PHYS 1110 Lecture 8 Professor Stephen Thornton September 20, 2012.

PHYS 1110

Lecture 8

Professor Stephen Thornton

September 20, 2012

If there is a current in If there is a current in

the loop in the direction the loop in the direction

shown, the loop will:shown, the loop will:

A) move upA) move up

B) move downB) move down

C) rotate clockwiseC) rotate clockwise

D) rotate counterclockwiseD) rotate counterclockwise

E) both rotate and moveE) both rotate and move

BB field out of North field out of NorthBB field into South field into South

Reading QuizReading Quiz

Look at the north pole: here the

magnetic field points to the rightright and

the current points out of the pageout of the page.

The right-hand rule says that the force

must point upup. At the south pole, the

same logic leads to a downwarddownward force.

Thus the loop rotates clockwiseclockwise.

A) move upA) move up

B) move downB) move down

C) rotate clockwiseC) rotate clockwise

D) rotate counterclockwiseD) rotate counterclockwise

E) both rotate and moveE) both rotate and move

If there is a current in If there is a current in

the loop in the direction the loop in the direction

shown, the loop will:shown, the loop will:

Reading QuizReading Quiz

Discuss when Midterm Exam 1 will be. Then determine homework 2 due date.

When would you prefer Exam 1?

A)Thursday, September 27B)Tuesday, October 2C)Thursday, October 4

Helical Motion in a Magnetic Field

Remember

F qv B

What happens if we form a loop with the current carrying wire?

• Do demo with wire loop in magnetic field B.

(galvanometer demo)• We find that the loop rotates in opposite directions depending on the direction of the current!

Magnetic Forces on a Current Loop

Forces cause a torque

F I L B

Ftotal = 0

F IhB=

Magnetic Torque on a Current Loop

Top viewWire: width w height h

left right

Torque

/ 2 / 2

sin (if at angle)

IhBw IhBwIhwB IABIAB

Area = A = hw

F I L B

Wires affected are into screen

Magnetic Force on Current Loops• Consider a rectangular loop in a constant

magnetic field. Can also have N loops.

• Can easily find the force on each side of the loop

• Forces cancel but, depending on orientation, there may be a torque

We define the magnetic dipole moment of the coil to be .IA NIA

We showed the torque to be

We can rewrite this in vector form to be

where we have used N loops and the magnetic dipole moment,

sinIAB

B N I A B B

.N I A

Potential Energy of Magnetic Dipole

We can show (but not going to do) that the potential energy of a magnetic dipole moment in a magnetic field is

The torque rotates loop until vectors are parallel to

0B N I A B

Loopof wire

A galvanometer takes advantage of the torque on a current loop to measure current; the spring constant is calibrated so the scale reads in amperes. Remember that all analog ammeters use a galvanometer.

Galvanometer

An electric motor uses the torque on a current loop in a magnetic field to turn magnetic energy into kinetic energy.

Do electric motor demo

A) leftA) left

B) rightB) right

C) zeroC) zero

D) into the pageD) into the page

E) out of the pageE) out of the page

Conceptual QuizConceptual Quiz

A vertical wire carries a current A vertical wire carries a current

and is in a vertical magnetic field. and is in a vertical magnetic field.

What is the direction of the force What is the direction of the force

on the wire? on the wire?

When the current is parallelparallel to

the magnetic field lines, the force

on the wire is zerozero. B

A) leftA) left

B) rightB) right

C) zeroC) zero

D) into the pageD) into the page

E) out of the pageE) out of the page

A vertical wire carries a current A vertical wire carries a current

and is in a vertical magnetic field. and is in a vertical magnetic field.

What is the direction of the force What is the direction of the force

on the wire?on the wire?

F I B= ´

Hans Oersted, a Danish physicist, discovered this in 1820 while entertaining students and friends at home. He was preparing a physics lecture.

Experimental observation

shows 2

for a long straight wire

Now we can imagine combining some of these effects.

• We know that a current carrying wire produces a magnetic field.• We also know that a current carrying wire feels a force in a magnetic field.• If we have two wires, can we use one wire to produce a magnetic field at the position of the second wire? Yes!• If the second wire carries a current,

then it should feel a force!• Do demo – next slide.

We have to look closely at fields and forces to see how the forces occur.

2 2 1F I B

We do this experiment to show that current carrying wires exert forces on each other.

Magnetic field due to current moving through a coil of wire.

Note similarity between B of a bar magnet and B of a coil of wire.

The Solenoid

Magnetic Resonance Imaging

Almost 200 years ago, Michael Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:

Induced EMF

Do demos about induced currents.

Push and pull magnet in and out of coils of wire to show current production.

Induced Current Produced by a Moving Magnet

Therefore, a changing magnetic field induces an emf/current.

Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.

Induced EMF

Magnetic Induction

B linked by iron bar.

We conclude that it is the change in magnetic flux that causes induced current.

