Biot and Savart conducted experiments on the force exerted by
an electric current on a nearby magnet They arrived at a
mathematical expression that gives the magnetic field at some point
in space due to a current The magnetic field is at some point P The
length element is The wire is carrying a steady current of I
Slide 6
Experiments show that the magnetic field at a point near a long
straight wire is This relationship is valid as long as r, the
perpendicular distance to the wire, is much less than the distance
to the ends of the wire Calculation of B near a wire Where o is
called the permeability of free space. Permeability is a measure of
the effect of a material on the magnetic field by the
material.
Slide 7
dB the magnetic field produced by a small section of wire ds a
vector the length of the small section of wire in the direction of
the current r the positional vector from the section of wire to
where the magnetic field is measured I the current in the wire
angle between ds & r
Slide 8
Biot-Savart Law Observations The vector is perpendicular to
both and to the unit vector directed from toward P The magnitude of
is inversely proportional to r 2, where r is the distance from to P
The magnitude of is proportional to the current and to the
magnitude ds of the length element The magnitude of is proportional
to sin where is the angle between the vectors and
Slide 9
The observations are summarized in the mathematical equation
called the Biot-Savart law: The magnetic field described by the law
is the field due to the current-carrying conductor Dont confuse
this field with a field external to the conductor Permeability of
Free Space The constant o is called the permeability of free space
o = 4 x 10 -7 T. m / A
Slide 10
Total Magnetic Field is the field created by the current in the
length segment ds To find the total field, sum up the contributions
from all the current elements I The integral is over the entire
current distribution
Slide 11
Biot-Savart Law Final Notes The law is also valid for a current
consisting of charges flowing through space represents the length
of a small segment of space in which the charges flow For example,
this could apply to the electron beam in a TV set
Slide 12
Compared to Distance The magnitude of the magnetic field varies
as the inverse square of the distance from the source The electric
field due to a point charge also varies as the inverse square of
the distance from the charge Direction The electric field created
by a point charge is radial in direction The magnetic field created
by a current element is perpendicular to both the length element
and the unit vector
Slide 13
Compared to Source An electric field is established by an
isolated electric charge The current element that produces a
magnetic field must be part of an extended current distribution
Therefore you must integrate over the entire current
distribution
Slide 14
for a Long, Straight Conductor, Special Case If the conductor
is an infinitely long, straight wire, = /2 and = - /2 The field
becomes r
Slide 15
for a Long, Straight Conductor, Direction The magnetic field
lines are circles concentric with the wire The field lines lie in
planes perpendicular to to wire The magnitude of the field is
constant on any circle of radius r The right-hand rule for
determining the direction of the field is shown
Slide 16
Magnetic Force Between Two Parallel Conductors, final The
result is often expressed as the magnetic force between the two
wires, F B This can also be given as the force per unit
length:
Slide 17
Definition of the Ampere The force between two parallel wires
can be used to define the ampere When the magnitude of the force
per unit length between two long, parallel wires that carry
identical currents and are separated by 1 m is 2 x 10 -7 N/m, the
current in each wire is defined to be 1 A The SI unit of charge,
the coulomb, is defined in terms of the ampere When a conductor
carries a steady current of 1 A, the quantity of charge that flows
through a cross section of the conductor in 1 s is 1 C Definition
of the Coulomb
Slide 18
Magnetic Field of a Wire A compass can be used to detect the
magnetic field When there is no current in the wire, there is no
field due to the current The compass needles all point toward the
Earths north pole Due to the Earths magnetic field Here the wire
carries a strong current The compass needles deflect in a direction
tangent to the circle This shows the direction of the magnetic
field produced by the wire Use the active figure to vary the
current
Slide 19
Amperes Law The product of can be evaluated for small length
elements on the circular path defined by the compass needles for
the long straight wire Amperes law states that the line integral of
around any closed path equals o I where I is the total steady
current passing through any surface bounded by the closed
path:
Slide 20
Amperes law describes the creation of magnetic fields by all
continuous current configurations Most useful for this course if
the current configuration has a high degree of symmetry Put the
thumb of your right hand in the direction of the current through
the amperian loop and your fingers curl in the direction you should
integrate around the loop
Slide 21
Magnetic Field of a Wire, 3 The circular magnetic field around
the wire is shown by the iron filings
Slide 22
dB perpendicular to ds dB perpendicular to r |dB| inversely
proportional to |r| 2 |dB| proportional to current I |dB|
proportional to |ds| |dB| proportional to sin
Slide 23
We already used the Biot-Savart Law to show that, for this
case,. Lets show it again, using Amperes Law: First, we are free to
draw an Amperian loop of any shape, but since we know that the
magnetic field goes in circles around a wire, lets choose a
circular loop (of radius r ). Then B and ds are parallel, and B is
constant on the loop, so Amperes Law
Slide 24
Magnetic Field Produced by Long Straight Current- Carrying Wire
Any time an electric current flows in a wire, it produces a
magnetic field.In a long, straight wire, the direction of the
magnetic field can be determined using the right hand rule. Point
the thumb of your right hand in the direction of the current in the
wire and curl your fingers. Your fingers now point in the direction
of the magnetic field caused by the current in the wire. The
strength of the field is directly proportional to the current in
the wire and inversely proportional to the distance from the wire.
