9. Parallel circuits

Definition

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In a parallel circuit, the elements are conected so that the voltage is equal across each parallel component. If you look at how they are connected in the figure, you can see that, in this case, the wire is not a single continuous line with inserted elements, as it was in the series circuit. Now, the wire splits off into several lines, so that the electric current has more of one independent path to travel. These lines are usually represented in electrical diagrams as parallel to each other, hence its name. 

The red arrows in the figure represent the current intensity. 

PhET Interactive Simulations, University of Colorado Boulder, https://phet.colorado.edu , CC-BY-4.0 

Batteries in parallel

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Batteries connected in parallel must have the same voltage. The positive pole of one is connected to the positive pole of the other and the negative poles are also connected to each other. If this were done the other way around, such an intense electric current would be produced between the two that it would destroy them. 

 

In this connection, the electrons find two possible branches to circulate, so that they either pass through one cell or the other. Therefore:

  • The current intensity of the main branch is the sum of the intensities that pass through each of the branches :

I = I1 + I2

  • The potential difference across any element of the circuit is equal (all electrons will gain the same energy, when passing through one cell or the other):

Batteries connected in parallel last longer, since only half of the electrons pass through each one of the total that would pass through if there were one single battery.

Resistors in parallel

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In a parallel circuit, all the resistors must be connected to the two terminals of the battery. The cable coming out of the battery splits off, offering two possible paths for the electrons: some of them pass through one resistor and the rest pass through the other. Therefore:

  • The current intensity equals the adition of the intensities that pass through each branch:

I = I1 + I2

  • The potential difference across resistors is equal.

V = V1 = V2

                                                                   Parallel resistor diagram                             

The whole circuit resistance is called equivalent resistance. To calculate the equivalent resistance of a parallel circuit, we use this equation:

1 divided by R equals 1 divided by r1 + 1 divided by r2

Audio of the equation: 

Solved example

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In the circuit in the figure, calculate :

a) the voltage across each resistor

In parallel circuits the voltage is equal across all the elements V = 12 V

b) the current that flows through each resistance

Knowing V and R we can calculate I with Ohm's law:

I 1  = V/R 1  = 12/3 = 4 A

                                         I 2  = V/R 2  = 12/5 = 2.4 A

c) the equivalent resistance of the circuit

                              

R = 1 / 0.53 = 1.9 Ω

Audio:  

Exercise. Drop-down activity

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Three resistors, of 2, 3 and 6 Ω, are connected in parallel to a 12 V battery. Calculate the current that flows through the battery .

Choose the correct option in each case to solve the problem.

Known quantities: R1 = 2 Ω,  R2 = 3 Ω,  R3=  Ω

           V = V1 = V2 = V3 = (circuit in parallel)

Unknown quantitty:

Equation: To calculate I, we have these two equations:

I = I1 + I2 + I3 (parallel circuit)

I = / (Ohm's law)

In this case, we are going to use the first one. First, we need to calculate the current intensities that pass through each branch:

I1  = V1  / R1  = 12 / 2 = 6 A

I2 = V2 / R2 =   / =

I3  = V3 / R3  = 12 / 6 = 2 A 

And then we can add them to calculate the total I:

I = I1 + I2 + I3 = + + = 12 A

Audio: 

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Solving problems autonomy exercise

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Solve this problem in your notebook following all the steps and write the solution in the blank space provided. Then check with the feedback that you have solved it correctly. If there are mistakes or missed steps, write them down in the exercise in your notebook with a red pen.

Calculate the equivalent resistance of a parallel circuit with two 4 Ω resistors . For a current of 6 A to flow through them, what  should be the voltage of the battery they are connected to?

Audio: 

Answer:

R =  Ω

V =

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