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PHYSICS - GIANCOLI 7E

CH 21: ELECTROMAGNETIC INDUCTION AND FARADAY'S LAW

CONCEPT: ELECTROMAGNETIC INDUCTION A coil/loop of wire with a VOLTAGE across each end will have a current in it

- Voltage source isn’t always a battery, voltage can be created → ____________ 3 common ways to INDUCE a voltage / current on a coil of wire:

In all 3 cases, the _______________________ (B) is changing!

- Interaction between magnetism & electricity known as ELECTROMAGNETIC INDUCTION The magnitude of the induced current depends on how ____________ these changes happen.

- Bar magnet moving into coil → Faster it goes, larger the induced current

- Current changing in electromagnet near a coil → Faster the current changes, larger the induced current

INDUCTION 1) Moving a bar magnet 2) Varying current 𝒊 in electromagnet (solenoid) 3) Turning electromagnet on & off

Bar Moving: [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ] 𝒊 varying : [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ] Turn on/off: [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ]

Not Moving: [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ] 𝒊 constant: [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ] Kept on/off: [ 𝒊𝒊𝒏𝒅 | 𝐍𝐎 𝒊𝒊𝒏𝒅 ]

V

𝒊



Page 2

CONCEPT: MAGNETIC FLUX Remember: Electric flux is just the amount of Electric Field (E) passing through a surface.

- MAGNETIC FLUX is just the amount of _____________ Field (B) passing through a surface.

Remember, 𝜽 → angle between B and the _________________ of the surface!

𝚽𝐁 is always positive (or zero).

EXAMPLE: What is the magnetic flux through the square surface depicted in the following figure, if B = 0.05 T? The side length of the square is 5 m.

Normal

θ

A

𝚽𝐄 = E A cos Θ → Units: 𝐍⋅𝐦𝟐

𝐂

𝚽𝐁 = __________ → Units: 1 Wb = 𝐓 ⋅ 𝐦𝟐

Electric Flux

Magnetic Flux

30o

Surface

B

𝐄ሬԦ

θ

A Normal

𝐁ሬሬԦ



Page 3

PRACTICE: MAGNETIC FLUX THROUGH A RING

A ring of radius 0.5m lies in the xy-plane. If a magnetic field of magnitude 2T points at an angle of 22o above the x-axis,

what is the magnetic flux through the ring?

EXAMPLE: ROTATING RING

A ring of radius 2 cm is in the presence of a 0.6 T magnetic field. If the ring begins with its plane parallel to the magnetic field, and ends with the plane of the ring perpendicular to the magnetic field, what is the change in the magnetic flux?



Page 4

CONCEPT: FARADAY’S LAW Changing magnetic field through conducting loops creates an ___________________.

- This is actually due to a changing MAGNETIC FLUX →

- Faster changes → Higher induced EMFs & currents! →

Remember! 𝚽𝐁 = 𝐁𝐀𝐜𝐨𝐬 𝛉

- In problems, one variable will always ___________ while the other two remain ____________.

EXAMPLE: a) What is the induced EMF in the following circuit, with an area of 50 cm2, if the magnetic field changes from

3T to 6T in 5s? b) What is the induced current, if the resistor in the circuit has a resistance of 2 Ω?

B A Cos Θ

Changing Constant Constant

B A Cos Θ

Constant Changing Constant

B A Cos Θ

Constant Constant Changing

Faraday’s Law: Induced EMF is the rate at which the magnetic flux changes with time.

Ɛ𝒊𝒏𝒅 = 𝒊𝒊𝒏𝒅𝑹 = ____________ → Units: ___

N

Changing _______

Changing _______

Changing _______

𝑩ሬሬԦ 𝑩ሬሬԦ

𝑩ሬሬԦ



Page 5

PRACTICE: FARADAY’S LAW AND SOLENOIDS

A tightly-wound 200-turn rectangular loop has dimensions of 40cm by 70cm. A constant magnetic field of 3.5T points in the

same direction as the normal of the loop. If the dimensions of the loop change to 20cm by 35cm over 0.5s, with the number

of turns remaining the same, what is the induced EMF on the rectangular loop?



Page 6

EXAMPLE: FARADAY’S LAW AND TWO CIRCULAR LOOPS

A small circular loop of wire with radius r = 5cm and resistance 10mΩ is centered inside a larger circular loop of wire with

radius r = 5m. The larger loop carries an initial current of 6A. The larger loop is then disconnected from its voltage source,

and the current steadily decreases to 0 over a time of 20µs.

a) What is the change in the magnetic flux through the smaller circular loop during this time?

b) What is the magnitude of the induced EMF on the smaller loop?

c) What is the induced current on the smaller loop?