The equation is: B = 0 I/2r where 0 is a proportionality constant
called the permeability of free space. Its value is 4 x 10 -7
Tm/A.
Slide 25
Draw some magnetic field lines (loops in this case) along the
wire. Using xs and dots to represent vectors out of and into the
page, show the magnetic field for the same wire. Note B diminishes
with distance from the wire. B out of page B into page Direction of
the Field of a Long Straight Wire Right Hand Rule Grasp the wire in
your right hand Point your thumb in the direction of the current
Your fingers will curl in the direction of the field
Slide 26
The force on an electric current in a wire can be calculated
much the same way since a current consists of charges moving
through a wire. F = qvBsin F = (q/t)(vt)sin Since q/t = I and vt =
L(length of the wire), the equation becomes: F = ILBsin
Slide 27
Magnetic forces on moving charges When a charge moves through a
B-field, it feels a force. It has to be moving It has to be moving
across the B-field Direction of force is determined by another
right-hand rule
Slide 28
A current-carrying wire produces a magnetic field The compass
needle deflects in directions tangent to the circle The compass
needle points in the direction of the magnetic field produced by
the current
Slide 29
Magnitude of the Field of a Long Straight Wire Magnetic field I
e For a long straight wire Amperes law The magnitude of the field
at a distance r from a wire carrying a current of I is given by the
formula above, where o = 4 x 10 -7 T m / A o is called the
permeability of free space
Slide 30
Ampres Law Andr-Marie Ampre found a procedure for deriving the
relationship between the current in a arbitrarily shaped wire and
the magnetic field produced by the wire Ampres Circuital Law B || =
o I Integral (=sum) over the closed path
Slide 31
Choose an arbitrary closed path around the current Sum all the
products of B || d s around the closed path; B || is the component
of B parallel to ds. Wire surface Amperian Loop Direction of
Integration
Slide 32
Rules for finding direction of B field from a current flowing
in a wire
Slide 33
Permanent magnets arent the only things that produce magnetic
fields. Moving charges themselves produce magnetic fields. We just
saw that a current carrying wire feels a force when inside an
external magnetic field. It also produces its own mag-netic field.
A long straight wire produces circular field lines centered on the
wire. To find the direction of the field, we use another right hand
rule: point your thumb in the direction of the current; the way
your fingers of your right hand wrap is the direction of the
magnetic field. B diminishes with distance from the wire. The pics
at the right show cross sections of a current carrying wire. I B I
out of page, B counterclockwise I into page, B clockwise
Slide 34
The positive charge is moving straight out of the page. What is
the direction of the magnetic field at the position of the dot? 1.
Right 2. Left 3. Up 4. Down
Slide 35
The magnetic field of a straight, current-carrying wire is 1.
parallel to the wire. 2. inside the wire. 3. perpendicular to the
wire. 4. around the wire. 5. zero.
Slide 36
Example: Use Amperes Law to find B near a very long, straight
wire. B is independent of position along the wire and only depends
on the distance from the wire (symmetry). r i i By symmetry
SupposeI = 10 A R = 10 cm
Slide 37
blue - into figure ( - ) red - out of figure (+) Amperes Law -
a line integral
Slide 38
Two Parallel Wires Two long parallel wires suspended next to
each other will either attract or repel depending on the direction
of the current in each wire. B-field produced by each wire
interacts with current in the other wire Produces magnetic
deflecting force on other wire. Wires exert equal and opposite
forces on each other.
Slide 39
1.Parallel wires with current flowing in the same direction,
attract each other. 2.Parallel wires with current flowing in the
opposite direction, repel each other. Note that the force exerted
on I 2 by I 1 is equal but opposite to the force exerted on I 1 by
I 2.
