Phy2049: Magnetism

• Last lecture: Biot-Savart’s and Ampere’s law:– Magnetic Field due to a straight wire– Current loops (whole or bits)and solenoids

• Today: reminder and Faraday’s law.

hittTwo long straight wires pierce the plane of the paper at vertices of an equilateral triangle as shown. They each carry 3A but in the opposite direction. The wire on the left has the current coming out of the paper while the wire on the right carries the current going into the paper. The magnetic field at the third vertex (P) has the magnitude and direction(North is up):

(1) 20 μT, east (2) 17 μT, west (3) 15 μT, north (4) 26 μT, south (5) none of these

X

4 cm

Magnetic Field Units• From the expression for force on a current-carrying wire:

– B = Fmax / I L– Units: Newtons/Am Tesla (SI unit)– Another unit: 1 gauss = 10-4 Tesla

• Some sample magnetic field strengths:– Earth: B = 0.5 gauss = 0.5 x 10-4 T = 5 x 10-5 T– Galaxy: B 10-6 gauss = 10-10 T– Bar magnet: B 100 – 200 gauss– Strong electromagnet: B = 2 T– Superconducting magnet: B = 20 – 40 T– Pulse magnet: B 100 T– Neutron star: B 108 – 109 T– Magnetar: B 1011 T

Force Between Two Parallel Currents• Force on I2 from I1

– RHR Force towards I1

• Force on I1 from I2

– RHR Force towards I2

• Magnetic forces cause attraction between two parallel currents

I1I2

0 1 0 1 22 2 1 2 2 2

I I IF I B L I L L

r r

I1I2

0 2 0 1 21 1 2 1 2 2

I I IF I B L I L L

r r

Force Between Two Anti-Parallel Currents

• Force on I2 from I1

– RHR Force away from I1

• Force on I1 from I2

– RHR Force away from I2

• Magnetic forces repel two antiparallel currents

I1I2I1I2

0 1 0 1 22 2 1 2 2 2

I I IF I B L I L L

r r

0 2 0 1 21 1 2 1 2 2

I I IF I B L I L L

r r

Parallel Currents (cont.)• Look at them edge on to see B fields more clearly

Antiparallel: repel

F F

Parallel: attract

F F

B

BB

B2 1

2

2

2

1

11

B Field @ Center of Circular Current Loop

• Radius R and current i: find B field at center of loop

– Direction: RHR #3 (see picture)

• If N turns close together

02

iB

R

02

N iB

R

Current Loop Example

• i = 500 A, r = 5 cm, N=20

70

20 4 10 5001.26T

2 2 0.05i

B Nr

Challenge: How much heat is produced?

B Field of Solenoid• Formula found from Ampere’s

law– i = current– n = turns / meter

– B ~ constant inside solenoid– B ~ zero outside solenoid– Most accurate when L>>R

• Example: i = 100A, n = 10 turns/cm– n = 1000 turns / m

0B in

7 34 10 100 10 0.13TB

Field at Center of Partial Loop

• Suppose loop covers angle

• Use example where = (half circle)– Define direction into page as positive

02 2

iB

R

0 0

1 2

0

1 2

2 2 2 2

1 14

i iB

R R

iB

R R

Partial Loops (cont.)

• Note on problems when you have to evaluate a B field at a point from several partial loops– Only loop parts contribute, proportional to angle

(previous slide)– Straight sections aimed at point contribute nothing– Be careful about signs, e.g.in (b) fields partially cancel,

whereas in (a) and (c) they add

2

2

Consider the magnetic field generated by a wire coil of radius R which carries a current i. The magnetic fieldat a point P on the z-axis is given by:

2oiRB

R

The magnetic field of a magnetic dipole.

3/ 22

2

3

2

3 3 3

Here is the distance between

P and the coil center. For points far from the loop

( ) we can use the approximation: 2

Here is the magnetic2 2 2

dipole m

o

o o o

zz

iRz R Bz

i R iABpz z z

3

oment of the loop. In vector form:

( )2

The loop generates a magnetic field that has the sameform as the field generated by a bar magnet.

oB zz

3( )2

oB zz

(29 – 14)

Chapter 30 Induction and Inductance

In this chapter we will study the following topics:

-Faraday’s law of induction -Lenz’s rule -Electric field induced by a changing magnetic field

Electric current without a battery

-Inductance and mutual inductance - RL circuits -Energy stored in a magnetic field

(30 – 1)

In a series of experiments Michael Faraday in England and Joseph Henry in the US were able to generate electric currents without the use of batteries Below we describe some of the

Faraday's experiments

se experiments thathelped formulate whats is known as "Faraday's lawof induction"

The circuit shown in the figure consists of a wire loop connected to a sensitiveammeter (known as a "galvanometer"). If we approach the loop with a permanent magnet we see a current being registered by the galvanometer. The results can be summarized as follows:

A current appears only if there is relative motion between the magnet and the loop Faster motion results in a larger current If we

1.2.3. reverse the direction of motion or the polarity of the magnet, the currentreverses sign and flows in the opposite direction. The current generated is known as " "; the emf that appears induced current is known as " "; the whole effect is called " "induced emf induction

(30 – 2)

In the figure we show a second type of experimentin which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. Whenthe current in loop 1 is constant no induced current is observed in loop 2. The conclusion is that the magnetic field in an induction experiment can begenerated either by a permanent magnet or by anelectric current in a coil.

loop 1loop 2

Faraday summarized the results of his experiments in what is known as" "Faraday's law of induction

An emf is induced in a loop when the number of magnetic field lines thatpass through the loop is changing

Faraday's law is not an explanation of induction but merely a description of of what induction is. It is one of the four " of electromagnetism"all of which are statements of experim

Maxwell's equationsental results. We have already encountered

Gauss' law for the electric field, and Ampere's law (in its incomplete form)

(30 – 3)

B

n̂

dA

The magnetic flux through a surface that bordersa loop is determined as follows:

BMagnetic Flux Φ

1 we divide the surface that has the loop as its borderinto area elements of area . dA.

For each element we calculate the magnetic flux through it: cos

ˆHere is the angle between the normal and the magnetic field vectorsat the position of the element.

We integrate a

Bd BdA

n B

2.

3.

2: T m known as the Weber (symbol

ll the terms. cos

We can express Faraday's law of induction in the folowin W

g b)

form:

B BdA B dA

SI magnetic flux unit

B

The magnitude of the emf induced in a conductive loop is equal to rateat which the magnetic flux Φ through the loop changes with time

E

B B dA

Bddt

E(30 – 4)

B

n̂

dA

cosB BdA B dA

Change the magnitude of within the loop Change either the total area of the coil or

the portion of the area within the magnetic field

Change the angle

BBMethods for changing Φ through a loop

1.2.

3.

ˆ between and

Problem 30-11cos cos

sin

22 sin 2

B

B

B n

NAB NabB td NabB t

dtffNabB t

An Example.

E

E

B

n̂

loop(30 – 5)

Bddt

EWe now concentrate on the negative sign in the equation that expresses Faraday's law.The direction of the flow of induced currentin a loop is acurately predicted by what is known as Lenz's

Lenz's Rule

rule.

An induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current

Lenz's rule can be implemented using one of two methods:

In the figure we show a bar magnet approaching a loop. The induced current flowsin the direction indicated becaus

1. Opposition to pole movement

e this current generates an induced magnetic fieldthat has the field lines pointing from left to right. The loop is equivalent to a magnet whose north pole haces the corresponding north pole of the b

B

ar magnetapproaching the loop. The loop the approaching magnet and thus opposes the change in which generated the induced current.

repels

(30 – 6)

N S

magnet motion

Bar magnet approaches the loop with the north pole facing the loop.

2. Opposition to flux changeExample a :

B

As the bar magnet approaches the loop the magnet field points towards the leftand its magnitude increases with time at the location of the loop. Thus the magnitudeof the loop magnetic flux also

B

increases. The induced current flows in the

(CCW) direction so that the induced magnetic field opposes

the magnet field . The net field . The induced current is ti

net i

B

B B B B

counterclockwise

B B

hus trying to from increasing. Remember it was the increase in that generatedthe induced current in the first place.

prevent

(30 – 7)

N S

magnet motion

Bar magnet moves away from the loop with north pole facing the loop.

2. Opposition to flux changeExample b :

B

As the bar magnet moves away from the loop the magnet field points towards the leftand its magnitude decreases with time at the location of the loop. Thus the magnitudeof the loop magnetic flux

B

also decreases. The induced current flows in the

(CW) direction so that the induced magnetic field adds to

the magnet field . The net field . The induced current is thusi

net i

B

B B B B

clockwise

B B

trying to from decreasing. Remember it was the decrease in that generatedthe induced current in the first place.

prevent

(30 – 8)

S N

magnet motion

Bar magnet approaches the loop with south pole facing the loop.

2. Opposition to flux changeExample c :

B

As the bar magnet approaches the loop the magnet field points towards the rightand its magnitude increases with time at the location of the loop. Thus the magnitudeof the loop magnetic flux als

B

o increases. The induced current flows in the

(CW) direction so that the induced magnetic field opposes

the magnet field . The net field . The induced current is thus tryi

net i

B

B B B B

clockwise

B B

ing to from increasing. Remember it was the increase in that generatedthe induced current in the first place.

prevent

(30 – 9)

S N

magnet motion

Bar magnet moves away from the loop with south pole facing the loop.