Page 7

PRACTICE: INDUCTION IN A ROTATING LOOP

A square conducting wire of side length 4 cm is in a 2 T magnetic field. It rotates such that the angle of the magnetic field to

the normal of the square increases from 30o to 60o in 2 s. What is the induced current on the wire if its resistance is 5 mΩ?



Page 8

CONCEPT: LENZ’S LAW Faraday’s Law gives us the magnitude of the induced EMF / Current.

- To find ________________ of induced current, we use Lenz’s Law.

You may see Faraday’s Law represented as: Ɛ = 𝑵𝚫𝚽𝑩

𝚫𝒕

: ____ : ____

ΔΦ𝐵: ____ ΔΦ𝐵: ____ EXAMPLE: In the following scenarios, find the direction of the current induced on the conducting wires.

`

Induced 𝑖𝑛𝑑 is always directed [ ALONG | OPPOSITE ] increasing B-Field.

Induced 𝑖𝑛𝑑 is always directed [ ALONG | OPPOSITE ] decreasing B-Field.

Moving Bar Magnet

Lenz’s Law: The direction of induced current creates an induced B-field to ____________ CHANGES in magnetic flux.

- Remember your Right-Hand Rule for circular currents! Thumb → 𝑖𝑛𝑑𝑢𝑐𝑒𝑑; Fingers → 𝑖𝑖𝑛𝑑

Moving Loop In/Out of Magnetic Field

: ____ 𝑖𝑛𝑑: ____

ΔΦ: ____ 𝑖𝑖𝑛𝑑: [CW | CCW]

: ____ 𝑖𝑛𝑑: ____

ΔΦ: ____ 𝑖𝑖𝑛𝑑: [CW | CCW]



Page 9

PRACTICE: DIRECTION OF INDUCED CURRENT IN A RING An outer ring is connected to a variable voltage source. If the battery’s voltage is continuously INCREASING, what is the direction of the induced current in the inner ring, centered inside of the outer ring?

EXAMPLE: LENZ’S LAW FOR LONG STRAIGHT WIRE

A long straight wire on a horizontal surface in the xy-plane carries a constantly increasing current in the +y direction. A square loop of wire lies flat on the surface to the right of the wire. When viewed from above, what is the direction of the induced current in the square loop?

𝒊



Page 10

CONCEPT: MOTIONAL EMF Remember! A changing magnetic flux produces an INDUCED EMF.

- When this happens through _______________, this is called MOTIONAL EMF.

1) Conducting rod moves through a B-Field with v, charges feel a ___________________

2) (+) charges feel force [ UPWARD | DOWNWARD ] → Charges separate

3) Charges produces E-Field to eventually balance B-Field → 𝐅𝐄 ___ 𝐅𝐁

If we attach this moving conducting rod to a U-shaped wire, we can use Faraday’s Law on the circuit it makes!

- As the rod slides, the [ B-Field | Area | Angle ] changes

𝚫𝚽𝐁

𝚫𝒕= __________ = ___________

EXAMPLE: In the circuit below, if the wire has a resistance of 10 mΩ, a) what is the current induced if the length of the bar is 10 cm, the speed of the bar is 25 cm/s, and the magnetic field is 0.2 T? b) What about the power generated by the circuit?

𝑩ሬሬԦ

𝒗ሬሬԦ

𝑭ሬሬԦ𝑩

𝑳

- Induced EMF Ɛ = ______

𝑩ሬሬԦ 𝒗ሬሬԦ

𝑳

𝒙


- Induced EMF Ɛ = ______



Page 11

PRACTICE: BAR MOVING IN UNKNOWN MAGNETIC FIELD A thin rod moves perpendicular to a uniform magnetic field. If the length of the rod is 10 cm and the induced EMF is 1 V when it moves at 5 m/s, what is the magnitude of the magnetic field?

a

b




Page 12

EXAMPLE: FORCES ON LOOPS EXITING MAGNETIC FIELD

A rectangular loop with length L = 20 cm and resistance R = 0.40Ω is pulled out of a magnetic field B = 0.5 T at a constant velocity of 12m/s. a) What is the magnitude and direction of the induced current in the loop at the instant when the loop is halfway out of the field? b) What is the magnitude of the external force needed to keep this loop exiting at constant velocity?

