Download ppt - Magnetic fields and magnetic forces Han Christian Oersted/Andr é Ampere – discovered the relationship between moving charges and magnetism Michael Faraday/Joseph

Magnetic fields and magnetic forces

• Han Christian Oersted/André Ampere – discovered the relationship between moving charges and magnetism

• Michael Faraday/Joseph Henry – discovered that moving a magnet near a conducting loop can cause a current in the loop

• Magnetic fields are produced by electric currents.• The Lorentz force: F=qvxB• SI unit: Testa (T)• If the charge is moving in a region where E and B fields are

present:

Magnetic field sources

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

The magnetic field lines around a

long wire


The magnetic field lines of a current loop

Soleniod Bar magnet


The bar magnetThe earth


The bar magnet and the earth


http://www.timed.jhuapl.edu/WWW/science/objectives.php

http://www.timed.jhuapl.edu/WWW/science/objectives.php

Imager of Sprites and Upper Atmospheric Lightning (ISUAL)

Imager of Sprites and Upper Atmospheric Lightning (ISUAL)

The Lorentz force


Magnetic force on moving charge


Mass spectrometer


e/m experiment

http://physics.csustan.edu/GENERAL/Ian/GeneralPhysicsIIlabs/EoverM/movies/Path.htm

Magnetic force on a current-carrying wire


What is generated in the wire?


Magnetic flux

• Magnetic flux is the product of the average magnetic field and the perpendicular area that it penetrates.

AB d

dA

dB

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlMagnetic flux density

Gauss’ law of magnetism

The net magnetic flux out of any closed surface is zero.

What is the physical significance of this statement?

Units:

1. Magnetic field B: 1T=1Ns/Cm

2. Magnetic flux : 1W=1Tm2

Motion of a charged particles in a magnetic field

• Two positive ions having the same charge q, but different masses m1 and m2, are accelerated from rest through a potential difference V. They then enter a region where there is a uniform magnetic field B normal to the plane of the trajectory. Show that if the beam entered the magnetic field along the x-axis, the value of the y-coordinate for each ion at any time t is approximately

provided y is remains much smaller than x. • Can this be used for isotope separation?

2

1

2

8

mV

qBxy

Magnetic force on a current-carrying conductor

BlF I

lBAnqvBqvnAlF dd The force on all of the moving charges:

vd

A

l

but dnqvJ IlBF

In general

BlF Idd differential form

An electromagnetic rail gun• A conducting bar with mass m and length L slides over a horizontal

rails that are connected to a voltage source. The voltage source maintains a constant current I in the rails and bar; and a constant, uniform, vertical magnetic field B fills the region between the rails. – Find the magnitude and direction of the net force on the conducting bar.

Ignore friction, air resistance and electrical resistance.– If the bar has a mass m, find the distance d that the bar must move

along the rails from rest to attain speed v.– It has been suggested that rail guns based on this principle could

accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth. Let B=0.5T, I=2000A, m=25kg and L=0.5m.

I B

Force and torque on a current loop


Hall effect


The magnetic dipole moment (magnetic moment)

IA

TorqueBμτ

Where is the direction of the solenoid’s tendency of rotation?

BI


Sources of magnetic field

20

sin

4 r

vqB

20 ˆ

4 r

q rvB

• The magnetic field produced by a moving charge is proportional to the

– charge

– velocity of the charge

– inverse of the square of the distance

r̂

vqfield point

rsource point

170 104 TmA (permeability constant)

Magnetic field of a current element

r̂

2

0 ˆ

4 r

Id rlB

r

20

sin

4 r

AdlvqndB d

20 sin

4 r

IdldB

nqAdldQ

I

20 ˆ

4 r

Idd

rlB

Biot-Savart law

Bd

field point

source pointld

• For an infinitely long straight wire, the magnetic field at a distance x from the wire is given by

x

I

2

0B

• Ampere’s lawId 0 lB

Force between parallel conductors

r

I

2

0B

x x

Br

lIIlBIF

2

'' 0 I

I’

r

II

l

F

2

'0One ampere is that unvarying current that, if present in each of tow parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2x10-7 Nm-1.

r

Magnetic field of a circular current loop

20 ˆ

4 r

Idd

rlB

I

dl

x

y

z

dBz

dBy

r̂

Let a be the radius of the ring.

222

0

4 ax

dlIdB

sin

4 220

ax

dlIdBy

cos

4 220

ax

dlIdBz

dl

ax

a

ax

IBy

2

122

220 1

4

23

22

20

2 ax

IaBy

What is B at the center of the ring?

