Parallel circuit rules

The parallel circuit rules show how to use Ohm's law when the circuit has more than one device receiving electrical energy. It will also help us see how the resistance, the current, and the voltage change in the parallel circuit. A parallel circuit is shown below. Notice that the circuit has 2 switches (green), 1 voltage source of 24 volts, and 2 resistors. 

To be more precise, the circuit has 2 pathways or 2 branches.

Simply put, the current has the ability to "make choices" as to where it would go. If the current can either go this way or that way, then the circuit is a parallel circuit.

Look carefully at the circuit below and you will see that the current can either go to the 8 Ohms resistor or the 4 Ohms resistor. This is what we mean by "making choices."

Take a close look at the two red dots. Did you notice that both devices are connected to the same two red dots? This means that both devices are fed by the same voltage of 24 volts. 

Let V be the voltage feeding the circuit

Let V₁ be the voltage feeding the device with the 4 Ohms resistor.

Let V₂ be the voltage feeding the device with 8 Ohms resistor.

Then, V = V₁ = V₂

Now, let us talk about the current. Let us pretend that the resistance is the same in both branches.

Since there is a resistance, the current going there will be less.

Since the resistance is the same for both branches, it makes sense to say that both branches will receive the same amount of current.

The current I is split into I₁ and I₂. Half of the current will go to the branch with the 4 Ohms resistor and the other half to the branch with the 4 Ohms resistor.

If one resistor is equal to 4 Ohms and the other 8 Ohms like the first figure, more current will go to the branch with 4 Ohms. In fact twice as much current will go to the branch with a lesser resistance.

However, the amount of current that will go to the two branches must equal the amount that came from the source or the current I.

I = I₁ + I₂

Now we can find the resistance for the circuit using I = I₁ + I₂

Let R₁ and R₂ be the resistances in the 2 branches. Let R_p be the resistance of the circuit. The resistance of the circuit is usually called equivalent resistance. Ohm's law will apply separately to each branch.

I = I₁ + I₂

I = V / R1

+   V / R2

I = ( 1 / R1

+   1 / R2 ) V    equation 1

Comparing equation 1 and equation 2, we get

Parallel circuit rules generalization

The parallel circuit rules can be summarized and generalized as below

Let V be the voltage feeding the circuit

Let V₁, V₂,..., V_n be the voltage feeding other devices in the branches

Then, V = V₁ = V₂ = ... = V_n

Let I be the current in the circuit

Let I₁, I₂,..., I_n be the current going to other devices in the branches

Then, I = I₁ + I₂ + ... + I_n

Let R₁, R₂, ... , R_n be the resistances in the branches. Let R_p be the equivalent resistance.

1 / Rp

=   1 / R1

+   1 / R2 + ...

+   1 / Rn

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