Series and Parallel Connections
When several devices are connected end to end in such a way that there is only one path for the current to flow and so, the same current flows through each, then such a circuit is called a "series circuit". When several devices are connected to a common voltage in a certain manner that they provide alternative paths for the current. Where the current in each path (device) will depend on its resistance. Such a circuit is called parallel circuit.
Resistances in Series
When three resistors, R1, R2 and R3 are connected end to end as shown in the figure below, then it would be referred as resistances in series. In case of series connection, the equivalent resistance of the combination, is sum of these three electrical resistances.
V1 = IR1, V2 = IR2 and V3 = IR3.
From figure, the total voltage is the sum of voltage drop across R1, R2 and R3, is V1, V2 and V3 respectively,
V = V1 + V2 + V3
V = IR1+ IR2 + IR3
V = I (R1 + R2 + R3)
Now, if we consider the total combination of resistances as a single resistor of electric resistance value R, then according to Ohm's law ,
V = IR
IR = II (R1 + R2 + R3)
R = R1 + R2 + R3
So the above proof shows that equivalent resistance of a combination of resistances in series is equal to the sum of individual resistance. If there were n number of resistances instead of three resistances, the equivalent resistance will be
R = R1 + R2 + R3 +.........+Rn
Resistances in Parallel
Let's three resistors of resistance value R1, R2 and R3 are connected in such a manner, that right side terminal of each resistor are connected together as shown in the figure below, and also left side terminal of each resistor are also connected together. This combination is called resistances in parallel.If a voltage, V is applied across this combination, then it will draw a current I. As this current will get three parallel paths through these three electrical resistances, the electric current will be divided into three parts. Say currents I1, I1 and I1 pass through resistor R1, R2 and R3 respectively. The total source current will be sum f branch currents,
I = I1 + I2 + I3
Hence,
I1 = V/R1, I2 = V/R2 and I3 = V/R3,
And
I = V / R
where R is the equivalent resistance of the combination.
V / R = V/R1 + V/R2 + V/R3
1 / R = 1/R1 + 1/R2 + 1/R3
The above expression represents equivalent resistance of resistor in parallel. If there were n number of resistances connected in parallel, instead of three resistances, the expression of equivalent resistance would be
1/R = 1/R1 + 1/R2 + 1/R3 + .......... + 1/Rn
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