Slide 2 Sources of the Magnetic Field & Magnetic Induction
Fall 2006 Slide 3 Remember the wire? Slide 4 Try to remember Slide
5 The Coulombs Law of Magnetism A Vector Equation duck Slide 6 For
the Magnetic Field, current elements create the field. This is the
Law of Biot-Savart Slide 7 Magnetic Field of a Straight Wire We
intimated via magnets that the Magnetic field associated with a
straight wire seemed to vary with 1/d. We can now PROVE this! Slide
8 From the Past Using Magnets Slide 9 Right-hand rule: Grasp the
element in your right hand with your extended thumb pointing in the
direction of the current. Your fingers will then naturally curl
around in the direction of the magnetic field lines due to that
element. Slide 10 Lets Calculate the FIELD Note: For ALL current
elements in the wire: d s X r is into the page Slide 11 The Details
Slide 12 Moving right along 1/d Slide 13 A bit more complicated A
finite wire Slide 14 P1P1 r ds Slide 15 More P 1 Slide 16 P2 Slide
17 APPLICATION: Find the magnetic field B at point P Slide 18
Center of a Circular Arc of a Wire carrying current Slide 19 More
arc ds Slide 20 The overall field from a circular current loop
Sorta looks like a magnet! Slide 21 Iron Slide 22 Howya Do Dat?? No
Field at C Slide 23 Force Between Two Current Carrying Straight
Parallel Conductors Wire a creates a field at wire b Current in
wire b sees a force because it is moving in the magnetic field of
a. Slide 24 The Calculation Slide 25 Definition of the Ampere The
force acting between currents in parallel wires is the basis for
the definition of the ampere, which is one of the seven SI base
units. The definition, adopted in 1946, is this: The ampere is that
constant current which, if maintained in two straight, parallel
conductors of infinite length, of negligible circular cross
section, and placed 1 m apart in vacuum, would produce on each of
these conductors a force of magnitude 2 x 10-7 newton per meter of
length. Slide 26 Amperes Law The return of Gauss Slide 27 Remember
GAUSSS LAW?? Surface Integral Slide 28 Gausss Law Made calculations
easier than integration over a charge distribution. Applied to
situations of HIGH SYMMETRY. Gaussian SURFACE had to be defined
which was consistent with the geometry. AMPERES Law is the Gauss
Law of Magnetism! (Sorry) Slide 29 The next few slides have been
lifted from Seb Oliver on the internet Whoever he is! Slide 30
Biot-Savart The Coulombs Law of Magnetism Slide 31 Invisible
Summary Biot-Savart Law (Field produced by wires) Centre of a wire
loop radius R Centre of a tight Wire Coil with N turns Distance a
from long straight wire Force between two wires Definition of
Ampere Slide 32 Magnetic Field from a long wire I B r ds Using
Biot-Savart Law Take a short vector on a circle, ds Thus the dot
product of B & the short vector ds is: Slide 33 Sum B. ds
around a circular path I B r ds Sum this around the whole ring
Circumference of circle Slide 34 Consider a different path Field
goes as 1/r Path goes as r. Integral independent of r i Slide 35
SO, AMPERES LAW by SUPERPOSITION: We will do a LINE INTEGRATION
Around a closed path or LOOP. Slide 36 Amperes Law USE THE RIGHT
HAND RULE IN THESE CALCULATIONS Slide 37 The Right Hand Rule..
AGAIN Slide 38 Another Right Hand Rule Slide 39 COMPARE Line
Integral Surface Integral Slide 40 Simple Example Slide 41 Field
Around a Long Straight Wire Slide 42 Field INSIDE a Wire Carrying
UNIFORM Current Slide 43 The Calculation Slide 44 R r B Slide 45
Procedure Apply Amperes law only to highly symmetrical situations.
Superposition works. Two wires can be treated separately and the
results added (VECTORIALLY!) The individual parts of the
calculation can be handled (usually) without the use of vector
calculations because of the symmetry. THIS IS SORT OF LIKE GAUSSs
LAW WITH AN ATTITUDE! Slide 46 The figure below shows a cross
section of an infinite conducting sheet carrying a current per unit
x-length of l; the current emerges perpendicularly out of the page.