L

𝒗



Page 13

CONCEPT: TRANSFORMERS Power in North America is delivered to outlets in homes at 120 V.

- This is too large to operate many delicate electronics, such as computers.

Remember! A coil with a changing magnetic field can induce an EMF on a second coil

- This induced EMF can be as small as needed. A TRANSFORMER does exactly this – it uses Faraday’s law to convert a large voltage to a small EMF:

EXAMPLE: You need to build a transformer that drops the 120 V of a regular North American outlet to a much safer 15 V.

You already have a solenoid with 50 turns made, but you need to make a second solenoid to complete your transformer.

What is the least number of turns the second solenoid could have?

V1 V2

The ratio of the VOLTAGES in a transformer depends upon the ratio of the TURNS:

𝑽𝟐

𝑽𝟏=

𝑵𝟐

𝑵𝟏



Page 14

PRACTICE: OPERATING A LAPTOP

An outlet in North America outputs electricity at 120 V, but a typical laptop needs to operate at around 20 V. In order to do

so, a transformer is placed in a laptop’s power supply. If the coil in the circuit connected to the laptop has 20 turns, how

many turns must the coil in the circuit with the outlet have?



Page 15

CONCEPT: MUTUAL INDUCTANCE Mutual Inductance: For two nearby conducting coils, a current changing through one coil induces an EMF on the other.

- The coil with the changing current is known as the _______________, the other is the _________________.

Total Flux 𝚽𝟐 depends on N2 & Magnetic Field , which depends on _____

- 𝚽𝟐 is __________________ to 𝑖1 → _____________________

EXAMPLE: What is the mutual inductance of two solenoids of length L and area A, one with N1 turns and the other with N2?

EXAMPLE: A solenoid of 25 turns, with an area of 0.005 m2 is wound around a 10 cm solenoid with 50 turns, as shown in the figure below. If, at some instant in time, the current through the 10 cm solenoid is 0.5 A and changing at 50 mA/s, what’s the induced EMF on the 25 turn solenoid?

Coil 1 N1

Coil 2 N2

𝑖1

M is a proportionality constant called the MUTUAL INDUCTANCE

𝑴 = __________ → UNITS: Henry [H] → 1 H = 1 ____ / ____

- depends only on the # of turns and the shape of the coils! (𝑖1 will cancel out)

L

N1

N2

The EMF on the secondary coil is

Ɛ = _________ = _________

L

N1

N2



Page 16

PRACTICE: MUTUAL INDUCTANCE OF TWO SOLENOIDS

An outer solenoid with 30 turns is wound tightly around an inner coil 25cm long with a diameter of 4cm and 300 turns. The current in the inner solenoid is 0.12 A and is increasing at a rate of 1.75×103A/s. a) What is the average magnetic flux through each turn of the outer coil? b) If the resistance of the outer coil is 20mΩ, what is the magnitude of the induced current through the outer coil?



Page 17

CONCEPT: SELF INDUCTANCE A current-carrying wire can induce an EMF _______________ through changes in magnetic flux!

- 𝚽𝐓𝐨𝐭𝐚𝐥 depends on N and magnetic field , which depends on ____.

- 𝚽𝐁 is ________________ to 𝑖. → _____________________

We can write the self-induced EMF using Faraday’s Law OR in terms of the self-inductance 𝑳:

EXAMPLE: What is the expression for the self-inductance of a single current-carrying loop of wire with radius r?

N

𝒊

L is a proportionality constant called the SELF INDUCTANCE

𝑳 = __________ → UNITS: Henry [H] → 1 H = 1 ____ / ____

- depends only on the # of turns and the shape of the coil! (𝑖 cancels out)

Ɛ = −𝑵𝚫𝚽𝑩

𝚫𝒕 =

𝒊



Page 18

PRACTICE: SELF-INDUCTING COIL OF WIRE

A single loop of wire with a current of 0.3A produces a flux of 0.005 Wb. If the self-induced EMF on this loop is 10 mV, how

quickly must the current be changing?



Page 19

EXAMPLE: SELF-INDUCTANCE OF A TOROIDAL SOLENOID

A toroidal solenoid has 500 turns, cross-sectional area of 6.25cm2, and mean radius of 4cm. a) What is the self-inductance

of this toroidal solenoid? b) If the current decreases constantly from 5A to 2A in 6ms, what is the induced EMF in the coil?



