
Using equation (4.18), you get
$$ \frac{1}{R_{P1}} = \frac{1}{R_1} + \frac{1}{R_2} $$
$$ \frac{1}{R_{P2}} = \frac{1}{R_3} + \frac{1}{R_4} $$
Finally, using equation (4.12), the net effective resistance is given by Rtotal = RP1 + RP2
4. Parallel Connection of Series Resistors
If you consider a connection of a set of series resistors connected in a parallel circuit, you get a parallel-series circuit. Let R1 and R2 be connected in series to give an effective resistance of RS1. Similarly, let R3 and R4 be connected in series to give an effective resistance of RS2. Then,both of these serial segments are connected in parallel (Figure 4.9).
Figure 4.9 Parallel Connection of Series Resistors
Using equation (4.12), you get
RS1 = R1 + R2,
RS2 = R3 + R4
Finally, using equation (4.18), the net effective resistance is given by
$$ \frac{1}{R_{total}} = \frac{1}{R_{S1}} + \frac{1}{R_{S2}} $$
5. Comparison between series and parallel connections
The difference between series and parallel circuits may be summed as follows in Table 4.3
Table 4.3 Difference between series and parallel circuit
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