(a) Use the BiotSavart law and symmetry to show that for all points
P above the sheet, and all points P below it, the magnetic field B
is parallel to the sheet and directed as shown. (b) Use Ampere's
law to find B at all points P and P. Slide 47 FIRST PART Vertical
Components Cancel Slide 48 Apply Ampere to Circuit Infinite Extent
B B L Slide 49 The Math Infinite Extent B B B ds=0 Slide 50 A
Physical Solenoid Slide 51 Inside the Solenoid For an INFINITE
(long) solenoid the previous problem and SUPERPOSITION suggests
that the field OUTSIDE this solenoid is ZERO! Slide 52 More on Long
Solenoid Field is ZERO! Field is ZERO Field looks UNIFORM Slide 53
The real thing.. Weak Field Stronger - Leakage Fairly Uniform field
Finite Length Slide 54 Another Way Slide 55 Application Creation of
Uniform Magnetic Field Region Minimal field outside except at the
ends! Slide 56 Two Coils Slide 57 Real Helmholtz Coils Used for
experiments. Can be aligned to cancel out the Earths magnetic field
for critical measurements. Slide 58 The Toroid Slightly less dense
than inner portion Slide 59 The Toroid Slide 60 Induction Slide 61
Magnetic Flux For a CLOSED Surface we might expect this to be equal
to some constant times the enclosed poles but there aint no such
thing! Slide 62 Examples S N Slide 63 Consider the poor little
capacitor i i CHARGING OR DISCHARGING . HOW CAN CURRENT FLOW
THROUGH THE GAP?? Slide 64 Through Which Surface Do we measure the
current for Amperes Law? I=0 Slide 65 In the gap DISPLACEMENT
CURRENT Slide 66 Slide 67 From The Demo.. Slide 68 Faradays
Experiments ???? Slide 69 Insert Magnet into Coil Slide 70 Remove
Coil from Field Region Slide 71 Thats Strange .. These two coils
are perpendicular to each other Slide 72 Definition of TOTAL
ELECTRIC FLUX through a surface: Slide 73 Magnetic Flux: THINK OF
MAGNETIC FLUX as the AMOUNT of Magnetism passing through a surface.
Dont quote me on this!!! Slide 74 Consider a Loop Magnetic field
passing through the loop is CHANGING. FLUX is changing. There is an
emf developed around the loop. A current develops (as we saw in
demo) Work has to be done to move a charge completely around the
loop. xxxxxxxxxxxxxxx Slide 75 Faradays Law (Michael Faraday) For a
current to flow around the circuit, there must be an emf. (An emf
is a voltage) The voltage is found to increase as the rate of
change of flux increases. xxxxxxxxxxxxxxx Slide 76 Faradays Law
(Michael Faraday) xxxxxxxxxxxxxxx We will get to the minus sign in
a short time. Slide 77 Faradays Law (The Minus Sign)
xxxxxxxxxxxxxxx Using the right hand rule, we would expect the
direction of the current to be in the direction of the arrow shown.
Slide 78 Faradays Law (More on the Minus Sign) xxxxxxxxxxxxxxx The
minus sign means that the current goes the other way. This current
will produce a magnetic field that would be coming OUT of the page.
The Induced Current therefore creates a magnetic field that OPPOSES
the attempt to INCREASE the magnetic field! This is referred to as
Lenzs Law. Slide 79 How much work? xxxxxxxxxxxxxxx A magnetic field
and an electric field are intimately connected.) emf Slide 80 The
Strange World of Dr. Lentz Slide 81 MAGNETIC FLUX This is an
integral over an OPEN Surface. Magnetic Flux is a Scalar The UNIT
of FLUX is the weber 1 weber = 1 T-m 2 Slide 82 We finally stated
FARADAYs LAW Slide 83 From the equation Lentz Slide 84 Flux Can
Change If B changes If the AREA of the loop changes Changes cause
emf s and currents and consequently there are connections between E
and B fields These are expressed in Maxwells Equations Slide 85
Maxwells Equations (Next Course.. Just a Preview!) Gauss Faraday
Slide 86 Another View Of That damned minus sign again ..SUPPOSE
that B begins to INCREASE its MAGNITUDE INTO THE PAGE The Flux into
the page begins to increase. An emf is induced around a loop A
current will flow That current will create a new magnetic field.
THAT new field will change the magnetic flux. xxxxxxxxxxxxxxx Slide
87 Lenzs Law Induced Magnetic Fields always FIGHT to stop what you
are trying to do! i.e... Murphys Law for Magnets Slide 88 Example
of Nasty Lenz The induced magnetic field opposes the field that
does the inducing! Slide 89 Slide 90 Dont Hurt Yourself! The
current i induced in the loop has the direction such that the
currents magnetic field Bi opposes the change in the magnetic field
B inducing the current. Slide 91 Lets do the Lentz Warp again !
