physics - clutch ch 28: induction and...

! www.clutchprep.com

!

PHYSICS - CLUTCH

CH 28: INDUCTION AND INDUCTANCE

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

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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]

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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 = ______

PHYSICS - CLUTCH


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


PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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?

PHYSICS - CLUTCH


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

PHYSICS - CLUTCH


CONCEPT: LR CIRCUITS LR Circuits are, as the name implies, circuits containing ___________________ and ___________________

There are two steps needed to analyze this circuit: - CURRENT GROWTH: When S1 is closed, but S2 is open, the battery produces a current in the circuit - CURRENT DECAY: When S1 is open and S2 is closed, the current decays because the battery is removed

CURRENT GROWTH in an LR circuit does not occur instantly – the inductor resists changes to currents

𝑖(𝑡) =𝑉

𝑅(1 − 𝑒−𝑡/𝜏)

CURRENT DECAY in an LR circuit does not occur instantly – the inductor resists changes to currents

𝑖(𝑡) =𝑉

𝑅𝑒−𝑡/𝜏

The TIME CONSTANT, 𝝉 =𝑳

𝑹, determines the how quickly growth and decay occurs

S1

S2

L R

V

L R

V

L R

PHYSICS - CLUTCH


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?

PRACTICE: TIME TO HALF MAXIMUM CURRENT

An LR circuit with L = 0.1 H and R = 10 Ω are connected to a battery with the circuit initially broken. When the circuit is

closed, how much time passes until the current reaches half of its maximum value?

PHYSICS - CLUTCH


PRACTICE: UNKNOWN CURRENT IN AN LR CIRCUIT Consider the LR circuit shown below. Initially, both switches are open. First, switch 1 is closed, and current is allowed to grow to its maximum value. Then, switch two is closed and switch 1 is open, and current is allowed to decay for 0.05 s. What is the maximum current in the circuit? What is the final current in the circuit if V = 10 V, L = 0.02 H, and R = 5 Ω?

S1

S2

L R

V

PHYSICS - CLUTCH


CONCEPT: LC CIRCUITS LC Circuits are made up of __________________ and __________________, as their name implies

The current in this circuit OSCILLATES:

𝜙 is known as the PHASE ANGLE, and it determines what part of the oscillation you begin at EXAMPLE: An LC circuit with an inductor of 0.05 H and a capacitor of 35 µF begins with the current at half its maximum value. What is the phase angle of this oscillation?

+q -q

i

q = 0

i

+q

i

-q q = 0

i

+q -q

i

MATHEMATICALLY, the current and charge are represented by

- 𝒊(𝒕) = _____________________ (𝝎 = √𝟏/𝑳𝑪 is the angular frequency of oscillation)

- 𝒒(𝒕) = _____________________

PHYSICS - CLUTCH


EXAMPLE: OSCILLATIONS IN AN LC CIRCUIT

An LC circuits, with L = 0.05 H and C = 50 mF, begins with the capacitor fully charged. After 0.1 s, the current is 0.2 A.

Under these conditions, how many seconds does it take for a fully charged plate to transfer all of its charge to the other

plate?

PRACTICE: MAXIMUM CURRENT IN LR CIRCUIT

An LR circuit has a 0.5 mF capacitor initially charged to 1 mC. If it is connected to a 0.04 H inductor, what is the maximum

current in the circuit?

PHYSICS - CLUTCH


CONCEPT: ENERGY IN AN LC CIRCUIT An inductor is just a coil of wire – we don’t assume it has any resistance

So when the capacitor loses its charge, and therefore its energy, where does it go?

So long as the wires don’t have any resistance, energy is conserved in an LC circuit:

EXAMPLE: An LC circuit has an 0.1 H inductor and a 15 nF capacitor, and begins with the capacitor maximally charged. After 0.1 s, how much energy is stored by the inductor? If the initial charge on the capacitor were 50 mC, what is the maximum current in the circuit?

An inductor can store MAGNETC ENERGY

- 𝑼 = ___________________

+q -q

i

q = 0

i

+q

i

-q q = 0

i

+q -q

i

E UC UL E UC UL E UC UL E UC UL E UC UL

PHYSICS - CLUTCH


PRACTICE: ENERGY LOSS DUE TO RESISTANCE Let’s say an LC circuit begins with the capacitors carrying a maximum charge of 10 mC. After the capacitor has lost half of its charge, what is the current in the circuit if L = 0.01 H and C = 50 mF? If during the time for the capacitor to lose half its charge, resistance within the circuit dissipated 0.2 mJ, what then would the current in the circuit be?

PHYSICS - CLUTCH


CONCEPT: LRC CIRCUITS As the name implies, an LRC circuit contains _________________, _________________, and _________________

In and LRC circuit, with the capacitor initially charged, we have: - 𝚺𝑽 = _____________________________ = 0 _________________________________ = 0 _________________________________ = 0 There are 3 solutions to the equation above: the UNDERDAMPED, CRITICALLY DAMPED, and OVERDRAMPED

UNDERDAMPED CRITICALLY DAMPED OVERDAMPED

- 𝑞(𝑡) = 𝑄𝑒−(𝑅/2𝐿)𝑡cos(𝜔′𝑡 + 𝜙)

- Occurs for small R

- Looks almost like an LC circuit

- But R is sapping energy

- 𝑞(𝑡) = 𝑄𝑒−(𝑅/2𝐿)𝑡

- Occurs when 𝑹𝟐 = 𝟒𝑳/𝑪

- Looks like an RC Circuit

- No simple equation

- Occurs for large R

- Looks like an RC Circuit

The new angular frequency is

- 𝝎′ = √ 𝟏

𝑳𝑪−

𝑹𝟐

𝟒𝑳𝟐

+q -q

q

t

PHYSICS - CLUTCH


EXAMPLE: AMPLITUDE DECAY IN AN LRC CIRCUIT An LRC circuit has an inductance of 10 mH, a capacitance of 100 µF, and a resistance of 20 Ω. What type of LRC circuit is this? How long will it take for the maximum charge stored on the capacitor to drop by half?

PHYSICS - CLUTCH


physics - clutch ch 28: induction and...

Documents