Page 20

CONCEPT: INDUCTORS IN CIRCUITS A coil of wire placed in a circuit is known as an INDUCTOR → OR

Because inductors are circuit elements, we use Kirchhoff’s Rules on them as we go around in a circuit. - Remember: Inductors only do something if the current is [ CONSTANT | CHANGING ] → Ɛ𝐿 = __________

Use Lenz’s Law to find the direction of the induced EMF.

- If the direction of the induced EMF points along your Kirchoff Loop, the voltage is [ + | - ] EXAMPLE: Write out Kirchhoff’s Loop rule for the following circuit, assuming the battery’s voltage is increasing.

a b

𝒊 [Constant]

V

a b

𝒊 [Increasing]

V

a b

𝒊 [Decreasing]

V



Page 21

CONCEPT: LR CIRCUITS LR (or RL) Circuits are circuits containing ___________________ and ___________________

Depending on the switch positions, there are two processes happening in this circuit:

- CURRENT GROWTH: S2 open and S1 closed, current [ INCREASES | DECREASES ]

- CURRENT DECAY: S1 open and S2 closed, current [ INCREASES | DECREASES ]

CURRENT GROWTH: Current starts from [ ZERO | MAX ]; inductor resists growing current, eventually reaches _____

CURRENT DECAY: Current starts from [ ZERO | MAX ]; inductor resists decreasing current, eventually reaches ____

The TIME CONSTANT, 𝝉 =𝑳

𝑹, determines the how quickly growth and decay occurs

- Lower time constants → [ FASTER | SLOWER ] changes in current - Higher time constants → [ FASTER | SLOWER ] changes in current

S1

S2

L R

V

𝑖(𝑡) =𝑉

𝑅(1 − 𝑒−𝑡/𝜏) L R

V

𝑖(𝑡) =𝑉

𝑅𝑒−𝑡/𝜏

L R



Page 22

EXAMPLE: UNKOWN RESISTANCE IN LR CIRCUIT An LR circuit has a time constant of 0.025 s and is initially connected to a 10 V battery. If after 0.005 s of being

disconnected from the battery, the current is 0.5 A, what is the resistance of the circuit?



Page 23

PRACTICE: UNKNOWN CURRENT IN AN LR CIRCUIT Consider the LR circuit shown below. Initially, both switches are open. Switch 1 is closed. a) What is the maximum current in the circuit after a long time? Then, S1 is opened and S2 is closed. b) What is the current in the circuit after 0.05s?

S1

S2

L = 0.02H R = 5 Ω

V =10V



Page 24

EXAMPLE: TIME TO HALF MAXIMUM CURRENT

An 0.1 H inductor and 10Ω resistor are disconnected from a battery for a long time. When the switch is closed and the

battery is connected again, how much time passes until the current reaches half of its maximum value?



Page 25

CONCEPT: ALTERNATING VOLTAGES AND CURRENTS BEFORE, we only considered DIRECT CURRENTS, currents that only move in _________________________________

- NOW we consider ALTERNATING CURRENTS, currents that move in __________________________________ Alternating currents are produced by ALTERNATING VOLTAGES

- ONLY alternating voltage we will consider is 𝒗(𝒕) = 𝑽𝒎𝒂𝒙𝐜𝐨𝐬(𝝎𝒕)

EXAMPLE: In North America, the frequency of AC voltage coming out of household outlets is 60 Hz. If the maximum voltage

delivered by an outlet is 120 V, what is the voltage at 0.04 s?

This alternating voltage produces an ALTERNATING CURRENT of

- 𝒊(𝒕) = ___________________ (𝝎 is the angular frequency of alternations)

V

t

Vmax

I

t

Imax



Page 26

PRACTICE: ALTERNATING CURRENT

An AC source produces an alternating current in a circuit with the function 𝑖(𝑡) = (1.5𝐴) cos[(250𝑠−1)𝑡]. What is the

frequency of the source? What is the maximum current in the circuit?

EXAMPLE: AC CIRCUIT GRAPHS

Current and voltage in an AC circuit are graphed in the following figure. What are the functions that describe these values?

I, V

t

11 V

- 2.5 A

0.05 s



Page 27

PRACTICE: ANGULAR FREQUENCY OF ALTERNATING CURRENT The current in an AC circuit takes 0.02 s to change direction. What is the angular frequency of the AC source?



Page 28

CONCEPT: RMS CURRENT AND VOLTAGE In alternating current circuits, what is the average of the voltage and the current?