Slide 92 Lenzs Law An induced current has a direction such that the
magnetic field due to the current opposes the change in the
magnetic flux that induces the current. (The result of the negative
sign!) OR The toast will always fall buttered side down! Slide 93
An Example The field in the diagram creates a flux given by B =6t 2
+7t in milliWebers and t is in seconds. (a)What is the emf when t=2
seconds? (b) What is the direction of the current in the resistor
R? Slide 94 This is an easy one Direction? B is out of the screen
and increasing. Current will produce a field INTO the paper (LENZ).
Therefore current goes clockwise and R to left in the resistor.
Slide 95 Figure 31-36 shows two parallel loops of wire having a
common axis. The smaller loop (radius r) is above the larger loop
(radius R) by a distance x >> R. Consequently, the magnetic
field due to the current i in the larger loop is nearly constant
throughout the smaller loop. Suppose that x is increasing at the
constant rate of dx/dt = v. (a) Determine the magnetic flux through
the area bounded by the smaller loop as a function of x. (Hint: See
Eq. 30-29.) In the smaller loop, find (b) the induced emf and (c)
the direction of the induced current. v Slide 96 B is assumed to be
constant through the center of the small loop and caused by the
large one. Slide 97 The calculation of B z Slide 98 More Work In
the small loop: dx/dt=v Slide 99 Which Way is Current in small loop
expected to flow?? Slide 100 What Happens Here? Begin to move
handle as shown. Flux through the loop decreases. Current is
induced which opposed this decrease current tries to re- establish
the B field. Slide 101 moving the bar Slide 102 Moving the Bar
takes work v Slide 103 What about a SOLID loop?? METAL Pull Energy
is LOST BRAKING SYSTEM Slide 104 Back to Circuits for a bit . Slide
105 Definition Current in loop produces a magnetic field in the
coil and consequently a magnetic flux. If we attempt to change the
current, an emf will be induced in the loops which will tend to
oppose the change in current. This this acts like a resistor for
changes in current! Slide 106 Remember Faradays Law Lentz Slide 107
Look at the following circuit: Switch is open NO current flows in
the circuit. All is at peace! Slide 108 Close the circuit After the
circuit has been close for a long time, the current settles down.
Since the current is constant, the flux through the coil is
constant and there is no Emf. Current is simply E/R (Ohms Law)
Slide 109 Close the circuit When switch is first closed, current
begins to flow rapidly. The flux through the inductor changes
rapidly. An emf is created in the coil that opposes the increase in
current. The net potential difference across the resistor is the
battery emf opposed by the emf of the coil. Slide 110 Close the
circuit Slide 111 Moving right along Slide 112 Definition of
Inductance L UNIT of Inductance = 1 henry = 1 T- m 2 /A is the flux
near the center of one of the coils making the inductor Slide 113
Consider a Solenoid n turns per unit length l Slide 114 So. Depends
only on geometry just like C and is independent of current. Slide
115 Inductive Circuit Switch to a. Inductor seems like a short so
current rises quickly. Field increases in L and reverse emf is
generated. Eventually, i maxes out and back emf ceases. Steady
State Current after this. i Slide 116 THE BIG INDUCTION As we begin
to increase the current in the coil The current in the first coil
produces a magnetic field in the second coil Which tries to create
a current which will reduce the field it is experiences And so
resists the increase in current. Slide 117 Back to the real world i
Switch to a Slide 118 Solution Slide 119 Switch position b Slide
120 Max Current Rate of increase = max emf V R =iR ~current Slide
121 Solve the loop equation. Slide 122 IMPORTANT QUESTION Switch
closes. No emf Current flows for a while It flows through R Energy
is conserved (i 2 R) WHERE DOES THE ENERGY COME FROM?? Slide 123
For an answer Return to the Big C We move a charge dq from the (-)
plate to the (+) one. The (-) plate becomes more (-) The (+) plate
becomes more (+). dW=Fd=dq x E x d +q -q E= 0 A/d +dq Slide 124 The
calc The energy is in the FIELD !!! Slide 125 What about POWER??
power to circuit power dissipated by resistor Must be dW L /dt
Slide 126 So Energy stored in the Capacitor Slide 127 WHERE is the
energy?? l Slide 128 Remember the Inductor?? ????????????? Slide
129 So Slide 130 ENERGY IN THE FIELD TOO! Slide 131 IMPORTANT
CONCLUSION A region of space that contains either a magnetic or an
electric field contains electromagnetic energy. The energy density
of either is proportional to the square of the field strength.
Slide 132