If there are N rings, what is B at the center of the rings?

.B

dl Irr

IdlBBdld 0

0 22

lB

The magnetic field at a distance r from a conductor has a magnitude

r

IB

2

0

.B

dl

IdlBdlBd 01 lB

Id 0 lB

.

1

2

3

4

a

d

d

c

c

b

b

addddd lBlBlBlBlB

ab

c

d

a

d

d

c

c

b

b

adlBdldlBdld 13 00lB

a

d

d

c

c

b

b

adlBdldlBdld 13 00lB

r1

r2

022

22 2

2

01

1

0

rr

Ir

r

Id

lB

Take note:

Even though there is a magnetic field everywhere along the integration path, the line integral is zero if there is no current passing through the area bounded by the path.

Ampere’s law

enclosedId 0 lBx

.

.x

x

. x path of integration

Curl your fingers of your right hand around the integration path so that they curl in the direction of integration. Then your right thumb indicates the positive current direction.

Applications of Ampere’s law

kJ ˆ12

2

20

a

r

a

I

A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is J. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relation

0J

for ra

for ra

where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and Io is a constant having units of amperes.

a) Show that Iois the total current passing through the entire cross section the wire.

b) Using Ampere’s law, derive an expression for the magnitude of the magnetic field B in the region ra.

c) Obtain and expression for the current I contained in the circular cross section of radius ra and centered at the cylinder axis.

d) Using Ampere’s law, derive an expression for the magnitude of the magnetic field B in the region ra.


The figure below shows an end view of two long, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite direction. Derive the expression for the magnitude of B at any point on the x-axis.

.

x

a

ax

P


A circular loop has radius R and carries current I2 in a clockwise direction. The center of the loop is a distance D above a long, straight wire. What are the magnitude and direction of the current I1 in the wire if the magnetic field at the center of the loop is zero?

I2R

D

I1

Study the examples in the book!

Field inside a long cylindrical conductor

I

Amperian loop

Rr

enclosedId 0 lB

2

2

02R

rIrB

B

20

2 R

rIB

r

B

R

Field of a solenoid


nLIBL 0enclosedId 0 lB

Let n be the number of turns.

nIB 0

Find the magnetic field of a toroidal solenoid.

Magnetic materials

e

I

L

IA

r

ev

T

eI

2

22

2 evrr

r

ev

But vrL mmvrL

Lm

e

2

The dipole moment’s component in a particular direction is an integral multiple of

2

h

Js10626.6 34hWhen we speak of the magnitude of a magnetic moment, we mean the “maximum component in a given direction”. aligned with B means that has its maximum possible component in the direction of B.

2

hL If 224 Am10274.9

4

m

eh

The Bohr magneton

Vi

i

μM

ParamagnetismThe magnetization of the material is proportional to the applied magnetic field in which the material is placed.

Magnetization

I

MμBB 00

Curie’s law

TC

BM Mmagnetization Ttemperature

(K)

CCurie’s constant Bmagnetic field

For a given ferromagnetic material the long range order abruptly disappears at a certain temperature called the Curie temperature.

MaterialCurie temperature

(K)

Fe 1043

Co 1388

Ni 627

Gd 293

Dy 85

CrBr3 37

Au2MnAl 200

Cu2MnAl 630

Cu2MnIn 500

EuO 77

EuS 16.5

MnAs 318

MnBi 670

GdCl3 2.2

Fe2B 1015

MnB 578Data from F. Keffer, Handbuch der Physik, 18, pt. 2, New York: Springer-Verlag, 1966 and P. Heller, Rep. Progr. Phys., 30, (pt II), 731 (1967)

The relative permeability:

0 mK

For common paramagnetic materials, Km varies from 1.00001 to 1.003.

The magnetic susceptibility:

1 mm KThe magnetic susceptibility is the amount by which the relative permeability differs from unity.

Material m

Iron ammonium alum 66

Uranium 40

Platinum 26

Aluminum 2.2

Sodium 0.72

Oxygen gas 0.19

Bismuth -16.6

Mercury -2.9

Silver -2.6

Carbon (diamond) -2.1

Lead -1.8

Sodium chloride -1.4

Copper -1.0Young &Freedman, University Physics 11th ed., p.1089

Diamagnetism

The orbital motion of electrons creates tiny atomic current loops, which produce magnetic fields. When an external magnetic field is applied to a material, these current loops will tend to align in such a way as to oppose the applied field.

Diamagnetism is the residual magnetic behavior when materials are neither paramagnetic nor ferromagnetic.

Ferromagnetism

Strong interactions between magnetic moments cause to line up parallel to each other in regions called magnetic domains.

Hysteresis


A piece of iron has a magnetization M=6.5x104 A/m. Find the average magnetic dipole moment per atom in this piece of iron. Express your answer in Bohr magnetons and in Am2. The density of iron is 7.8x103kg/m3. The atomic mass of iron is 55.845g/mol.

Average magnetic dipole

moment per atom of iron

Electromagnetic induction

• Results from experiments– When there is no current in the

electromagnet, so that B=0, the galvanometer shows no current.

– When the electromagnet is turned on, there is a momentary current through the meter as B increases.

– When B levels off at a steady value, the current drops to zero, no matter how large B is.

galvanometer

N

S

electromagnet


• Results from experiments– With the coil in a horizontal plane,

we squeeze it so as to decrease the cross sectional area of the coil. The meter detects current only during the deformation, not before or after. When we increase the area to return the coil to its original shape, there is current in the opposite direction, but only while the area of the coil is changing.

galvanometer

N

S

electromagnet


• Results from experiments– If we rotate the coil a few degrees

about a horizontal axis, the meter detects a current during the rotation, in the same direction as when we decreased the area. When we rotate the coil back, there is a current in the opposite direction during this rotation.

– If we jerk the coil out of the magnetic field, there is a current during the motion, in the same direction as when we decreased the area.

galvanometer

N

S

electromagnet

Electromagnetic induction• Results from experiments

– If we decrease the number of turns in the coil by unwinding one or more turns, there is a current during the unwinding, in the same direction as when we decreased the area. If we wind more turns onto the coil, there is a current in the opposite direction during the winding.

galvanometer

N

S

electromagnet


• Result from experiments– When the magnet is turned off, there

is a momentary current in the direction opposite the current when it was turned on.

– The faster we carry out any of these changes, the greater is the current.

– If all these experiments are repeated with a coil that has the same shape but different material and different resistance, the current in each case is inversely proportional to the total circuit resistance.

What is common in these results?

galvanometer

N

S

electromagnet

Faraday’s law: The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop.

Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.


dt

d B

AB dB

dt

dN B

For N turns,

Direction of induced emf

• Define the direction of the vector area A.

• From the directions of A and the magnetic field B, determine the sign of the magnetic flux B and its rate of change.

• Determine the sign of the induced emf or current. If the flux is increasing, so dB/dt is positive, then the induced emf or current is negative.

• If the flux is decreasing, dB/dt is negative and the induced emf or current is positive.

AIncreasingB

<0

ADecreasing B

>0

Direction of induced emf

• Determine the direction of the induced emf or current using your right hand. Curl the fingers of your right hand around the A vector, with your right thumb in the direction of A. If the induced emf or current in the circuit is positive, it is in the same direction as your curled fingers. If the induced emf or current is negative, it is in the opposite direction.

A

IncreasingB

>0

ADecreasing B

<0

Faraday’s law


Faraday’s law

Lenz’s law: The direction of any magnetic induction effect is such as to oppose the cause of the effect.







ExamplesTwo coupled circuits, A and B, are situated as shown below. Use Lenz’s law to determine the direction of the induced current in resistor ab when (a) coil B is brought closer to coil A with the switch closed, (b) the resistance of R is decreased while the switch remains closed, and (c) switch S is opened.

R

S a b

An alternator is a device that generates an emf. A rectangular loop is made to rotate with a constant angular velocity about the axis shown below. The magnetic field B is uniform and constant. At time t=0, =0, determine the induced emf.

A B

Consider a motor with a square coil 10cm on a side with 500 turns of wire. If the magnitude of the B field is 0.2T, at what rotation speed will the average back emf of the motor be 112V? The back emf of a motor is the emf induced by changing magnetic flux through its rotating coil.

A conducting disk with radius R lies in the xy-plane and rotates with constant velocity about the z-axis. The disk is in a uniform, constant B field parallel to the z-axis. Find the induced emf between the center and the rim of the disk.

Examples

ExamplesConsider a U-shaped conductor in a uniform B-field. If a metal with length L is put across the arms of the conductor, forming a circuit, and move the rod to the right with constant velocity v, find the magnitude and direction of the resulting emf.

vxx

xx

x x xxx x

xx

xxx

x x x x x

xx

x

xx

xxxx

x

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions. Terminals a and b of winding A may be connected to a battery through a reversing switch. Where is the direction of the induced current in the resistor R if a) the current in winding A is from a to b and is increasing? b) the current is from b to a and decreasing? c) the current is from b to a and increasing? a b Winding A

Winding B

A long wire carries a constant current I. A metal bar with length L is moving at constant velocity v. Calculate the emf induced in the bar? Which point is at higher potential? What is the magnitude of the induced current if the bar is replaced by a rectangular wire loop?

Examples

v v

I I

Motional electromotive force

v

b

ax 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 x x x x x x x x x x x x x x x x x x x x x

I

vBLMotional electromotive force:

L

LBv dd

LBv dThis equation can be used for non-stationary conductors in changing magnetic fields.

B

A conducting disk with radius R lies in the xy-plane and rotates with constant angular velocity about the z-axis. The disk is in a uniform, constant B field parallel to the z-axis. Find the induced emf between the center and the rim of the disk.

Induced electric fields

G

If the area vector A points in the same direction as B set up by the solenoid, then

nIABAB 0

dt

dInA

dt

d B0

I, dI/dt

What force makes the charges move around the loop?

Maxwell’s equations:

0enclosedQ

d AE

0 AB d

dt

did Ec 00 lB

dt

dd B

lE

dt

dd B

lE

Displacement current

enclosedId 0 lB

EEdEdd

ACVq

dt

d

dt

dqi EC

Conduction current

Displacement current:

dt

di ED

dt

did Ec 00 lB

Generalized Ampere’s law:

Electromagnetic properties of superconductors

Kammerlingh Onnes (1911) discovered superconductivity.

The critical temperature for superconductors is the temperature at which the electrical resistivity of a metal drops to zero.

http://scienceworld.wolfram.com/biography/photo-credits.html

Type 1 semiconductors:Mat.

Tc

Be 0

Rh 0

W 0.015

Ir 0.1

Lu 0.1

Hf 0.1

Ru 0.5

Os 0.7

Mo 0.92

Mat. Tc

Al 1.2

Pa 1.4

Th 1.4

Re 1.4

Tl 2.39

In 3.408

Sn 3.722

MaZr

Tc0.546

Cd 0.56

U 0.2

Ti 0.39

Zn 0.85

Ga 1.083

Type 2 semiconductors:

http://www.physnet.uni-hamburg.de/home/vms/reimer/htc/pt3.html

The Meissner Effect

Mixed-State Meissner Effect

“Magnetism and superconductivity are natural enemies”. - Lindenfeld

Macroscopic magnetization depends upon aligning the electron spins parallel to one another, while superconductivity depends upon pairs of electrons with their spins antiparallel.

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/maglev2.html

Movies:

The Eddy currents

The Eddy currents are induced currents due to masses of metal moving in magnetic fields or located in changing magnetic fields.


A “proper” complete circuit is not necessary for currents caused by induced emf’s to flow. Microscopic currents can flow within conductors and they are eddy currents!

Some applications of that make use of eddy currents:

1. Electromagnetic damping: eddy currents flow in such a way as to oppose the motion that causes them, acting like a brake on a moving body.

2. Induction heating: current flows cause heating effects and eddy currents are no different.

Transformers



InductanceInductance is typified by the behavior of a coil of wire in resisting any change of electric current through the coil.

A changing current in a coil induces an emf in that same coil!


The coil is the inductor and the relationship between the current and the emf is described by inductance (self-inductance).

dt

dILemf

When there are 2 or more inductors present, the coupling between the coils is described by their mutual inductance.

A coil is a reactionary device, not liking any change! The induced voltage will cause a current to flow in the secondary coil which tries to maintain the magnetic field which was there.

Mutual inductanceThe induced emf in coil 1 is due to self inductance L.

The induced emf in coil 2 is caused by the change in the current I1:

t

iM

t

BANemf

121

122

tNemf

222

dt

diM

dt

dN 1

212

2

1

2221 i

NM B

Show that M12=M21.

Unit of inductance: 1H=1Wb/A=1J/A2

12122 iMN B Define:B field from coil 1 passing through coil 2

Self-inductance and inductors

An inductor (or a choke) is a devise that opposes any current variations throughout the circuit and is designed to have a particular inductance.

i

NL B (Self inductance)

dt

diLSelf induced emf:

Inductor:

Consider a single isolated coil. When a varying current is present in a circuit, it sets up a changing magnetic field that causes a changing magnetic flux through the same circuit. The resulting emf is called a self-induced emf.

Define:


http://www.rsphilippines.com/

Wire ended toroidal RFI suppression chokes designed for use with phase angle control equipment applications operating at 240V ac.

RF chokes consisting of a ferrite based coil former encapsulated in a polypropylene outer case.

A general purpose RF chokes suitable for power decoupling in logic circuits, IF tuned circuit applications and filters etc.

Offers high resonance frequency; suitable for RF blocking and filtering, interference suppression in small size equipment, decoupling and telecoms and entertainment electronics

Explain what happens to the bulb when the switch is closed.

Variable source of emf

L

a

bWhat is the potential difference between points a and b?

Magnetic field energy

dt

diLiiVP ab

LididUPdt

2

2

1LiU

Does energy flow into a resistor whenever a steady current passes through it?Does energy flow into a resistor whenever a varying current passes through it?Does energy flow into an ideal zero-resistance inductor when a steady current through it?Does energy flow into an ideal zero-resistance inductor when an increasing current passes through it?

Show that the self-inductance of an ideal toroidal solenoid of mean radius r and cross-sectional area A is:

r

ANL

2

20

The RL circuit

Show that the energy density of an ideal toroidal solenoid is

222

022

1

2 r

iN

rA

Uu

Show that for an ideal toroidal solenoid

0

2

2B

u

The magnetic field H is defined as

0B

H

(magnetic energy density in a vacuum)

(magnetic energy density in a material)2

2Bu

Again, an inductor in a circuit makes it difficult for rapid changes in current to occur!

In steady state condition, what is the potential difference between the ends of the inductor?

http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html

dt

diLiRV

Suppose that the switch is initially open, but is suddenly closed at t=0.

0dt

diLiRV

0dt

diLiRV

What is the rate of change of the current at t=0?

What is the current at the steady state condition?

dtL

R

RV

i

di

tidt

L

R

RV

i

di00

''

'

tL

R

eR

Vi 1

The time constant for an RL circuit is L/R.

http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html

Current decay in an RL circuit

0dt

diLiR

tL

R

eIi

0

The current in a coil can not increase (or decrease) much faster than L/R.

Note that Vs/R=I0

Suppose you want to send a square wave down a wire. How does the output signal look like?

Vs/R=I0

http://www.sweethaven.com/sweethaven/ModElec/acee/lessonMain.asp?iNum=0402

In terms of energy considerations,

02 dt

diLiRiVi

02 dt

diLiRi

When an RL circuit is decaying, what is the expression of the energy stored in the inductor as a function of time?

The LC circuit

Consider a charged capacitor that is connected to an inductor. Assume no resistance and no energy losses to radiation. What happens to the current in the circuit?

What is the current in the circuit?What is the stored potential energy in the capacitor?What is the stored potential energy in the inductor?

http://www.greenandwhite.net/~chbut/LC_Oscillator/LC_Oscillator.swf

0C

q

dt

diL

Kirchoff’s voltage law:

01

2

2

qLCdt

qd

A solution to the 2nd order differential equation is:

tQq cos

The current is

tQi sin

What is the physical significance of the proposed solution?Are there any possible solutions? What are they?Can you consider an LC circuit as a conservative system?

In terms of energy considerations,

C

Q

C

qLi

222

1 222

221qQ

Lci

The RLC circuit

Kirchoff’s voltage law:

0C

q

dt

diLiR

01

2

2

qLCdt

dq

L

R

dt

qd

0'1

0

tidt

Cdt

diLiR

Similarities between SHM and LC circuit

2

2

1mvKE 2

2

1LiME

2

2

1kxPE

C

qEE

2

2

1

222

2

1

2

1

2

1kAkxmv

C

Q

C

qLi

222

2

1

2

1

2

1

22 xAm

k

dt

dxv 221

qQLcdt

dqi

m

k

LC

1

tAx cos tQq cos

012 LC

mL

Rm

Auxiliary equation:

LCL

R

L

Rm

14

2

1

2

2

LCL

R

L

Rm

14

2

1

2

2

1

LCL

R

L

Rm

14

2

1

2

2

2

LCL

Rtt

L

R

LCL

Rtt

L

R

BeAeq1

422

14

22

22

General solution:

When R2<<4/LC, (underdamped case),

221

422

14

22Re L

R

LC

tit

L

R

L

R

LC

tit

L

R

BeAeq

22

14

22

14

22Re L

R

LC

tit

L

RL

R

LC

tit

L

R

eBeeAeq

Euler’s equation: sincos iei

221

42 1

42

sin1

42

cos

2

L

R

LC

ti

L

R

LC

te L

R

LC

ti

2

2 14

2cos

L

R

LC

tAeq

tL

R

2

22

4

1cos

L

R

LCtAeq

tL

R

2

2

4

1

L

R

LC

Overdamped:

LCL

Rtt

L

R

LCL

Rtt

L

R

BeAeq1

422

14

22

22

01

42

LCL

R

Critical damping: 01

42

LCL

R

Underdamped: 01

42

LCL

R

Overdamped

Critically damped

Underdamped

http://en.wikipedia.org/wiki/Image:RLC-serial-Over_Damping.PNG

http://en.wikipedia.org/wiki/Image:RLC-serial-Critical_Damping.PNG

http://en.wikipedia.org/wiki/Image:RLC-serial-Under_Damping.PNG

Alternating current~Symbol:

Phasors are rotating vectors.

tIi cos

I

t

How do we measure sinusoidally varying current?

http://www.allaboutcircuits.com/vol_3/chpt_3/4.html

max

2II rav

Rectified average value of a sinusoidal current:

Root-mean-square current:-Square the instantaneous current-Take the average of the sum of the squares-Take the square root

tIi cos

tIi 222 cosBut

tt 2cos12

1cos2

tIi 2cos

2

122

2

II rms

02cos t

Note that I is the maximum current!

2

VVrms

The normal voltage source from local outlets is 220VAC. Is this Vrms?

What are the maximum and minimum voltages that a TV can have if it is rated at 110VAC? 110VDC?

Resistance and reactance

Resistor in an AC circuit:

Phasor diagram:

The current and voltage are in phase!

Inductor in an AC circuit:tI sin


Phasor diagram:

The voltage leads the current by 90o in phase!

dt

diL

tIdt

dLV sin

tLIV cos

tI sin

otLIV 90sin

Note: The phase of the voltage is defined relative to the current!

For a pure resistor, the phase is 0. For a pure inductor, the phase is 90o.

The inductive reactance is defined as

LX L

LIXV The voltage difference between the inductor:

Capacitor in an AC circuit:


tIdt

dqi sin

tI

q

cos

tI sin

tC

I

C

qV

cos

otC

IV 90sin

Phasor diagram:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlThe voltage lags the current by 90o in phase!

The capacitive reactance is defined as

CXC

1

The voltage difference between the capacitor:

CIXV

R

LX

CX

Mnemonic for the phase relations of current and voltage: ELI the ICE man!

CC IXV

LL IXV RR IXV

CC IXV

LL IXV RR IXV

CL VV

RR IXV

CL VV

IZVR

22CLR VVVV

22CL XXRIV

Define: 22CL XXRZ (Impedance)

The impedance is the ratio of the voltage amplitude across the circuit to the current amplitude in the circuit.

For an RLC circuit:

2

2 1

C

LRZ

The angle is the phase angle of the source with respect to the current.

RC

L

V

VV

R

CL

1

tan

At resonance, the phase becomes 0! Thus,

LC

1

RC

L

1

tan0


Power in AC circuits

For any sinusoidally varying quantity, the rms value is always 0.707 times the amplitude:

ZIV

22

ZIV rmsrms

The instantaneous power delivered to a circuit element is

vip

tItVp coscos

ttVItVIp sincossincoscos 2

cos2

1VIPave

cosrmsrmsave IVP

The power factor: cos

A low power factor (large angle of lag or lead) means that for a given potential difference, a large current is needed to supply a given amount of power.

-high I2R losses in transmission

-To correct: connect a capacitor in parallel with the load. WHY?

Transformers:




Derive and expression for Vout/Vin as a function of the angular frequency of the source.

R L

C

~

Vout

In an LRC series circuit, the magnitude of the phase angle is 54o, with the source voltage lagging the current. The reactance of the capacitor is 350and the resistor resistance is 180. The average power delivered by the source is 140W. Find the reactance of the inductor, the rms current and the rms voltage.

Derive and expression for Vout/Vin as a function of the angular frequency of the source.

R L

C~ Vout

In the circuit shown below, switch S is closed at time t=0. Find the reading of each meter just after S is closed. What does each meter read long after S is closed?

40

5

20mH

1015

A2 A3 A4A1

10mH

S

25V

Electromagnetic waves


Speed of light ≡ 299,792,458 m/s

Maxwell’s equations:

0enclosedQ

d AE

0 AB d

dt

did Ec 00 lB

dt

dd B

lE