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.

**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
â€˜system of resistorsâ€™ or â€˜grouping of resistorsâ€™. 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.

A series circuit
connects the components one after the other to form a â€˜single loopâ€™. 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**.

Let, three resistances R_{1},
R_{2} and R_{3} be connected in series (Figure 4.6). Let the current
flowing through them be I. According to Ohmâ€™s Law, the potential differences V_{1},
V_{2} and V_{3} across R_{1}, R_{2} and R_{3}
respectively, are given by:

V_{1} = I R_{1}
(4.7)

V_{2} = I R_{2} (
4.8)

V_{3} = I R_{3}
(4.9)

The sum of the potential differences
across the ends of each resistor is given by:

V = V_{1} + V_{2} +
V_{3}

Using equations (4.7),
(4.8) and (4.9), we get

V = I R_{1} + I R_{2}
+ I R_{3} (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 R_{S}. Then,

V = I R_{S}
(4.11)

Combining equations
(4.10) and (4.11), you get,

I R_{S} = I R_{1}
+ I R_{2} + I R_{3}

R_{S} = R_{1}
+ R_{2} + R_{3} (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 â€˜nâ€™ resistors of equal resistance R are connected in series,
the equivalent resistance is â€˜n Râ€™.

i.e., R_{S} = 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:**

R_{1} = 5 â„¦, R_{2} = 3 â„¦, R_{3} = 2 â„¦, V = 10 V

Rs = R_{1} + R_{2} + R_{3}, Rs = 5 + 3 + 2 = 10,
hence

Rs = 10 â„¦

The current, I = V/Rs = 10/10 = 1 A

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.

Consider that three
resistors R_{1}, R_{2} and R_{3} 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 I_{1}, I_{2} and I_{3} passing through R_{1},
R_{2} and R_{3} respectively.

According to the Ohmâ€™s
law, you have,

The total current
through the circuit is given by

I = I_{1} + I_{2}
+ I_{3}

Using equations (4.13),
(4.14) and (4.15), you get

Let the effective
resistance of the parallel combination of resistors be R_{P}. Then,

Combining equations
(4.16) and (4.17), you have

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 â€˜nâ€™ resistors of equal resistances R are connected
in parallel, the equivalent resistance is R/n.

*The equivalent
resistance in a parallel combination is less than the lowest of the individual
resistances.*

If you consider the
connection of a set of parallel resistors that are connected in series, you get
a series â€“ parallel circuit. Let R_{1} and R_{2} be connected
in parallel to give an effective resistance of R_{P1}. Similarly, let R_{3} and
R_{4} be connected in parallel to give an effective resistance of R_{P2}.
Then, both of these parallel segments are connected in series (Figure 4.8).

Using equation (4.18),
you get

Finally, using equation (4.12), the
net effective resistance is given by R_{total} = R_{P1} + R_{P2}

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).

Using equation (4.12),
you get

R_{S1} = R_{1}
+ R_{2},

R_{S2} = R_{3}
+ R_{4}

Finally, using equation
(4.18), the net effective resistance is given by

The difference between
series and parallel circuits may be summed as follows in Table 4.3

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10th Science : Chapter 4 : Electricity : System of Resistors |

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