PHYSICS - CLUTCH CH 24: CAPACITORS &...

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PHYSICS - CLUTCH

CH 24: CAPACITORS & DIELECTRICS

CONCEPT: CAPACITORS AND CAPACITANCE

● CAPACITOR: Two nearby surfaces of _________&____________ charge → stores [ POTENTIAL | KINETIC ] energy

● Connecting a capacitor to a battery produces a simple CIRCUIT

- “Source” of moving charges comes from a __________________

- Voltage of battery = voltage of capacitor

● CHARGE on capacitor → 𝑄 = _____

- CAPACITANCE (C) measures the ___________ of the capacitor.

- Larger capacitance → [ LARGER | SMALLER ] the charge stored

EXAMPLE: What is the charge on the capacitor in the following figure?

● CAPACITANCE is defined as 𝐶 = __________ - Units are F (Farads)

9 V

3 F

Battery Capacitor

CIRCUIT

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CONCEPT: PARALLEL PLATE CAPACITORS

- Electric Field BETWEEN plates is ______________.

- Electric Field OUTSIDE plates is ______________.

● The magnitude of the UNIFORM Electric Field within a capacitor: 𝐄 = _______ → 𝐄 =𝐐

𝛜𝟎𝐀

● Equipotential surfaces between plates: EXAMPLE: A parallel plate capacitor has an area of 5 cm2, a plate separation of 10 mm, and a voltage across the plates of 100 V. a) What is the charge of the capacitor? b) What is the magnitude of the electric field between the plates?

+Q

-Q Electric Field

Equipotential

Surfaces

● Capacitance for ANY Capacitor is → C = Q / V

● Capacitance for PARALLEL PLATE Capacitor is: → C = __________

- A is area, d is distance between plates, 𝜖0 = 8.85 × 10−12 [𝐹

𝑚]

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PRACTICE: CAPACITANCE OF PARALLEL CIRCULAR PLATES

Two circular plates of radius 2cm are brought together so their separation is 5mm. What is the capacitance of these plates? EXAMPLE: POINT CHARGE IN CAPACITOR

Two 1 cm by 1 cm plates, separated by 10 mm, form a capacitor. If each plate is charged to 30 nC, (a) What is the potential difference between the plates? (b) What is the electric field between the plates? (c) How much energy does it take to move a – 5 nC charge from the positive plate to the negative plate? PRACTICE: CHARGING A CAPACITOR

A 3 F capacitor is given a potential difference across its plates of 10 V. What is the charge built up on its plates? If the source of the potential difference across the plates is removed, but the plates maintain their charge, what is the new potential difference across the capacitor if the distance between the plates is doubled?

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CONCEPT: ENERGY STORED BY CAPACITOR

● Remember: Capacitors separate charges, and this separation leads to potential energy stored. But HOW MUCH energy?

● ENERGY DENSITY (𝑢) = Energy per unit volume → 𝒖 = __________ - Volume of a parallel plate capacitor → 𝒗𝒐𝒍𝒖𝒎𝒆 = __________ → 𝒖 = __________ = ___________ EXAMPLE: Two parallel plates of area 50 cm2, with a separation of 10 mm, have a voltage across them of 20 V. What is the energy stored? The energy density? EXAMPLE: What is the strength of the electric field in a capacitor storing 2.5 mJ per cubic-centimeter?

● Energy stored by ANY capacitor → 𝐔 = _________ = __________ = _________ - Use Q = CV to change between all 3 forms

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PRACTICE: DEFIBRILLATOR

A cardiac defibrillator can be modeled as a parallel plate capacitor. When it is charged to a voltage of 2 kV, it has a stored energy of 1 kJ. What is the capacitance of the defibrillator? PRACTICE: ENERGY RELEASED BY FLASHBULB

Typically, a flashbulb will have a capacitance of 1000 mF. If the bulb were charged to a voltage of 500 V, how much energy is released when the flash goes off, if the bulb loses 80% of its charge in a single flash?

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EXAMPLE: SPHERICAL CAPACITOR (USING CALCULUS) What is the capacitance of two concentric spherical shells, one of radius a and one of radius b, with a < b? Consider the charge on each sphere to be +/- Q.

EXAMPLE: CYLINDRICAL CAPACITOR (USING CALCULUS) What is the capacitance per unit length of two concentric, infinitely long cylindrical shells, one of radius a, and one of radius b, with a < b. Consider the charge on each cylinder to be +/- Q.

a

b

+Q

-Q

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CONCEPT: COMBINING CAPACITORS IN SERIES AND PARALLEL

● In Circuit problems, we can COLLAPSE / COMBINE capacitors into a SINGLE ___________________ capacitor.

SERIES CONNECTION PARALLEL CONNECTION

- Direct connection, - Equivalent Capacitance:

𝟏

𝐂𝐞𝐪= __________________

- Splits off, forms a loop - Equivalent Capacitance:

𝐂𝐞𝐪 = _________________

● For circuits with combinations, find Ceq’s from inside → outside. EXAMPLE: What is the equivalent capacitance of the following capacitors?

● For TWO capacitors in SERIES, 𝐶𝑒𝑞 = ________

EXAMPLE: What is the equivalent capacitance of the following capacitors?

1 F 3 F

4 F 2 F

2 F

2 F

1 F

4 F

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EXAMPLE: EQUIVALENT CAPACITANCE OF 4 CAPACITORS

What is the equivalent capacitance of the following combination of capacitors?

