Tutorial 2 - Capacitor Circuits

Please read the extension material for a more in depth treatment of the topic

Learning Objective

  • Recognise series and parallel capacitor circuits

  • Calculate the single capacitor equivalents

Parallel Capacitors

Here is a circuit consisting of two capacitors in parallel. They have values C1 and C2 and are connected to a battery of voltage V.

 

Ctot = C1 + C2

This is true for any number of parallel capacitors, so

 

Ctot = C1 + C2 + C3 + + Cn

Worked Example

What is the single capacitor equivalent of this circuit below? What is the charge on each capacitor?

  • Use Ctot = C1 + C2 to get the capacitance

  • Use Q = CV to get the charge.

Answer

Ctot = C1 + C2 = 4 mF + 6 mF = 10 mF

            Charge on C1 = 4 10-6 F 12 V = 4.8 10-5 C = 48 mC

            Charge on C2 = 6 10-6 F 12 V = 7.2 10-5 C = 72 mC

            Total charge = 48 mC + 72 mC = 120 mC

 

Question 1

What is the single capacitor equivalent of these parallel capacitors?

Answer

 

Series Capacitors

Here is a circuit consisting of two capacitors in series. They have values C1 and C2 and are connected to a battery of voltage V.

In any series circuit

 

Since Q = It, it is reasonable to say that the charge that has moved is the same all the way round. If a number of electrons of total charge of Q crowds onto the negative plates of C2 then the same number of electrons are repelled away from the positive plates. These crowd onto the negative plates of C1 and repel the same number away from the positive plates.

 

Now we know that V = Q/C and that Vtot = V1 + V2. So we can write:

This gives us a general relationship for any number of series capacitors:

We can tackle problems that involve both series and parallel capacitors in a similar way to the way we tackle problems with combined series and parallel arrays of resistors.

 

Worked Example

What is the single capacitor equivalent of this circuit below? What is the charge on each capacitor? What are the voltmeter readings?

 

Use

  • 1/Ctot = 1/C1 + 1/C2

  • Voltages add up to the battery voltage.

Answer

Work out the total capacitance:

           

Ctot = 12/5 = 2.4 mF        

Now we can work out the charge:

Q = CV = 2.4 mF 12 V = 28.8 mC

Now work out the voltages:

  • On capacitor 1, V = 28.8 mC 4 mF = 7.2 V

  • On capacitor 2, V = 28.8 mC 6 mF = 4.8 V

Voltages add up. Battery voltage = 7.2 + 4.8 = 12.0 V (Kirchhoff II)

 

 

Question 2

What is the single capacitor equivalent of these series capacitors?

Answer

 

When you tackle problems involving both series and parallel capacitors in the same circuit, you may find it helpful to adopt the following problem solving strategy:

 

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