B B AF =

Faraday’s Discovery and the Law of Induction

There are many ways to change the magnetic flux through a surface:

• Move the magnet

• Turning current on or off in one loop induces current in another

• Move the loop

• Change the shape (and the area) of the loop

The Magnetic Flux Through a Loop

Flux is maximum

Flux is zero. magnetic flux

Look at the mathematics.

This is called Faraday’s Law of Induction after Michael Faraday.

is number of turnsDon't worry about the sign. We willhave a better way to find this later.

Induced emf BNt

Do some more demos.

1) Magnet through coil again.2) Flash bulb3) LED coil

Lenz’s Law

The induced current will always be in the direction to oppose the change that produced it.

Induced emf Induced currentÛ

Applying Lenz’s Law to a Magnet Moving Toward and Away From a Current Loop

Induced current

Conceputal QuizIn order to change In order to change the magnetic flux the magnetic flux through the loop, through the loop, what would you what would you have to do?have to do?

A) drop the magnetA) drop the magnet

B) move the magnet upwardsB) move the magnet upwards

C) move the magnet sideways a lotC) move the magnet sideways a lot

D) Only A and BD) Only A and B

E) A, B, and CE) A, B, and C

Moving the magnet in any directionany direction would

change the magnetic field through the

loop and thus the magnetic flux.

Conceptual QuizIn order to change In order to change the magnetic flux the magnetic flux through the loop, through the loop, what would you what would you have to do?have to do?

A) drop the magnetA) drop the magnet

B) move the magnet upwardsB) move the magnet upwards

C) move the magnet sideways a lotC) move the magnet sideways a lot

D) only A and BD) only A and B

E) A, B, and CE) A, B, and C

If a North pole moves If a North pole moves toward the loop from above toward the loop from above the page, in what direction is the page, in what direction is the induced current? the induced current?

A) clockwiseA) clockwise

B) counterclockwiseB) counterclockwise

C) no induced currentC) no induced current

Conceptual Quiz

If a North pole moves toward If a North pole moves toward the loop from above the page, the loop from above the page, in what direction is the in what direction is the induced current? induced current?

The magnetic field of the moving bar

magnet is pointing into the pageinto the page and

getting largerlarger as the magnet moves

closer to the loop. Thus the induced

magnetic field has to point out of the out of the

pagepage. A counterclockwisecounterclockwise induced

current will give just such an induced

magnetic field.

Conceptual Quiz

Follow-up:Follow-up: What happens if the magnet is stationary but the loop moves? What happens if the magnet is stationary but the loop moves?

Motional emf

What happens when we push rod down?

Determining the Direction of an Induced Current

We exert force to push bar down.

Motional emf

B A B v t

B v tB v

DF = D = D

DF D= = =

Find force and energy

mechanical

2 2 2 22

electrical

B vF B B

B vP Fv

Bv B vP I R R

x x x x x x x x x x x x

A wire loop is being A wire loop is being pulled through a pulled through a uniform magnetic field. uniform magnetic field. What is the direction of What is the direction of the induced current? the induced current?

Conceputal Quiz

Since the magnetic field is uniform, the

magnetic flux through the loop is not magnetic flux through the loop is not

changingchanging. Thus no current is inducedno current is induced.

x x x x x x x x x x x x

A wire loop is being A wire loop is being pulled through a pulled through a uniform magnetic field. uniform magnetic field. What is the direction of What is the direction of the induced current? the induced current?

Conceptual Quiz

Follow-up:Follow-up: What happens if the loop moves out of the page? What happens if the loop moves out of the page?

A conducting rod slides on a

conducting track in a constant

B field directed into the page.

What is the direction of the

induced current?

x x x x x x x x x x x

A conducting rod slides on a

conducting track in a constant

B field directed into the page.

What is the direction of the

induced current?

x x x x x x x x x x x

The B field points into the pageinto the page. The flux is increasingincreasing since the area is increasing. The induced B field opposes this change and therefore points out of the pageout of the page. Thus, the induced current runs counterclockwise,counterclockwise, according to the right-hand rule.

Follow-up:Follow-up: What direction is the magnetic force on the rod as it moves? What direction is the magnetic force on the rod as it moves?

A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator:

The axle is rotated by an external force such as falling water or steam. The brushes are in constant electrical contact with the slip rings. See next slide.

Electric Generators

An Electrical Generator

Falling water,steam

Produces AC power

Magnetic flux changes!

Current is induced

A Simple Electric Motor/Generator

Do demo

If the loop is rotating with constant angular velocity ω, the induced emf is sinusoidal:

For a coil of N loops,

Induced power:

(cos ) sinBA t BA tt

w w we eD=- = =

( )222sin

NABP I t

ee= = =

A generator has a coil of A generator has a coil of wire rotating in a wire rotating in a magnetic field. If the magnetic field. If the rotation rate increasesrotation rate increases, , how is the how is the maximum maximum output voltageoutput voltage of the of the generator affected?generator affected?

