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### Capacitor Networks

Electricity

- Consider a force like gravity but a

billion-billion-billion-billion times stronger

And with two kinds of active matter:

electrons and protons

And one kind of neutral matter: neutrons

The phenomenon of charge

Problem

Invisibility

Common problem in

physics: have to believe in

invisible stuff and find ways

to demonstrate its existence.

Danger of sth invisible

If we rub electrons onto the plastic, is it feasible to say that we rub

protons on it in the second experiment?

No! If we move protons, we move electrons with them. But what if in the

second experiment we still moved electrons – in the other direction?

Why matter is usually electrically neutral:

- Like charges repel, unlike charges attract
- Mixed + and – are pulled together by enormous attraction
- These huge forces balance each other almost out so that matter is neutral
- Two important laws:

Conservation & quantization of charge

Where do the charges come from?

Electrons and protons carry charge.

Neutrons don’t.

Positive (proton), negative (electron)

Consider:

Why does the electron not fall into

the nucleus?

Why does the nucleus not fly apart?

Why does the electron not fly apart?

Consequence of QM uncertainty

relation

More forces, total of four

Short ranged – limit for nucleus size

Uranium almost ready to fly apart

Electric Properties of Matter (I)

- Materials which conduct electricity well are called ______________
- Materials which prohibited the flow of electricity are called ________________
- ‘_____’ or ‘______’ is a conductor with an infinite reservoir of charge
- ____________ are in between and can be conveniently ‘switched’
- _____________are ideal conductors without losses

Appears visibly

in conductors

- Induction : Conductors and Insulators

- Are charges
- present?

b) Why are there not

more ‘-’ charges?

c) Why do like charges

collect at opposite

side?

d) Why does the metal

sphere not stay charged forever?

Coulomb’s Law

- Concept of point charges
- Applies strictly in vacuum although in air deviations are small
- Applies for charges at rest (electrostatics)

Force on a charge by other charges ~ ___________

~ ___________

~ ___________

Significant constants:

e = 1.602176462(63) 10-19C i.e. even nC extremely good statistics

(SI) 1/4pe0

Modern Physics: why value?

how constant?

Principle of superposition of forces:

If more than one charge exerts a force on a target charge,

how do the forces combine?

Find F1

Luckily, they add as vector sums!

Consider charges q1, q2, and Q:

F1 on Q acc. to Coulomb’s law

Component F1x of F1 in x:

What changes when F2(Q) is

determined?

What changes when q1

is negative?

Charges –q and 4q are placed as shown.

Of the five positions indicated at

1-far left, 2 – ¼ distance, 3 – middle, 4 – ¾ distance

and 5 – same distance off to the right,

the position at which E is zero is: 1, 2, 3, 4, 5

Group task:

Find fields for

all combinations

of distances and

charge arrangements

at all charge positions.

Direction E at black point equidistant from chargesis indicated by a vector. It shows that:a) A and B are + b) A and B are - c) A + B –d) A – B + e) A = 0 B -

Find flux through each surface for q = 30°

and total flux through cube

What changes for case b?

n1:

n2:

n3:

n4:

n5,n6:

Gauss’s Law

Basic message:

http://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.htmlhttp://www.cco.caltech.edu/~phys1/java/phys1/EField/EField.html

http://www.falstad.com/vector3de/

potential differences

Dust particle m= 5 10-9 [kg], charge q0 = 2nC

Energy conservation: Ka+Ua = Kb+Ub

Moving charges: Electric Current

- Path of moving charges: circuit
- Transporting energy
- Current

http://math.furman.edu/~dcs/java/rw.html

Random walk does not mean ‘no progression’

Random motion fast: 106m/s

Drift speed slow: 10-4m/s

e- typically moves only few cm

Positive current direction:= direction flow + charge

charge through A per unit time

Work done by E on moving charges heat

(average vibrational energy increased i.e. temperature)

Unit [A] ‘Ampere’

[A] = [C/s]

Current and current density do

not depend on sign of charge

Replace q by /q/

Concentration of charges n [m-3] , all move with vd, in dt moves vddt,

volume Avddt, number of particles in volume n Avddt

What is charge that flows out of volume?

Resistivity and Resistance

Properties of material matter too:

For metals, at T = const. J= nqvd ~ E

Proportionality constant r is resistivityr = E/JOhm’s law

Reciprocal of r is conductivity

Unit r: [Wm] ‘Ohm’ = [(V/m) / (A/m2)] = [Vm/A]

Ask for total current in and

potential at ends of conductor:

Relate value of current to

Potential difference between ends.

- For uniform J,E
- I = JA and V =EL
- with Ohm’s law E=rJ
- V/L = rI/A

Const. r I ~ V

‘resistance’ R = V/I [W]

- vs. R

R =rL/A

R = V/I

V = R I

I = V/R

E, V, R of a wire

Typical wire: copper, rCu = 1.72 x 10-8 Wm

cross sectional area of 1mm diameter wire

is 8.2x10-7 m-2

current a) 1A b) 1kA

for points a) 1mm b) 1m c) 100m apart

E = rJ = rI/A = V/m (a) V/m (b)

V = EL = mV (a), mV (b), V (c)

R = V/I = V/ A = W

Resistance of hollow cylinder length L, inner and

outer radii a and b

Current flow radially! Not lengthwise!