- The average of the voltage and the current is _____________ A better “average” value is the RMS VALUE, the _____________ _____________ _____________ To find the RMS value, you take the square, then the average, then the square root

𝑿 → 𝑿𝟐 → (𝑿𝟐)𝒂𝒗 → √(𝑿𝟐)𝒂𝒗

EXAMPLE: If the RMS voltage of an outlet in the US is 120 V, what is the maximum voltage of an outlet? If you complete a simple circuit with this AC source by connecting a 12 Ω resistor, what is the RMS and maximum current in this circuit?

V

t

Vmax

I

t

Imax

The RMS CURRENT and VOLTAGE are defined by

- 𝑰𝑹𝑴𝑺 =𝑰𝒎𝒂𝒙

√𝟐 or 𝑰𝒎𝒂𝒙 = √𝟐𝑰𝑹𝑴𝑺

- 𝑽𝑹𝑴𝑺 =𝑽𝒎𝒂𝒙

√𝟐 or 𝑽𝒎𝒂𝒙 = √𝟐𝑽𝑹𝑴𝑺



Page 29

PRACTICE: RMS CURRENT IN AN AC CIRCUIT An AC source operates with a 0.05 s period. 0.025 s after the current is at a maximum, the current is measured to be 1.4 A. What is the RMS current of this AC circuit?



Page 30

CONCEPT: POWER IN AC CIRCUITS In AC circuits, the only element to have an average power not equal to zero is the ____________________

- Whatever energy enters a(n) __________________ / __________________ equals the energy that leaves The MAXIMUM power of a resistor is

Since the power of a resistor is 𝑝(𝑡) = 𝑖(𝑡)2𝑅, we have the following graphs of current and power through a resistor:

EXAMPLE: An AC source operating at a maximum voltage of 120 V is connected to a 10 Ω resistor. What is the average

power emitted by this circuit? Is this equivalent to the RMS power, which would be 𝑖𝑅𝑀𝑆2 𝑅?

I t

P

The AVERAGE POWER emitted by an AC circuit is

𝑷𝒂𝒗 = _________________ = __________________

𝑷𝑴𝑨𝑿 = _________________



Page 31

CONCEPT: PHASORS A PHASOR is just a rotating vector, whose information lies in its X-COMPONENT. - Phasors make representing oscillating information, like voltage and current, easy:

EXAMPLE 1: For the following voltage phasor, is the voltage positive or negative?

Phasors obey all the same rules as vectors, such as addition, subtraction, etc. - To find the magnitude of a phasor, you can sum its components using the Pythagorean theorem, as with vectors. EXAMPLE 2: In the following phasor diagram, find the direction of the “net phasor” for the three phasors shown. Is the resulting quantity the phasor describes positive or negative?

V

t

𝜔



Page 32

PRACTICE: ANGULAR FREQUENCY OF A PHASOR The following phasor diagram shows an arbitrary phasor during its first rotation. Assuming that it begins with an angle of 0o, if the phasor took 0.027 s to get to its current position, what is the angular frequency of the phasor?

EXAMPLE: CONVERTING BETWEEN A FUNCTION AND A PHASOR

The current in an AC circuit is given by 𝑖(𝑡) = (1.5 𝐴) cos[(377 𝑠−1)𝑡]. Draw the phasor that corresponds to this current at 𝑡 = 15 𝑚𝑠, assuming the phasor begins at 0o.

30o



Page 33

PRACTICE: DRAWING A VOLTAGE PHASOR An AC source oscillates with an angular frequency of 120 s-1. If the initial voltage phasor is shown in the following phasor diagram, draw the voltage phasor after 0.01 s.

PRACTICE: INSTANTANEOUS VALUE FROM A PHASOR

A phasor of length 4 begins at 0o. If it is rotating at 𝜔 = 250 𝑠−1, what is the value of the phasor after 0.007 s?

V



Page 34

CONCEPT: RESISTORS IN AC CIRCUITS Remember! In an AC circuit, the current produced by the AC source is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋cos(𝜔𝑡) Ohm’s Law will give us the voltage across the resistor at any point in time:

- 𝑣𝑅(𝑡) = 𝑖(𝑡)𝑅

EXAMPLE: A 10 Ω resistor is plugged into an outlet with an RMS voltage of 120 V. What is the maximum current in the circuit? What about the RMS current? For MULTIPLE resistors in an AC circuit, you would just combine them into a single, equivalent resistor, as before.

