sp212 lesson 12 - united states naval academy · well as the energy stored! sp212 lesson 12...

SP212 Lesson 12Ch. 25, 4-5 – Capacitor Energies and Dielectrics

February 1, 2018

SP212 Lesson 12 February 1, 2018 1 / 15

Capacitors in Circuits

When electrical components (such as capacitors) are connected into circuits, weoften want to simplify the circuit by finding an equivalent capacitance for all (orat least most) of them.


Series vs. Parallel

Capacitors in Parallel have equalpotential drop V(and so does their equivalent)!

Capacitors in Series have an equal charge q (and so does their equivalent)!


Finding the Equivalent Capacitance

Capacitors in parallel... comes from q=CV and V’s are equal!:

Ceq =n∑

i=1

Ci

Capacitors in series... comes from q=CV and q’s are equal!:

1

Ceq=

n∑i=1

1

Ci


Equivalent Capacitance Example One

1st type of problem: For the capacitor network to the left, find itsequivalent capacitance.

1 Isolate some part of the system you know is in series or inparallel (like the top branch).

2 Compute its equivalent capacitance (series):1

Ceq(1)=

∑ni=1

1Ci

= 13µF + 1

6µF , Ceq(1) = 2µF

3 Now the circuit has a simpler, yet equivalent layout.

4 Pick another part of the system you know is in series orparallel (only one choice).

5 Compute its equivalent capacitance (parallel):Ceq(2) =

∑ni=1 Ci = 2µF + 2µF , Ceq(2) = 4µF

The system of capacitors on the top is electronically similar to asingle capacitor with Ceq(2) = 4µF .


Equivalent Capacitance Example Two

If a 12 Volt potential difference is placed across the circuit....

(a) Determine the charge on the 2µF capacitor.

(b) Determine the potential across the 3µF capacitor.

These problems use the fact that capacitors in series have thesame charge and capacitors in parallel share a potential drop!

Then q=CV for the win....

(a) So my equation is q=CV. If I know V across the capacitor Iknow the charge.

But the potential drop is 12V! So q=CV=(2µF )(12V)=(24µC ).

(b) It’s in a series branch but again q=CV. That is true for bothindividual capacitors and groups. Look at Ceq(1)!

The potential drop is 12V. So q=Ceq(1)V=(2µF )(12V)=(24µC ).

Capacitors in series have the same charge so q=(24µC ).


Energy Storage in Capacitors

Everything in physics comes down to conservation of energy. (that’s actually notmuch of an overstatement). Capacitors store energy in the form of their electricfield. The energy stored is:

U =q2

2C= (1/2)CV 2

The energy stored on the 2µF capacitor when it has 12 volts on it is:

U = (1/2)(2µF )(12V )2 = 144µJ = 1.44x10−4J


The Energy stored is in the ELECTRIC FIELD!

Energy is stored in the VOLUME of space between the plates – the E field!Demonstrate this for a parallel plate capacitor.

1 Energy Stored: U = 12CV

2

2 Parallel Plate Capacitor: C = ε0Ad

3 Volume between plates: Volume = Ad4 Constant E field between plates: V = Ed

Plug in.... U = 12 (ε0

Ad )(Ed)2 Rearrange.... U = 1

2 (ε0Ad)(E )2

Now make an energy density (Energy/volume) by dividing through by the volume:

u =1

2ε0E

2

Look - NO DEPENDENCE on any physical property of the capacitor! E only!SP212 Lesson 12 February 1, 2018 8 / 15

Dielectrics – the Good!

Remember....dipoles are places where potential energy is stored!

So the electric field stores energy and dipoles store energy....get TWO places tostore energy! Basically, the presence of local dipoles changes the effective value ofε0 in the equation to ε = κε0

That means all capacitances (all proportional to ε0) also grow by a factor of κ too!

C ′ = κC


Dielectrics – the Bad!

A vacuum filled capacitor doesn’t allow charge to pass from one plate to theother – the electric FIELD does the work!

This is because there are no conducting paths for the charge between theplates!

Unless you make one by filling the space with lots of dipole material.

Normally, no big, but dielectrics can turn into conductors with strong enoughelectric fields.

When the electric field exceeds a threshold....the “dielectric breakdownvalue” conducting paths appear and charge can move!

For air, electric fields of greater than 3000 Volts per millimeters causedielectric breakdown.


Boom! Dielectric Breakdown! 3 million volts per meter!


Dielectrics – the Ugly!

κ is a function of a complicated material and must be measured experimentally(why Material Sciences exists).

κ = 1. exactly for vacuum, κ = 1.000054 for air and κ = 310 for Strontiumtitanate. It varies! Table 25-1

Takeaway: Dielectrics boost the energy storage capacity of capacitors! But theelectric field changes too and it depends how the capacitor is wired up to a powersource.

Insert a dielectric? C ′ = κC but also E ′ = Eκ because E = σ

ε0

So there is an interplay between V = Ed and q = CV and U = 1/2CV 2


Battery disconnected?

The main idea is that a disconnected battery cannot provide more charge or moreenergy, right? So while the capacitance goes up, q stays constant so V mustdecrease.

This should not be a surprise....since V=Ed, and E is decreasing due to κ, V hadto decrease.


Battery left connected?

This time the battery is still there and can provide additional charge. It does sountil the capacitor reaches the same potential. So C increases and q increases, aswell as the energy stored!


Challenge – Capacitors

A parallel-plate capacitor is connected to a battery of electric potential V. If theplate separation is decreased while the battery is still connected do the followingcapacities increase, decrease or remain the same?

1 The capacitance?

2 The potential difference across the capacitor?

3 The charge on the capacitor?

4 The energy stored on the capacitor?

5 The magnitude of the electric field between plates?

6 The energy density of that electric field?


sp212 lesson 12 - united states naval academy · well as the energy stored! sp212 lesson 12...

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