Slide 40
Currents in Same Direction Wires Attract Currents in Opposite
Directions Wires Repel where a is the distance between the wires.
B-field produced by wire 2 at the position of wire 1 is given
by
Slide 41
B-field produced by wire 2 at the position of wire 1 is given
by where a is the distance between the wires.
Slide 42
41 Example 1 Determine the magnetic field at point A. A
Slide 43
42 Example 2 Determine the magnetic field at point A. A
Slide 44
43 Example 3 Two parallel conductors carry current in opposite
directions. One conductor carries a current of 10.0 A. Point A is
at the midpoint between the wires, and point C is a distance d/2 to
the right of the 10.0-A current. If d = 18.0 cm and I is adjusted
so that the magnetic field at C is zero, find (a) the value of the
current I and (b) the value of the magnetic field at A. d/2
Slide 45
44 Example 3 d/2 Two parallel conductors carry current in
opposite directions. One conductor carries a current of 10.0 A.
Point A is at the midpoint between the wires, and point C is a
distance d/2 to the right of the 10.0-A current. If d = 18.0 cm and
I is adjusted so that the magnetic field at C is zero, find (a) the
value of the current I and (b) the value of the magnetic field at
A.
Slide 46
45 Field due to a long Straight Wire Need to calculate the
magnetic field at a distance r from the center of a wire carrying a
steady current I The current is uniformly distributed through the
cross section of the wire
Slide 47
46 Field due to a long Straight Wire The magnitude of magnetic
field depends only on distance r from the center of a wire. Outside
of the wire, r > R
Slide 48
47 Field due to a long Straight Wire The magnitude of magnetic
field depends only on distance r from the center of a wire. Inside
the wire, we need I , the current inside the amperian circle
Slide 49
48 Field due to a long Straight Wire The field is proportional
to r inside the wire The field varies as 1/r outside the wire Both
equations are equal at r = R
Slide 50
Magnetic forces on moving charges F B = qvB F B = B IL If a
current I in a wire of length L passes through the field, the force
is:
Slide 51
Magnetic force on current Wire can be levitated by magnetic
force when a current passes through it F = B IL
Slide 52
Force on a current-carrying conductor The direction of magnetic
force always perpendicular to the direction of the magnetic field
and the direction of current passing through the conductor.
Slide 53
Magnetic Force Between Two Parallel Conductors F 1 =B 2 I 1 B 2
= 0 I 2 /(2 d) F 1 = 0 I 1 I 2 /(2 d) The field B 2 at wire 1 due
to the current I 2 in wire 2 causes the force F 1 on wire 1.
Slide 54
Two Parallel Wires Since the wires are parallel, the force
exerted by wire 2 on wire 1 is given by However, it is often more
appropriate to determine the force per unit length on the wire. a =
separation l = length of wire
Slide 55
Force Between Two Conductors, cont Parallel conductors carrying
currents in the same direction attract each other Parallel
conductors carrying currents in the opposite directions repel each
other
Slide 56
Defining Ampere and Coulomb The force between parallel
conductors can be used to define the Ampere (A) If two long,
parallel wires 1 m apart carry the same current, and the magnitude
of the magnetic force per unit length is 2 x 10 -7 N/m, then the
current is defined to be 1 A The SI unit of charge, the Coulomb
(C), can be defined in terms of the Ampere (A) If a conductor
carries a steady current of 1 A, then the quantity of charge that
flows through any cross section in 1 second is 1 C
Slide 57
If I 1 = 2 A and I 2 = 6 A in the figure below, which of the
following is true: (a) F 1 = 3 F 2, (b) F 1 = F 2, or (c) F 1 = F 2
/3? (b). The two forces are an action-reaction pair. They act on
different wires, and have equal magnitudes but opposite
directions.