The VOLTAGE ACROSS THE RESISTOR is

- 𝒗𝑹(𝒕) = ___________________



Page 35

PRACTICE: OSCILLATING VOLTAGE ACROSS A RESISTOR

The voltage across a resistor is found to be given by 𝑣𝑅(𝑡) = (10𝑉) cos[(120𝑠−1)𝑡]: a) At what frequency does the AC course operate? b) If the resistance is 12 Ω, what is the maximum current in this circuit? c) What is the RMS voltage of the AC source?

EXAMPLE: RESISTORS IN PARALLEL IN AN AC CIRCUIT

What is the current through the 10 Ω resistor in the following circuit?

(5 𝑉) cos[(200 𝑠−1)𝑡] 5 Ω 3 Ω 10 Ω



Page 36

CONCEPT: PHASORS FOR RESISTORS Remember! The voltage and current across a resistor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋cos(𝜔𝑡)

- 𝑣𝑅(𝑡) = 𝑖𝑀𝐴𝑋𝑅cos(𝜔𝑡)

Because both cosines have the same angle (𝜔𝑡), they are said to be IN PHASE.

- This is reflected in their phasors:

EXAMPLE: An AC source with an angular frequency of 20 s-1 is connected to a resistor with the circuit broken. 0.2 s after

the circuit is completed, draw the voltage phasor and the current phasor.

𝜔𝑡

𝐼

𝜔𝑡

𝑉𝑅

𝜔𝑡

𝑉𝑅 𝐼

Voltage across a resistor is IN PHASE with the current



Page 37

PRACITCE: RESISTOR VOLTAGE AND CURRENT PHASORS A 12 Ω resistor is connected to an AC source. If the resistor’s voltage phasor is initially at 0o, and the figure below shows the phasor after 0.04 s, answer the following: a) What is the angular frequency of the source? Assume the phasor is on its first rotation. b) What does the current phasor diagram look like?

c) What is the current in the circuit at this point (𝑡 = 0.04𝑠)?

42o

5 V



Page 38

CONCEPT: CAPACITORS IN AC CIRCUITS The current in an AC circuit at any time is

- 𝑖(𝑡) = ____________________ Remember! The voltage across a capacitor is 𝑣𝐶 = ___________

- Using calculus, one can show 𝑞(𝑡) =𝑖𝑀𝐴𝑋

𝜔cos (𝜔𝑡 −

𝜋

2)

This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LAGS the current by 90o:

The MAXIMUM voltage across the capacitor is 𝑉𝐶 = ________________

- This result looks A LOT like Ohm’s Law, if we have some resistance-like quantity 1/𝜔𝐶

We define the CAPACITIVE REACTANCE as EXAMPLE: An AC power source delivers a maximum voltage of 120 V at 60 Hz. What is the maximum current in a circuit

with this power source connected to a 100 µF capacitor?

The VOLTAGE ACROSS A CAPACITOR in an AC circuit is

- 𝑣𝐶(𝑡) = _______________________

I

t

V

𝑿𝑪 = 𝟏/𝝎𝑪



Page 39

PRACTICE: MAXIMUM CHARGE IN A CAPACITOR AC CIRCUIT An AC source operates at a maximum voltage of 120 V and a frequency of 60 Hz. If it is connected to a 175 µF capacitor, what is the maximum charge stored on the capacitor?

EXAMPLE: CURRENT IN A PARALLEL RC AC CIRCUIT

An AC source operating at 160 s-1 and a maximum voltage of 15 V is connected in parallel to a 5 Ω resistor and in parallel to a 1.5 mF capacitor. What is the RMS current through the capacitor?



Page 40

PRACTICE: OSCILLATION FREQUENCY OF A CAPCITOR CIRCUIT A 300 µF capacitor is connected to an AC source operating at an RMS voltage of 120 V. If the maximum current in the circuit is 1.5 A, what is the oscillation frequency of the AC source?



Page 41

CONCEPT: PHASORS FOR CAPACITORS Remember! The voltage and current across a capacitor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos (𝜔𝑡)

- 𝑣𝑐(𝑡) = 𝑖𝑀𝐴𝑋XC cos (𝜔𝑡 −𝜋

2)

Because both cosines have a DIFFERENT angle, they are said to be OUT OF PHASE – The voltage LAGS the current


EXAMPLE: An AC source is connected to a capacitor. At a particular instant in time, the voltage across the capacitor is positive and increasing in magnitude. Draw the phasors for voltage and current that correspond to this time.