A) increasesA) increases

B) decreasesB) decreases

C) stays the sameC) stays the same

D) varies sinusoidallyD) varies sinusoidally

Conceptual Quiz

The maximum voltage is the leading

term that multiplies sin(sin(tt)) and is

given by = = NBANBA. Therefore, if

increases increases, then must increase must increase

as well.

)sin( tNBA

A generator has a coil of A generator has a coil of wire rotating in a wire rotating in a magnetic field. If the magnetic field. If the rotation rate increasesrotation rate increases, , how is the how is the maximum maximum output voltageoutput voltage of the of the generator affected?generator affected?

A) increasesA) increases

B) decreasesB) decreases

C) stays the sameC) stays the same

D) varies sinusoidallyD) varies sinusoidally

Conceptual Quiz

Conceptual Quiz:Look at the demonstration of the large electromagnet. Observe what happens (spark) when the switch is opened. What best explains this?A) The battery voltage is leaking through.B) The steady current passing through the magnet.C) Induces a large back current (back emf).D) in this case.V

Answer: C

Nature doesn’t want the magnetic flux to change, so it induces a large current (back emf) to produce a magnetic field. This emf results in the spark across the switch.

Inductance and Inductors• Faraday’s Law: Changing current

in a circuit will induce emf in that circuit as well as others nearby

• Self-Inductance: Circuit induces emf in itself (source of back emf)

• Mutual Inductance: Circuit induces emf in second circuit

Inductance magnetic flux depends on current

is called inductance (actually self inductance here).

The inductance L is a proportionality constant that depends on the geometry of the circuit

Changing Current in an Inductor

Switch open. No current flowing.

Switch closed. Inductor opposes magnetic flux change. Induces current to oppose battery current; current rises more slowly.

Inductor

Magnetic field energyWe know that a battery has to do work to cause current to flow. Similarly an inductor has to do work to cause an induced current to flow from 0 to I in time T. This energy comes from the magnetic field.

2The work done is the energy stored in the inductor

LIP I V I

U P T LI W

There will be a magnetic flux in Loop 1 due to current I1 flowing in Loop 1 and due to current I2 flowing in Loop 2.

1 1 12 2

2 2 21 1

Similarly,(2)

L I M I

Now it is clearer why we call L self inductance and M mutual inductance.

For example, two nearby coils

Mutual InductanceA long thin solenoid of length ℓ and cross-sectional area A contains N1 closely packed turns of wire. Wrapped around it is an insulated coil of N2 turns. All the flux from coil 1 (the solenoid) passes through coil 2. The magnetic flux between the two coils is linked. We call this mutual inductance.

Unit of inductance: the henry, H:

1 H = 1 V·s/A = 1 Ω·s.

A transformer is an example of mutual inductance.

Solenoid Self-Induction

Area A

Only depends on geometry.

20, so

NBA nNIA An I

LI L An

Consider an inductorFor a solenoid

2 2 2 2 2 20 0

1 1 1( )

1 1But , so

magnetic energy energy density

volume 2

U LI n A I n I A

B nI U B A B V

General energy density

1 general result

Bu u u E

We can produce an emf by using AC voltage and coils.

Show demo of AC coils and light bulb

Do transformer demo

Transformer equation

Primary coil:

Secondary coil:

For good transformer,

If resistance is small,

step-up and step-down transformers

This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:

Lots of applications for transformers,the bug zapper.

Power distribution

Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.

120 V120 V

What is the voltage What is the voltage

across the lightbulb? across the lightbulb?

A) 30 VA) 30 V

B) 60 VB) 60 V

C) 120 VC) 120 V

D) 240 VD) 240 V

E) 480 VE) 480 V

The first transformerfirst transformer has a 2:1 ratio2:1 ratio

of turns, so the voltage doublesvoltage doubles.

But the second transformersecond transformer has a

1:2 ratio1:2 ratio, so the voltage is halvedvoltage is halved

again. Therefore, the end result is

the same as the original voltagesame as the original voltage.

120 V120 V 240 V240 V 120 V120 V

What is the voltage What is the voltage

across the lightbulb? across the lightbulb?

A) 30 VA) 30 V

B) 60 VB) 60 V

C) 120 VC) 120 V

D) 240 VD) 240 V

E) 480 VE) 480 V

A) greater than 6 VA) greater than 6 V

B) 6 VB) 6 V

C) less than 6 VC) less than 6 V

D) zeroD) zero

A 6 V battery is connected to one

side of a transformer. Compared

to the voltage drop across coil A,

the voltage across coil B is:

A B6 V

Conceptual Quiz

The voltage across B is zeroThe voltage across B is zero.

Only a changingchanging magnetic flux

induces an emf. Batteries can

provide only dc currentdc current.

A) greater than 6 VA) greater than 6 V

B) 6 VB) 6 V

C) less than 6 VC) less than 6 V

D) zeroD) zero

A B6 V

Conceptual Quiz

A 6 V battery is connected to one

side of a transformer. Compared

to the voltage drop across coil A,

the voltage across coil B is:

PHYS 1110 Lecture 8 Professor Stephen Thornton September 20, 2012.

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