2. Opposition to flux changeExample d :

As the bar magnet moves away from the loop the magnet field points towards the right and its magnitude decreases with time at the location of the loop. Thus the magnitude of the loop magnetic flux

B

B also decreases. The induced current

flows in the (CCW) direction so that the induced magnetic field

adds to the magnet field . The net field . The induced curi

net i

B

B B B B

counterclockwise

B B

rent is thus trying to from decreasing. Remember it was the decrease in that generated the induced current in the first place.

prevent

(30 –10)

By Lenz's rule, the induced current always opposesthe external agent that produced the induced current.Thus the external agent must always on theloop-magnetic fie

Induction and energy transfers

do workld system. This work appears as

thermal energy that gets dissipated on the resistance of the loop wire.

Lenz's rule is actually a different formulation ofthe principle of energy conservationConsid

R

er the loop of width shown in the figure. Part of the loop is located in a region where a uniform magnetic field exists. The loop is beingpulled outside the magnetic field region with constantsp

L

B

eed . The magnetic flux through the loop The flux decreases with time

B

B

vBA BLx

d dx BLvBL BLv idt dt R R

EE(30 –11)

2 2 2 22

2 3

The rate at which thermal energy is dissipated on

( )

The magnetic forces on the wire sides are shown

in the figure. Forces and cancel each other.

Force

th

R

BLv B L vP i R RR R

F F

F

eqs.1

1 1

2 2

1

2 2 2

1

sin 90

The rate at which the external agent is

producing mechanical work ( )

If we compare equations 1 and 2 we see that indeed theme

ext

BLviL B F iLB iLB LBR

B L vFR

B L vP F vR

eqs.2

chanical work done by the external agent that movesthe loop is converted into thermal energy that appearson the loop wires.

(30 –12)

We replace the wire loop in the previous example with a solid conducting plate and move the plate out of the magnetic field as shown in the figure.

The motion between the plate and indB

Eddy currents

uces a

current in the conductor and we encounter an opposing force. With the plate the free electrons do not followone path as in the case of the loop. Instead the electronsswirl around the plate. These currents are known as " ". As in the case of the wire loop the netresult is that mechanical energy that moves the plate is transformed into thermal energy that heats up the pla

eddy currents

te.

(30 –13)

Consider the copper ring of radius shown in thefigure. It is paced in a uniform magnetic field pointing into the page, that icreases as function of time. The resulting chan

rB

Induced electric fields

ge in magnetic fluxinduces a current i in the counterclock-wise(CCW) direction.

The presence of the current in the conducting ring implies that an induced

electric field must be present in order to set the electrons on motion.Using the argument above we can reformulate Farada

i

E

y's law as follows:

A changing magnetic field produces an electric field

The induced electric field is generated even in the absense of the copper ring. Note :

(30 –14)

Consider the circular closed path of radius shown in the figure to the left. The picture is the same as thatin the previous page except that the copper ring has been removed. The path is now an a

r

bstract line.The emf along the path is given by the equation:

( )

The emf is also given by Faraday's law:

( ) If we compare eqs.1 with eqs.2

we get:

B

E ds

ddt

dE ds

eqs.1

eqs.2

E

E

2 2

2

cos0 2

22

B

BB

E ds Eds E ds rE

d dBr B rdt dt

dBrE rd

dt

r dBEdtt

(30 –15)

Consider a solenoid of length that has loops of

area each, and windings per unit length. A current

flows through the solenoid and generates a unifrom magnetic field ino

NNA n

iB ni

Inductance

side the solenoid.The solenoid magnetic flux B NBA

B

2The total number of turns The result we got for the

special case of the solenoid is true for any inductor. . Here is a constant known as the of the solenoid

B o

B

N n n A i

Li L

inductance

22

. The inductance depends on the geometry of the particular inductor.

For the solenoid oBo

n AiL n Ai i

Inductance of the solenoid

2 oL n A

(30 –16)

loop 1loop 2In the picture to the right we

already have seen how a changein the current of loop 1 results in a change in the flux through loop 2, and thus creates an induced emf in loop 2

Self Induction

If we change the current through an inductor this causesa change in the magnetic flux through the inductor

according to the equation: Using Faraday's

law we can determined the re

B

B

Lid diL

dt dt

the Henry (symbol: H)An inductor has inductance

sulting emf known as

em

1 H if a current change of 1 A/s results in a self-induced emf of

f

1 V

.

.

B

L

d diLdt dt

SI unit for L :

Eself induced

diLdt

E

(30 –17)

Consider the circuit in the upper figure with the switchS in the middle position. At 0 the switch is thrownin position a and the equivelent circuit is shown inthe lower figure. It cont

t

RL circuits

ains a battery with emf , connected in series to a resistor and an inductor (thus the name " circuit"). Our objective is to calculate the current as function of time . We write Kirchhoff's

R LRL

i t

E

loop rule starting at point x and moving around the loop in the clockwise direction.

0 diL iRddt

iiR Ldt

E E

/

The initial condition for this problem is: (0) 0. The solution of the differentialequation that satisfies the initial condition is:

The constant is known as th( 1 e) "t Li t eR

i

R

E time cons " of

the RL circuit.

tant

/( ) 1 ti t eR

E L

R

/

/

/

( ) 1 Here

The voltage across the resistor 1 .

The voltage across the inductor

The solution gives 0 at 0 as required by the initial conditio

t

tR

tL

Li t eR R

V iR e

diV L edt

i t

E

E

E

n. The solution gives ( ) /The circuit time constant / tells us how fastthe current approaches its terminal value. ( ) 0.632 /

( 3 ) 0.950 /

( 5 ) 0.993 /

If we wait only

i RL R

i t R

i t R

i t R

a few time c

E

EEE

the current, for all

practical purposes has reached its terminal value / . Ronstants

E .

(30 –19)

We have seen that energy can be stored in the electric fieldof a capacitor. In a similar fashion energy can be stored inthe magnetic field of an inductor. Consider t

Energy stored in a magnetic field

he circuitshown in the figure. Kirchhoff's loop rules gives:

2

2

If we multiply both sides of the equation we get:

The term describes the rate at which the batter delivers energy to the circuit

The term is the rate at which

di diL iR i Li i Rdt dt

i

i R

E EE

thermal energy is produced on the resistor

Using energy conservation we conclude that the term is the rate at which

energy is stored in the inductor. We integrate

both

BB

diLidt

dU diLi dU Lididt dt

2 2

sides of this equation: 2 2

ii

oB

o

L i LiU Li di

2

2B

LiU

(30 –20)

B Consider the soleniod of length and loop area

that has windings per unit length. The solenoidcarries a current i that generates a uniform magnetic field o

An

B ni

Energy density of a magnetic field

inside the solenoid. The magnetic fieldoutside the solenoid is approximately zero.

2 221The energy stored by the inductor is equal to

2 2This energy is stored in the empty space where the magnetic field is present

We define as energy density where is the volume in

oB

BB

n A iU Li

Uu VV

22 2 2 2 2 2 2

side

the solenoid. The density 2 2 2 2

This result, even though it was derived for the special case of a uniformmagnetic field, holds true in general.

o o oB

o o

n A i n i n i BuA

2

2B

o

Bu

(30 –21)

N2

N1 Consider two inductorswhich are placed close enough so that the magnetic field of onecan influence the other.

Mutual Induction

1 1

2 21 1

1

In fig.a we have a current in inductor 1. That creates a magnetic field in the vicinity of inductor 2. As a result, we have a magnetic flux through inductor 2. If curent varies wit

i BM i

i

2 12 21 21

h time, then we have a time varying flux through inductor 2 and therefore an induced emf across it.

is a constant that depends on the geometry

of the two inductors as we

d diM Mdt dt

Ell as their relative position. (30 –22)

12 21

diMdt

E

N2

N1

2 2

1 12 2

2

In fig.b we have a current in inductor 2. That creates a magnetic field in the vicinity of inductor 1. As a result, we have a magnetic flux through inductor 1. If curent varies wit

i BM i

i

1 21 12 12

h time, then we have a time varying flux through inductor 1 and therefore an induced emf across it.

is a constant that depends on the geometry

of the two inductors as we

d diM Mdt dt

Ell as their relative position.

21 12 diM

dtE

(30 –23)

N2

N1

12 21

diMdt

E

21 12 diM

dtE

12 21 12 21It can be shown that the constants and are equal. The constant is known as the " " between the two coils.Mutual inductance is a constant that depends on the geometr

M M M M MM

mutual inductance

the Hey of th

nry (H)e two

inductors as well as their relative position. The expressions for the induced emfs across the two inductors become:

The SI unit for M :

21 diM

dtE 1

2diMdt

E

(30 –24)

1 2Mi 2 1Mi

Electrons are going around a circle in a counterclockwise direction as shown. At the center of the circle they produce a magnetic field that is:

A. into the pageB. out of the pageC. to the leftD. to the rightE. zero

e

Long parallel wires carry equal currents into or out of the page. Rank according to the magnitude of the net magnetic field at the center of the square, highest first. (Parenthesis means the same magnitude).

1. C,D, (A,B) 2. A, B, (C,D) 3. B, A, C, D4. D, (A, B), C

Long, straight, parallel wires carry equal currents into or out of page. Rank according to the magnitude of the force on the central wire.

1. d, c, a, b 2. a, b, c, d 3. b, c, d, a 4. c, a, b, d5. b, d, c, a