𝜔𝑡

𝐼

𝜔𝑡 −𝜋

2

𝑉𝐶

𝐼

𝑉𝐶

Voltage across a capacitor LAGS the current



Page 42

PRACTICE: PHASORS IN A CAPACITOR CIRCUIT An AC source operates at a maximum voltage of 60 V and is connected to a 0.7 mF capacitor. If the current across the

capacitor is 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos[(100 𝑠−1)𝑡],

a) What is 𝑖𝑀𝐴𝑋? b) Draw the phasors for voltage across the capacitor and current in the circuit at 𝑡 = 0.02 𝑠. Assume that the current phasor begins at 0o.



Page 43

CONCEPT: INDUCTORS IN AC CIRCUITS Remember! The current in an AC circuit at any time is

- 𝑖(𝑡) = ____________________ Remember! The voltage across an inductor is 𝑣𝐿 = ___________

- Using calculus, one can show Δ𝑖

Δ𝑡(𝑡) = 𝑖𝑀𝐴𝑋𝜔 cos (𝜔𝑡 +

𝜋

2)

This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LEADS the current by 90o:

The MAXIMUM voltage across the inductor is 𝑉𝐿 = ________________

- This result looks A LOT like Ohm’s Law, if we have some resistance-like quantity 𝜔𝐿

We define the INDUCTIVE REACTANCE as EXAMPLE: An AC power source delivers a maximum voltage of 120 V at 60 Hz. If an unknown inductor is connected to this

source, and the maximum current in the circuit is found to be 5 A, what is the inductance of the inductor?

The VOLTAGE ACROSS AN INDUCTOR in an AC circuit is

- 𝑣𝐿(𝑡) = _______________________

I

t

V

𝑿𝑳 = 𝝎𝑳



Page 44

EXAMPLE: INDUCTORS AND GRAPHS The voltage across, and the current through, an inductor connected to an AC source are shown in the following graph. Given the information in the graph, answer the following questions: a) What is the peak voltage of the AC source? b) What is the frequency of the AC source? c) What is the inductive reactance of the circuit?

PRACTICE: CURRENT IN INDUCTOR AC CIRCUITS AT DIFFERENT FREQUENCIES

Will a frequency 𝑓 = 60 𝐻𝑧 or 𝜔 = 75 𝑠−1 produce a larger max current in an inductor connected to an AC source?

t

10 V

-2.5 A

0.1 s



Page 45

CONCEPT: PHASORS FOR INDCUTORS Remember! The voltage and current across an inductor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos (𝜔𝑡)

- 𝑣𝐿(𝑡) = 𝑖𝑀𝐴𝑋XL cos (𝜔𝑡 +𝜋

2)

Because both cosines have a DIFFERENT angle, they are said to be OUT OF PHASE – The current LAGS the voltage


EXAMPLE: An AC source is connected to an inductor. At a particular instant in time, the current in the circuit is negative and increasing in magnitude. Draw the phasors for voltage and current that correspond to this instant in time.

𝜔𝑡

𝐼

𝜔𝑡 +𝜋

2

𝑉𝐿 𝐼

𝑉𝐿

Voltage across an inductor LEADS the current



Page 46

PRACITCE: PHASORS IN A INDUCTOR CIRCUIT An AC source operates at a maximum voltage of 75 V and is connected to a 0.4 H inductor. If the current across the

inductor is 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos[(450 𝑠−1)𝑡],

a) What is 𝑖𝑀𝐴𝑋? b) Draw the phasors for voltage across the inductor and current in the circuit at 𝑡 = 4.2 𝑚𝑠. Assume that the current phasor begins at 0o.



Page 47

CONCEPT: IMPEDANCE IN AC CIRCUITS We know how to find the current in any AC circuit with ONE element

It’s just the maximum voltage divided by the ____________________

There are two types of circuits: series circuits and parallel circuits.

- Whenever an AC circuit has multiple elements in series, the __________________ phasors line up

- Whenever an AC circuit has multiple elements in parallel, the __________________ phasors line up

Consider an AC source connected in series to a resistor and a capacitor.

- In this case, the maximum voltage across the resistor and capacitor, 𝑉𝑅𝐶 , will NOT be equal to 𝑉𝑅 + 𝑉𝐶

- These maximum voltages, 𝑉𝑅 and 𝑉𝐶, occur at different times

- Instead, the maximum voltage 𝑉𝑅𝐶 will be the ______________________ of the voltage phasors

This leads us to 𝑉𝑅𝐶 = 𝐼𝑀𝐴𝑋√𝑅2 + 𝑋𝐶2 = 𝐼𝑀𝐴𝑋𝑍

EXAMPLE: What’s the impedance of an AC circuit with a resistor and inductor in series?

