Series and Parallel Resistance
Resistors are said to be connected in series, if the same current is flowing through each resistor when some potential difference is applied across the combination.
If the three resistances R1, R2 and R3 are connected as shown in they are said to be connected in series. let v be the potential difference applied across A and B using the battery E . in series combination, the same current (say) will be passing through each resistance. Let will be passing through each resistance. Let V1, V2, V3, be the potential difference across R1, R2, and R3, respectively. According to ohm’ law
V1 = iR1, V2, = lR2, V3, = lR3
Here, V = V1, + V2, + V3 = lR1 + lR2 + lR3 = I(R1 + R2 + R3)
If Rs is the equivalent resistance of the given series combination of resistances, them the potential difference across A and B is, V = IRs.
From above equations we have,
IRs = I(R1 + R2 + R3)
Or, Rs = R1 + R2 + R3
Thus the equivalent resistance of a number of resistors connected in series is equal to the sum of individual resistances.
It shows that the value of equivalent resistance of series combination of resistors is always greater than the resistance of individual resistors. Therefore, the resistors are to be connected in series if the effective resistance in the circuit is to be increased.
In a series resistance circuit, it should be noted that;
(i) The current is same in every resistor.
(ii) The current in the circuit is independent on the relative positions of the various resistors in the series.
(iii) The voltage across any resistor is directly proportional to the resistance of that resistor.
(iv) The total resistance in the circuit is equal to the sum of the individual resistances including the internal resistance of a cell if any.
(v) The total resistance in the series circuit is more than the greatest one in the circuit.
Resistances in parallel
The number of resistors are said to be connected in parallel if potential difference across each of them is the same and is equal to the applied potential difference.
If three resistances R1, R2, and R3 are connected as they are said to be connected in parallel. let V be the potential difference applied across A and B with the help of a battery E. let I be the main current in the circuit from battery and I1, I2, I3, be the currents through the resistances R1, R2, and R3 respectively. Then
I = I1 + I2 + I3
Here potential difference across each resistor is V, therefore V = I1 R1 = I2 R2 = I3 R3
Or Ii = V / R1 I2 = V / R2 I3 = V/ R3
Putting values in (19) we get
I = V / R1 + V / R2 + V / R3
If Rp is the equivalent resistance of the given parallel combination of resistances then
V = IRp or I = V/Rp
From above equations we have,
V / Rp = V / R1 + V / R2 + V / R3
Or, 1 / Rp = 1 / R1 + 1 / R2 + 1 / R3
Thus the reciprocal of equivalent resistance of number of resistors connected in parallel is equal to the sum of the reciprocals of the individual resistances.
It shows that the value of equivalent resistance of parallel combination of resistors is always less than the resistance of individual resistors. There froe, the resistors are to beconectedin parallel, if the effective resistance in the circuit is to be decreased.
In a parallel resistance circuit, it should be noted be
(i) The potential difference across each resistor is the same and is equal to the applied potential difference.
(ii) The current through each resistor is inversely proportional to the resistance of that resistor.
(iii) Total current through the parallel combination is the sum of the individual current through the various resistors.
(iv) The reciprocal of the total resistance of the parallel combination is equal to the sum of the reciprocals of the individual resistances.
(v) The total resistance in parole combination is less than the least resistance used in circuit.
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