PHYSICS - GIANCOLI CALC 4E CH 29: ELECTROMAGNETIC …lightcat-files.s3.amazonaws.com/packets/admin... · PHYSICS - GIANCOLI CALC 4E CH 29: ELECTROMAGNETIC INDUCTION Page 15. PRACTICE:

! www.clutchprep.com

!

PHYSICS - GIANCOLI CALC 4E

CH 29: ELECTROMAGNETIC INDUCTION

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: MAGNETIC FLUX WITH CALCULUS

If Magnetic Field vector OR Area vector changes along a surface, use ________________ to calculate Magnetic Flux.

- Magnetic Field may be a function of a position variable (like B = B0y x)

EXAMPLE: A magnetic field given by the equation = 1.5x2 passes through a square loop on the x-y plane with side length 2m. The sides of the loop run along the x and y axes. What is the magnetic flux through this square loop?

CHANGING MAGNETIC FLUX

𝚽𝐁 = _________ = __________

A

Normal

CONSTANT MAGNETIC FLUX

𝚽𝐁 = ⋅ = 𝐁𝐀𝐜𝐨𝐬𝛉

x

y

z

A

Normal



Page 5

EXAMPLE: MAGNETIC FLUX FROM STRAIGHT WIRE What is the magnitude of the magnetic flux through the rectangular loop of length L and width w from the current-carrying wire shown in the following figure?

i

d w

L



Page 6

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 7

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 8

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 9

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 10

CONCEPT: FARADAY’S LAW WITH CALCULUS

Remember; one variable will be changing (not uniformly) over time while the other two remain constant.

- For changing magnetic fields → 𝐝𝚽𝐁

𝒅𝒕= _______________

- For changing Area → 𝐝𝚽𝐁

𝒅𝒕= _______________

- For changing Angle → 𝐝𝚽𝐁

𝒅𝒕= _______________

EXAMPLE: A circular of radius 2m lies flat on a surface, with a magnetic field passing through the loop given by the

equation 𝐵(𝑡) = 4 + 3𝑡2. What is the magnitude of the induced EMF at t = 0.5s?

Faraday’s Law: constant rate of change in magnetic flux creates an average EMF:

- 𝜀𝑖𝑛𝑑 = 𝑁 |ΔΦ𝐵

Δ𝑡|

As Δt becomes very small, “change” → “derivative” & 𝜀𝑖𝑛𝑑 becomes an instantaneous EMF:

- 𝜀 =

=



Page 11

EXAMPLE: FARADAY’S LAW OF ROTATING RECTANGULAR LOOP A rectangular conducting loop with length a and width b is in the presence of a uniform magnetic field pointing into the page. The loop is then rotated with an angular speed 𝝎 about its axis. What is the induced EMF in the loop as a function of time?

a

b R



Page 12

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 13

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 14

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 15

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 16

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 17

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 18

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 19

Documents

PHYSICS - GIANCOLI CALC 4E CH 29: ELECTROMAGNETIC …lightcat-files.s3.amazonaws.com/packets/admin... · PHYSICS - GIANCOLI CALC 4E CH 29: ELECTROMAGNETIC INDUCTION Page 15. PRACTICE: