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. 

  In a parallel circuit, the current has more than 1 pathway.

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. 

Rule #1: Devices in parallel circuits have the same voltage as the voltage source.

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.

Rule #2: In parallel circuit, the current in the circuit is equal to the the sum of the currents in its parallel branches.

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 Rp 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

 =  
V / Rp

 =  
1 / Rp
V    equation 2

Comparing equation 1 and equation 2, we get

Rule #3:

1 / Rp
   =  
1 / R1
+  
1 / R2

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,..., Vn be the voltage feeding other devices in the branches

Then, V = V1 = V2 = ... = Vn

Let I be the current in the circuit

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

Then, I = I1 + I2 + ... + In

Let R1, R2, ... , Rn be the resistances in the branches. Let Rp be the equivalent resistance.

1 / Rp
   =  
1 / R1
+  
1 / R2
+ ...
+  
1 / Rn

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