Cross section is not constant:

2paL to 2pbL find resistance of

thin shell, then integrate

Area shell: 2prL with current path dr

and resistance dR between surfaces

dR= rdr/(2prL)

dV = I dR to overcome

And Vtot= SdV

R = int(dR) = r/(2pL) intab(dr/r)

= r/(2pL) ln(b/a)

Steady currents require circuits: closed loops of conducting material

otherwise current dies down after short time

Charges which do a full loop

must have unchanged potential

energy

Resistance always reduces U

A part of circuit is needed which

increases U again

This is done by the emf.

Note that it is NOT a force but

a potential!

First, we consider ideal sources (emf) : Vab = E = IR

I is not used up while flowing from + to –

I is the same everywhere in the circuit

Emf can be battery (chemical), photovoltaic (sun energy/chemical),

from other circuit (electrical), every unit which can create em energy

EMF sources usually possess Internal Resistance. Then,

Vab = E – Ir and I = E/(R+r)

(Ideal wires and am-meters have

Zero resistance)

No I through voltmeter

(infinite R) …

i.e. no current at all

Voltage is always measured in parallel, amps in series

Energy and Power in Circuits

Rate of conversion to electric energy: EI, rate of dissipation I2r –

difference = power output source

Resistor networks

Careful: opposite to capacitor series/parallel rules!

Find I25 and I20

A more general approach to analyze resistor networks

Is to look at them as loops and junctions:

This method works even

when our rules to reduce

a circuit to its Req

fails.

- Circuit with more
- than one loop!
- Apply both rules.
- Junction rule,
- point a:
- I = A

[Similarly point b:

]

- Loop rule: 3 loops
- to choose from
- ‘1’: decide direction
- in loop – clockwise
- (sets signs!)
- r = W

Find E from loop 2:

Lets do the loop counterclockwise: E = V

Group task: Find values of I1,I2,and I3!

5 currents

Use junction rule

at a, b and c

3 unknown currents

Need 3 eqn

Loop rule to 3 loops:

cad-source

cbd-source

cab

(3 bec.ofno.unknowns)

Let’s set R1=R3=R4=1W

And R2=R5=2W

I1 = A, I2 = A, I3 =

E ~ /Q/ Vab ~ /Q/

Double Q:

But ratio Q/Vab is constant

Capacitance is measure of ability of a capacitor to store energy!

(because higher C = higher Q per Vab = higher energy

Value of C depends on geometry (distance plates, size plates, and material

property of plate material)

E = s/e0

= Q/Ae0

For uniform field E and given plate distance d

Vab = E d

= 1/e0 (Qd)/A

Units: [F] = [C2/Nm] = [C2/J] … typically micro or nano Farad

In a series connection, the magnitude of charge on all plates is the same. The potential differences are not the same.

Or using concept of equivalent capacitance

1/Ceq =

In a parallel connection, the potential difference for all individual capacitors is the same. The charges are not the same.

Or Ceq =

Equivalent capacitance is used to simplify networks.

Group task: What is the equivalent capacitance

of the circuit below?

Step 1:

Find equivalent for C series at right

Hand.

Step 2:

Find equivalent for C parallel

left to right.

Solution: parallel branch …

series branch …

total:

Step 3:

Find equivalent for series.

24.63

C1 = 6.9 mF, C2= 4.6mF

Reducing the furthest right leg

(branch):

C=(

Combines parallel with nearest C2:

C =

Leaving a situation identical to what we have just worked out:

So the overall CEQ = mF

Charge on C1 and C2:

QC1 =

QC2 =

For Vab = 420[V], Vcd=?

Vac= Vbd = Vcd=

- 24.24: plate C 920 pF, charge on each plate 2.55 mC
- V between plates:
- V =
- b)For constant charge, if d is doubled, what will V be?
- c) How much work to double d?
- If d is doubled, …
- Work equals amount of extra energy needed which is mJ

Other common geometries

- Spherical capacitor

Need Vab for C, need E for Vab:

Take Gaussian surface as sphere

and find enclosed charge

- Cylindrical capacitor

Dielectric constant K

K= C/C0

For constant charge:

Q=C0V0=CV

And V = V0/K

Dielectrics are never

perfect insulators: material

leaks

If V changes with K, E must decrease too: E = E0/K

This can be visually understood considering that materials

are made up of atoms and molecules:

Induced charge: Polarization – Molecular View

Dielectric breakdown

- From now on
- instantaneous
- Quantities I and V
- in small fonts
- vab = i R
- vbc = q/C
- Kirchhoff:

As q increases

towards Qf,

i decreases 0

Charging a capacitor

- integral:

Take exponential of both sides:

Characteristic time constant t = RC !

Discharging a capacitor

Ex 26.37 - C= 455 [pF] charged with 65.5 [nC] per plate

- C then connected to ‘V’ with R= 1.28 [MW]
- iinitial?
- b) What is the time constant of RC?
- i = q/(RC)
- [C/(WF] =! [A]
- b) t = RC =

Group Task:

Find t

charged fully after 1[hr]? Y/N

Ex 26.80/82 C= 2.36 [mF] uncharged, then connected in series to

- R= 4.26 [W] and E=120 [V], r=0
- Rate at which energy is dissipated at R
- b) Rate at which energy is stored in C
- c) Power output source
- a) PR =
- b) PC=
- c) Pt =
- d) What is the answer to the questions after ‘a long time’?
- all zero

Group Task

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