Electromagnetic Induction Objective: TSW understand and apply the concept of magnetic flux in order...

Electromagnetic Induction

Objective:

TSW understand and apply the concept of magnetic flux in order to explain how induced emfs are created and calculate their value and polarity.

You will be responsible for the content contained in chapter 20, sections 1&2 of the textbook. Take the time to read these sections.

Chapter 20 Homework Problems:

4, 9, 15, 17, 19, 22, 23

Electromagnetic induction is the process by which an emf (voltage) is produced in a wire by a changing magnetic flux.

• Magnetic flux is the product of the magnetic field and the area through which the magnetic field passes.

• Electromagnetic induction is the principle behind the electric generator.

• The direction of the induced current due to the induced emf is governed by Lenz’s Law.

Here is a visual model of what we did in chapter 19:

Loop of wire

and an

External

magnetic field

Input

Current

Output

Force

(wire moves)

Electric Motor

Here is a visual model of what we will do in chapter 20:

Loop of wire

and an

External

magnetic field

Input

Force

(move wire)

Output

Current

Generator

The next two slides contain vocabulary and equations, you

should commit them to memory

Important Termsalternating current - electric current that rapidly reverses its directionelectric generator - a device that uses electromagnetic induction to convert mechanical energy into electrical energy electromagnetic induction - inducing a voltage in a conductor by changing the magnetic field around the conductorinduced current - the current produced by electromagnetic inductioninduced emf - the voltage produced by electromagnetic inductionFaraday’s law of induction - law which states that a voltage can be induced in a conductor by changing the magnetic field around the conductorLenz’s law - the induced emf or current in a wire produces a magnetic flux which opposes the change in flux that produced it by electromagnetic inductionmagnetic flux - the product of the magnetic field and the area through which the magnetic field lines pass.motional emf - emf or voltage induced in a wire due to relative motion between the wire and a magnetic field

IVPt

N

BAR

I

BLv

cos

Equations, Symbols, and Units

whereε = emf (voltage) induced by electromagnetic induction (V)v = relative speed between a conductor and a magnetic field (m/s)B = magnetic field (T)L = length of a conductor in a magnetic field (m)I = current (A)R = resistance (Ω)Φ = magnetic flux (Tm2 = Weber=Wb)A = area through which the flux is passing (m2) = angle between the direction of the magnetic field and the area through which it passes

Magnetic Flux

Consider a rectangular loop of wire of height L and width x which sits in a region of magnetic field of strength B. The magnetic field is directed into the page, as shown below:

L

w

The magnetic flux is given by the following equation:

cosBAΦ = The magnetic flux (Tm2 = Wb)

B = The magnetic field (T)

A = area of loop (m2)

Φ = angle between the field and the normal to the area

Faraday’s law states that an induced emf is produced by changing the flux, but how could the flux be changed?

• Turn the field off or on.

• Move the loop of wire out of the field

• Rotate the loop to change the angle between the field and the area of the loop.

Here is Faraday’s Law in equation form:

tN

Where

Є = The induced emf (voltage) (V)

ΔΦ = The change in flux (Wb)

Δt = The change in time (s)

N = number of loops.

*Note that an emf is only produced if the flux changes. The quicker the flux changes the larger the induced emf.

The induced emf in the wire will produce a current in the wire. The magnitude of the induced current is found using Ohm’s Law:

RI

IR

IRV

The direction of the induced current is found using Lenz’s Law (conservation of energy).

Lenz’s Law – The induced current will flow in a direction such that the magnetic field produced by it opposes the original change in flux. In simple terms the wire resists the change in flux and wants to go back to the way things were.

It is helpful to use RHR#2 when using Lenz’s Law.

Example 1: A circular loop of wire with a resistance of 0.5Ω and radius 30cm is placed in an external magnetic field of 0.2T. The magnetic field is turned off in .02 seconds.

a) Calculate the original flux of the loop.

b) Calculate the induced emf.

c) Calculate the current induced in the wire.

d) What direction does the induced current have?

e) What other way could the same emf be induced without turning the field off?

• • • •

• • • •

• • • •

• • • •

• • • •


a) Calculate the original flux of the loop.

• • • •

• • • •

• • • •

• • • •

• • • •

The flux (Φ) is the magnetic field times the cross sectional area

€

φ =BAcosθ

€

φ =(0.2T)(π )(.30m)2 cos0

φ = .057Wb


b) Calculate the induced emf.

• • • •

• • • •

• • • •

• • • •

• • • •

tN

€

=(1).057Wb

.02sε = 2.85V


c) Calculate the current induced in the wire.

• • • •

• • • •

• • • •

• • • •

• • • •

€

=IR

I =ε

R

I =2.85V

0.5ΩI = 5.7A


d) What direction does the induced current have?

• • • •

• • • •

• • • •

• • • •

• • • •

Use lenz’s law – The induced current will flow in a direction such that the magnetic field produced by it opposes the change in flux.Since the magnetic field is turned off it will be decreasing out of the page, so the induced current will produce a field out of the page in order to restore it.

Use the right hand rule – fingers out of the page inside the loop and grab the wire. Your thumb gives the direction of the current.