PRACTICE: EQUIVALENT CAPACITANCE OF 4 CAPACITORS

What is the equivalent capacitance of the following capacitors?

2 F 2 F

3 F

5 F

2 F

2 F

3 F 2 F

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CONCEPT: SOLVING CAPACITOR CIRCUITS

● In Circuit problems, you’ll be asked to find CHARGE and VOLTAGE across combinations of capacitors.

SERIES CONNECTION PARALLEL CONNECTION

- Equivalent Capacitance:

𝟏/𝐂𝐞𝐪 = 𝟏/𝐂𝟏 + 𝟏/𝐂𝟐 + 𝟏/𝐂𝟑

- Share [ CHARGE | VOLTAGE ] with EACH OTHER

- Share [ CHARGE | VOLTAGE ] with Ceq

- Equivalent Capacitance:

𝐂𝐞𝐪 = 𝐂𝟏 + 𝐂𝟐 + 𝐂𝟑

- Share [ CHARGE | VOLTAGE ] with EACH OTHER

- Share [ CHARGE | VOLTAGE ] with Ceq

STEPS FOR CAPACITOR CIRCUITS

1) Find SINGLE EQUIVALENT capacitor

2) Find V & Q for Ceq

3) Work backwards to find V & Q for each capacitor

EXAMPLE: What is the charge and voltage of each of the capacitors in the following circuit?

2 F

1 F

6 F

10 V

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PRACTICE: FIND CHARGE & VOLTAGE IN ALL CAPACITORS

What is charge and voltage across each capacitor below?

EXAMPLE: FIND CHARGE OF ONE CAPACITOR

What is the charge on the 3 F capacitor below?

2 F 2 F

3 F

10 V

5 V

1 F

3 F

4 F 2 F

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PRACTICE: FIND VOLTAGE OF THE BATTERY

What is the voltage of the battery below?

PRACTICE: FIND CHARGE OF CAPACITOR IN A COMPLEX ARRANGEMENT

What is the charge on the 5 F capacitor?

1 F

V = ?

3 F

1 F

2 F

3 C

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CONCEPT: INTRO TO DIELECTRICS

● Dielectric: Insulator between charged plates [ INCREASES | DECREASES ] capacitance: 𝐶 = 𝜅𝐶0

- DIELECTRIC CONSTANT ≥ 1 (no units!)

- Always [ STRENGTHEN | WEAKEN ] Electric Fields → 𝐸 = 𝐸0/𝜅

CONSTANT CHARGE (Q) CONSTANT VOLTAGE (V)

- No battery connected

- 𝑄 = 𝐶 𝑉 → V ___________

- 𝑈 =1

2𝑄2 / 𝐶 → U ___________

- 𝑢 =1

2𝜖0𝐸2 → u ___________

- Inserted when battery still connected

- 𝑉 = 𝑄 / 𝐶 → Q ___________

- 𝑈 =1

2𝐶 𝑉2 → U ___________

- 𝑢 =1

2𝜖0𝐸2 → u ___________

EXAMPLE: A capacitor is connected to a battery as shown below. What is the charge on the capacitor after a dielectric (𝜅 = 2) is inserted into the capacitor while it is still connected to the battery?

9 V

3 F

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EXAMPLE: CAPACITOR WITH A DIELECTRIC

A capacitor in a vacuum is charged to 64V between its plates, then disconnected. Initially, each plate has 32μC. An

insulating slab of dielectric glass with = 3 is placed between the plates. a) What is the capacitor’s new capacitance? b) What is the new voltage across the capacitor? PRACTICE: CIRCULAR PLATE CAPACITOR WITH DIELECTRIC

A parallel plate capacitor is formed by bringing two circular plates, of radius 0.5 cm, to a distance of 2 mm apart. The capacitor is made so that it has a dielectric of constant 𝜅 between the plates. When the charge on the capacitor is 3 nC, the voltage of the capacitor is 5000 V. What is the dielectric constant? EXAMPLE: CAPACITORS PARTIALLY FILLED WITH DIELECTRIC

What is the new capacitance of the two capacitors that are partially filled with dielectrics shown in the following figure?

(a) (b)

A

d/2

d/2

𝜅

A

d

L/2

𝜅

L/2

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CONCEPT: HOW DIELECTRICS WORK ● A DIELECTRIC is an insulating material that can ______________________

- At the atomic level:

● Dielectrics [ INCREASE / REDUCE ] electric field strength within capacitors

Alone In external 𝐸

+ -

+ -

+ -

+ -

+ -

+ - + -

+ - + -

+ - + -

____________ ____________

+

+

+

-

-

-

+Q -Q

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CONCEPT: DIELECTRIC BREAKDOWN ● Dielectric is an insulator charges can’t move … IDEALLY

- In REALITY, charges can move if voltage across insulator is large enough

- This is known as DIELECTRIC BREAKDOWN

● DIELECTRIC STRENGTH = ________________________________ ● Electrons “jump” from atom to atom:

● Lightning is a common example of dielectric breakdown EXAMPLE: A parallel plate capacitor is filled with air and connected to a power source of 100 V. What is the closest you can

put the plates together if dielectric breakdown of air occurs at an electric field of 3 × 106 V/m?

e e

+Q -Q

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PHYSICS - CLUTCH CH 24: CAPACITORS &...

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