The IMPEDENCE in an AC circuit, 𝒁, acts as the effective reactance in a circuit with multiple elements The MAXIMUM CURRENT output by the source is ALWAYS 𝐼𝑀𝐴𝑋 = __________________



Page 48

EXAMPLE: IMPEDANCE OF A PARALLEL LR AC CIRCUIT What’s the impedance of a parallel LR AC circuit?

PRACTICE: IMPEDANCE OF A PARALLEL RC AC CIRCUIT

What’s the impedance of a parallel RC AC circuit?



Page 49

PRACTICE: CURRENT IN A PARALLEL RC CIRCUIT An AC source operates at a maximum voltage of 120 V and an angular frequency of 377 s-1. If this source is connected in parallel to a 15 Ω resistor and in parallel to a 0.20 mF capacitor, answer the following questions: a) What is the maximum current produced by the source? b) What is the maximum current through the resistor? c) What is the maximum current through the capacitor?



Page 50

CONCEPT: LRC CIRCUITS IN SERIES In a series LRC circuit, the __________________ through each element is the same

In a DC circuit, we would simply say that 𝑉𝐿𝑅𝐶 = 𝑉𝐿 + 𝑉𝑅 + 𝑉𝐶 , since they are all in series

- In an AC circuit, this isn’t true, since the maximum voltages occur at different times

The IMPEDANCE, 𝒁, acts like the effective reactance of the circuit.

- In a series LRC circuit, the impedance is

The maximum current produced by the source is given by 𝑖𝑀𝐴𝑋 = ________________ EXAMPLE: A circuit is formed by attaching an AC source in series to an 0.5 H inductor, a 10 Ω resistor and a 500 µF capacitor. If the source operates at a VRMS of 120 V and a frequency of 60 Hz, what is the maximum current in the circuit?

In a series LRC circuit, the MAXIMUM voltage is

- 𝑉𝐿𝑅𝐶 = _____________________________

𝒁 = √𝑹𝟐 + (𝝎𝑳 −𝟏

𝝎𝑪)𝟐



Page 51

PRACTICE: VOLTAGE IN A SERIES LRC AC CIRCUIT An AC source operates at an RMS voltage of 70 V and a frequency of 85 Hz. If the source is connected in series to a 20 Ω resistor, a 0.15 H inductor and a 500 µF capacitor, answer the following questions: a) What is the maximum current produced by the source? b) What is the maximum voltage across the resistor? c) What is the maximum voltage across the inductor? d) What is the maximum voltage across the capacitor?



Page 52

CONCEPT: RESONANCE IN SERIES LRC CIRCUITS The impedance of an LRC circuit depends on the frequency of the AC source:

- The impedance is large at small 𝝎 and at large 𝝎 Recall that the impedance is 𝑍 = _________________________________

- The SMALLEST value of impedance, 𝑍 = 𝑅, occurs when 𝑋𝐶 = 𝑋𝐿

- When this occurs, the circuit is said to be in RESONANCE

Since resonance occurs when the impedance is SMALLEST, the current is LARGEST in resonance for series LRC EXAMPLE: An AC circuit is composed of a 10 Ω resistor, a 2 H inductor, and a 1.2 mF capacitor. If it is connected to a

power source that operates at a maximum voltage of 120 V, what frequency should it operate at to produce the largest

possible current in the circuit? What would the value of this current be?

In a series LRC circuit, the current is the same through the inductor and the capacitor

- In resonance, since 𝑋𝐿 = 𝑋𝐶 The voltage across the inductor and the capacitor is the same

The RESONANT FREQUENCY of an LRC circuit is

𝜔0 =1

√𝐿𝐶

𝜔

𝑋𝐶 =1

𝜔𝐶

𝑋𝐿 = 𝜔𝐿

𝑅

𝑍



Page 53

PRACTICE: VOLTAGES IN A SERIES LRC CIRCUIT IN RESONANCE A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µF capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor?



Page 54

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CH 21: ELECTROMAGNETIC INDUCTION AND …lightcat-files.s3.amazonaws.com/packets/admin_physics-3...An outer solenoid with 30 turns is wound tightly around an inner coil 25cm long with