The current is counter clockwise e) Rotate the loop

Let’s do some examples predicting the induced current

direction using Lenz’s Law

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •

• • • •X X X X

X X X X

X X X X

X X X X

X X X X

X X X X

X X X X

X X X X

X X X X

X X X X

Bout decreasing flux

Bin decreasing flux Bin increasing flux

Bout increasing flux

CCWCW

CCW CW

If you are really struggling applying Lenz’s Law, then memorize the following table:

decreasing Increasing

B out CCW CW

B in CW CCW

Example 2: A circuit with a total resistance of R is made using a set of metal wires and a copper bar. The magnetic field (B) is directed into the page as shown in the diagram. The bar of length L starts on the left and is pulled to the right at a constant velocity.

a) Calculate the induced emf in the circuit.

b) Calculate the current induced in the wire.

c) What direction does the induced current have?

d) What is the magnitude and direction of the magnetic force that opposes the motion of the bar?

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

vL

a) Calculate the induced emf in the circuit.

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

vL

€

=NΔΦ

Δt

€

=(1)(B)(Δx)(L)

Δt

€

=BLv

€

v =Δx

Δt

Δx

b) Calculate the current induced in the wire.

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

vL

€

V = IR

ε = IR

I =ε

R=BLv

R

c) What direction does the induced current have?

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

vL

Use Lenz’s Law – The flux is increasing into the page, so the induced current will produce a field that opposes this. So the induced magnetic field is out of the page. Using RHR put your fingers out of the page, inside the loop and grab the wire. Your thumb gives the direction of the induced current, which is CCW (Shown in green)

d) What is the magnitude and direction of the magnetic force that opposes the motion of the bar?

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

X X X X X X X

vL

€

FB = BIL

€

FB = BBLv

R

⎛

⎝ ⎜

⎞

⎠ ⎟L

€

FB =B2L2v

R

To find the direction of the magnetic force use the RHR on the copper bar. Fingers in the direction of the field, thumb in the direction of the current and the palm of your hand gives the direction of the magnetic force, which is to the left

FB

The last example led to the equation for the motional emf. The motional emf is the voltage induced in a wire as it moves in an external magnetic field. The induced emf will produce a current in the wire, which will in turn result in a force that opposes the motion of the wire. You don’t get something for nothing.

Motional emf

BLv

Where

Є = The induced emf (V)

B = The external magnetic field (T)

L = The length of the wire (m)

v = The velocity of the wire (m/s)

v

B (out of the page)

L

Example 3: A conducting rod of length 0.30 m and resistance 10.0 Ω moves with a speed of 2.0 m/s through a magnetic field of 0.20 T which is directed out of the page.

a) Find the emf induced in the rod.

b) Find the current in the rod and the direction it flows.

c) Find the power dissipated in the rod.

d) Find the magnetic force opposing the motion of the rod.

v

B (out of the page)

L


a) Find the emf induced in the rod.

€

=BLv

€

=(0.20T)(0.30m)(2.0m /s)

€

=.12v

v

B (out of the page)

L


b) Find the current in the rod and the direction it flows.

€

=IR

I =ε

R

I =.12V

10ΩI = .012A

The flux is increasing outward so the induced current will produce a magnetic field inward. The current will flow CW

v

B (out of the page)

L


c) Find the power dissipated in the rod.

€

P = IV

P = (.012A)(.12V )

P =1.44 ×10−3W

v

B (out of the page)

L


d) Find the magnetic force opposing the motion of the rod.

€

FB = BIL

FB = (0.20T)(.012A)(0.30m)

FB = 7.2 ×10−4N

Opposing the direction of the external force.

a

a

B

h

Example 4: A square loop of sides a = 0.4 m, mass m = 1.5 kg, and resistance 5.0 Ω falls from rest from a height h = 1.0 m toward a uniform magnetic field B which is directed into the page as shown.

(a) Determine the speed of the loop just before it enters the magnetic field.

As the loop enters the magnetic field, an emf ε and a current I is induced in the loop.

(b) Is the direction of the induced current in the loop clockwise or counterclockwise? Briefly explain how you arrived at your answer.

When the loop enters the magnetic field, it falls through with a constant velocity.

(c) Calculate the magnetic force necessary to keep the loop falling at a constant velocity.

(d) What is the magnitude of the magnetic field B necessary to keep the loop falling at a

constant velocity?(e) Calculate the induced emf in the loop as it enters and

exits the magnetic field.

a

a

B

h


(a) Determine the speed of the loop just before it enters the magnetic field.

€

Ug =K

mgh =1

2mv 2

v = 2gh

v = (2)(10m /s2)(1.0m)

v = 4.5m

s

a

a

B

h


As the loop enters the magnetic field, an emf ε and a current I is induced in the loop.

(b) Is the direction of the induced current in the loop clockwise or counterclockwise? Briefly explain how you arrived at your answer.

The flux is increasing into the page, so the induced current will produce a magnetic field out of the page. So the current will be CCW

a

a

B

h


When the loop enters the magnetic field, it falls through with a constant velocity.

(c) Calculate the magnetic force necessary to keep the loop falling at a constant velocity.

€

FB = FgFB = mg

FB = (1.5kg)(10m

s2 )

FB =15N

a

a

B

h


(d) What is the magnitude of the magnetic field B necessary to keep the loop falling at a

constant velocity?

€

FB = BIL

15N = BIL

€

=BLvIR = BLv

I =BLv

R

€

15N = BBLv

R

⎛

⎝ ⎜

⎞

⎠ ⎟L

15N =B2L2v

R

B =(15N)R

L2v=

(15N)(5.0Ω)

(0.4m)2(4.5m

s)

=10.2T

a

a

B

h


(e) Calculate the induced emf in the loop as it enters and exits the magnetic field.

€

=BLv

ε = (10.2V )(0.4m)(4.5m

s)

ε =18.4V

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