Chapter 19

DC Circuits

Objectives: The student will be able to:

• Determine the equivalent capacitance of capacitors arranged in series or in parallel or the equivalent capacitance of a series parallel combination.

• Determine the charge on each capacitor and the voltage drop across each capacitor in a circuit where capacitors are arranged in series, parallel, or a series parallel combination.

19.4 EMFs in Series and in Parallel; Charging a Battery

EMFs in series in the same direction: total voltage is the sum of the separate voltages

19.4 EMFs in Series and in Parallel; Charging a Battery

EMFs in series, opposite direction: total voltage is the difference, but the lower-voltage battery is charged.

EMF’s in Series and in Parallel: Charging a Battery

If you put batteries in series the “right way,” their voltages add:

+ =6 V 3 V 9 V

+ =6 V 3 V 3 V

If you put batteries in series the “wrong way,” their voltages add algebraically:

magnitudes only

chosen loop direction -6 V +3 V -3 V algebraically, using chosen loop direction

Algebraic addition of voltages for batteries in series comes directly from Kirchoff’s loop rule.

Why would you want to connect batteries in series?

More voltage! Brighter flashlights, etc. Chemical reactions in batteries yield a fixed voltage. Without changing the chemical reaction (i.e., inventing a new battery type), the only way to change voltage is to connect batteries in series.

This applies to any source of emf, not just batteries!

Go to www.howstuffworks.com to see how batteries work.They even expose the secret of the 9 volt battery!

Click on the picture above only if you are mature enough to handle this graphic exposé.

Go to www.howstuffworks.com to see how batteries work.They even expose the secret of the 9 volt battery!

Shocking!

Six 1.5 V batteries in series!

Why would you want to connect batteries in series the “wrong” way?

You probably don’t want to.

Use could use one battery to charge another—doesn’t seem too useful, although might be in special cases.

But remember, Kirchoff’s loop rule applies to all emf’s.

You could connect a source of emf – like the alternator in your car – so that it charges a battery.

Rechargeable batteries use an ac to dc converter as a source of emf for recharging.

Could you connect batteries (or sources of emf) in parallel?Sure!

3 V

3 V

You would still have a 3 V voltage drop across your resistor, but the two batteries in parallel would “last” longer than a single battery.

You could use Kirchoff’s rules to analyze this circuit and show that Vab = 3 V.

a b

19.4 EMFs in Series and in Parallel; Charging a Battery

EMFs in parallel only make sense if the voltages are the same; this arrangement can produce more current than a single emf.

It is used to provide more energy when large currentsare needed. Each of the cells in parallel has to produce only a fraction of the total current, so theenergy loss due to internal resistance is less than fora single cell; and the batteries will go dead less quickly.

ISeries and Parallel EMFs; Battery Charging

Example 26-10: Jump starting a car.A good car battery is being used to jump start a car with a weak battery. The good battery has an emf of 12.5 V and internal resistance 0.020 Ω. Suppose the weak battery has an emf of 10.1 V and internal resistance 0.10 Ω. Each copper jumper cable is 3.0 m long and 0.50 cm in diameter, and can be attached as shown. Assume the starter motor can be represented as a resistor Rs = 0.15 Ω. Determine the current through the starter motor (a) if only the weak battery is connected to it, and (b) if the good battery is also connected.

I Answer to Example 19-9

I Answer to Example 19-9

I Answer to Example 19-9

19.5 Circuits Containing Capacitors in Series and in Parallel

Capacitors in parallel have the same voltage across each one:

Circuits Containing Capacitors in Series and in Parallel

Capacitor:C

Capacitors connected in parallel:

C1

C2

C2

+ -

V

The voltage drop from a to b must equal V.

a b

Vab = V = voltage drop across each individual capacitor.

Vab

C1

C2

C3

+ -

V

a

Q = C V

Q1 = C1 V& Q2 = C2 V

& Q3 = C3 V

Now imagine replacing the parallel combination of capacitors by a single equivalent capacitor.

By “equivalent,” we mean “stores the same total charge if the voltage is the same.”

Ceq

+ -

V

a

Q1 + Q2 + Q3 = Ceq V = Q

Q3

Q2

Q1

+ -

Q

Q1 = C1 V Q2 = C2 V Q3 = C3 V

Q1 + Q2 + Q3 = Ceq V

Summarizing the equations on the last slide:

Using Q1 = C1V, etc., in the second line gives

C1V + C2V + C3V = Ceq V

C1 + C2 + C3 = Ceq (after dividing both sides by V)

Generalizing:

Ceq = Ci (capacitors in parallel)

Does this remind you of any of our resistor equations?

See Giancoli’s comment on why this makes sense, p. 533.

C1

C2

C2

+ -

V

a b

19.5 Circuits Containing Capacitors in Series and in Parallel

In this case, the total capacitance is the sum:

(19-5)

Capacitors connected in series:

C1 C2

+ -

V

C3

An amount of charge +Q flows from the battery to the left plate of C1. (Of course, the charge doesn’t all flow at once).

+Q -Q

An amount of charge -Q flows from the battery to the right plate of C3. Note that +Q and –Q must be the same in magnitude but of opposite sign.

C1 C2

+ -

V

C3

+QA

-QB

The charges +Q and –Q attract equal and opposite charges to the other plates of their respective capacitors:

-Q +Q

These equal and opposite charges came from the originally neutral circuit regions A and B.

Because region A must be neutral, there must be a charge +Q on the left plate of C2.

Because region B must be neutral, there must be a charge --Q on the right plate of C2.

+Q -Q

C1 C2

+ -

V

C3

+QA

-QB

Here’s the circuit after the charges have moved and a steady state condition has been reached:

-Q +Q+Q -Q

Q = C1 V1 Q = C2 V2 Q = C3 V3

The charges on C1, C2, and C3 are the same, and are

But we don’t know V1, V2, and V3 yet.

a b

We do know that Vab = V and also Vab = V1 + V2 + V3.

V3V2V1

Ceq

+ -

V

+Q -QV

Let’s replace the three capacitors by a single equivalent capacitor.

By “equivalent” we mean V is the same as the total voltage drop across the three capacitors, and the amount of charge Q that flowed out of the battery is the same as when there were three capacitors.

Q = Ceq V

Collecting equations:

Q = C1 V1 Q = C2 V2 Q = C3 V3

Vab = V = V1 + V2 + V3.

Q = Ceq V

Substituting for V1, V2, and V3:1 2 3

Q Q QV = + +

C C C

Substituting for V:eq 1 2 3

Q Q Q Q = + +

C C C C

Dividing both sides by Q:eq 1 2 3

1 1 1 1 = + +

C C C C

19.5 Circuits Containing Capacitors in Series and in Parallel

In this case, the reciprocals of the capacitances add to give the reciprocal of the equivalent capacitance:

(19-6)

Practice Problem 1 – Similar to p. 549 #35

Six 4.5 μF capacitors are connected in parallel and in series. Find equivalent capacitance for each case.

Practice Problem 1 – Answer

Six 4.5 μF capacitors are connected in parallel and in series. Find equivalent capacitance for each case.

Virtual Capacitor Lab

Homework:

Chapter 19: 34, 37, 41, 42, 43, 44Due in 2 days

Kahoot 19-4 and 19-5