Encyclopedia of Electrical Engineering

A parallel circuit can now be established by connecting a supply across
a set of parallel resistors as shown in Fig. No.1. The positive terminal of
the supply is directly connected to the top of each resistor, while the
negative terminal is connected to the bottom of each resistor. Therefore,
it should be quite clear that the applied voltage is the same across each
resistor.
**Fig.no.1: **Parallel Network.
In general,
Therefore, remember that
For the voltages of the circuit in Fig. No.1, the result is that
$$ E = V_1 = V_2 $$
Once the supply has been connected, a source current is established
through the supply that passes through the parallel resistors.
The smaller the total resistance, the greater is the current, as
occurred for series circuits also. The source current can then be determined
using Ohm's law:
$$I_s = {E \over R_T}$$
Since the voltage is the same across parallel elements, the current
through each resistor can also be determined using Ohm's law. That is,
$$ I_1 ={V_1 \over R_1} = {E \over R_1}$$
$$ I_2 = {V_2 \over R_2} = {E \over R_2}$$
The direction for the currents is dictated by the polarity of the voltage
across the resistors. Recall that for a resistor, current enters the positive
side of a potential drop and leaves the negative side. The result, as shown
in Fig.No.1, is that the source current enters point a, and currents I1 and
I2 leave the same point.
$$\bbox[5px,border:1px solid red] {\color{blue}{ I_s = I_1 + I_2}} $$
$$ {E \over R_T} = {V_1 \over R_1} + { V_2 \over R_2}$$
But $ E = V_1 = V_2$
$$ {E \over R_T} = {E \over R_1} + { E \over R_2}$$
$$ \bbox[5px,border:1px solid red] {\color{blue}{{1 \over R_T} = {1 \over R_1} + { 1 \over R_2}}}$$
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