Slide 58
Magnetic Field in Wires Moving current carries a magnetic field
Wires in your house generate a magnetic field Current in same
direction Force is attractive
Slide 59
Current in opposite directions Force is repulsive I1I1 I2I2 F
F
Slide 60
Example Two wires in a 2.0 m long cord are 3.0 mm apart. If
they carry a dc current of 8.0 A, calculate the force between the
wires. F = (4 X 10 -7 T m/A)(8.0A)(8.0A)(2.0m) (2 (3.0 X 10 -3 m) F
= 8.5 X 10 -3 N
Slide 61
Example The top wire carries a current of 80 A. How much
current must the lower wire carry in order to leviate if it is 20
cm below the first and has a mass of 0.12 g/m? Solve for I 2 I 2 =
15 A
Slide 62
Solenoids consist of a tightly wrapped coil of wire, sometimes
around an iron core. The multiple loops and the iron magnify the
effect of the single loop electromagnet. A solenoid behaves as just
like a simple bar magnet but only when current is flowing. The
greater the current and the more turns per unit length, the greater
the field inside. An ideal solenoid has a perfectly uniform
magnetic field inside and zero field outside. Solenoids are one of
the most common electromagnets Magnetic Field of a Solenoid
Slide 63
The cross section of a solenoid is shown. At point P inside the
solenoid, the B field is a vector sum of the fields due to each
section of wire. Note from the table that each section of wire
produce a field vector with a component to the right, resulting in
a strong field inside. In the ideal case the magnetic field would
be uniform inside and zero outside. I out of the page x x x x x x x
x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 I into the page P B B =
0
Slide 64
Length L A cross-sectional view of a tightly wound solenoid If
the solenoid is long compared to its radius, we assume the field
inside is uniform and outside is zero Apply Ampres Law to the red
dashed rectangle
Slide 65
64 Magnetic Field of a Solenoid An ideal solenoid is approached
when: the turns are closely spaced the length is much greater than
the radius of the turns Consider a rectangle with side parallel to
the interior field and side w perpendicular to the field The side
of length inside the solenoid contributes to the field This is path
1 in the diagram
Slide 66
65 Magnetic Field of a Solenoid Applying Amperes Law gives The
total current through the rectangular path equals the current
through each turn multiplied by the number of turns Solving Amperes
law for the magnetic field is n = N / is the number of turns per
unit length This is valid only at points near the center of a very
long solenoid
Slide 67
the magnetic field inside a solenoid is and So we have or
where. The magnetic field inside a solenoid is constant. If the
length of the solenoid is much greater than its diameter the
magnetic field outside the solenoid is nearly zero. The magnetic
field is strongest at the centre of the solenoid and becomes weaker
outside. numbers of loops / length of coil
Slide 68
Magnetic field at the center of a current- carrying solenoid (N
is the number of turns): B= 0 NI/L, where L is the length of the
solenoid and with n=N/L (number of turns per unit lengths) we get:
0 nI ( Ampres law) The longer the solenoid, the more uniform is the
magnetic field across the cross-sectional area with in the coil.
The exterior field is nonuniform, much weaker, and in the opposite
direction to the field inside the solenoid
Slide 69
Magnetic Field in a Solenoid, final The field lines of the
solenoid resemble those of a bar magnet
Slide 70
Solenoids and Bar Magnets A solenoid produces a magnetic field
just like a simple bar magnet. Since it consists of many current
loops, the resemblance to a bar magnets field is much better than
that of a single current loop.
Slide 71
70 Magnetic Field of a Solenoid The field lines in the interior
are approximately parallel to each other uniformly distributed
close together This indicates the field is strong and almost
uniform
Slide 72
71 Magnetic Field of a Solenoid The field distribution is
similar to that of a bar magnet As the length of the solenoid
increases the interior field becomes more uniform the exterior
field becomes weaker
Slide 73
When two current carrying wires are near each other, each one
exerts a force on the other. This can be a repulsive force or an
attractive force depending on the current direction. Right hand
rule # 1 when applied to wire 2 tells us that the force acting on
wire 2 is to the right, perpendicular to both the current direction
and the magnetic field direction. The magnitude of the force
between two current carrying wires can be calculated using the
equations: F 2 = I 2 LBsin Consider wire 1 to be fixed. In diagram
(a) wire 2 experiences a force due to the magnetic field generated
by wire 1 and the current flowing through wire 2. Right hand rule #
2, when applied to wire 1, tells us that the magnetic field at wire
2 acts upward.
Slide 74
Magnetic Flux Magnetic flux,, is a measure of the amount of
magnetic field lines going through an area. If the field is
uniform, flux is given by: B = B A = B A cos The area vector in the
dot product is a vector that points perpendicular to the surface
and has a magnitude equal to the area of the surface. Imagine youre
trying to orient a window so as to allow the maximum amount of
light to pass through it. To do this you would, of course, align A
with the light rays. With = 0, cos = 1, and the number of light
rays passing through the window (the flux) is a max. Note: with the
window oriented parallel to the rays, = 90 and B = 0 (no light
enters the window). The SI unit for magnetic flux is the
tesla-square meter: T m 2. This is also know as a weber ( Wb
).
Slide 75
Magnetic Flux The first step to understanding the complex
nature of electromagnetic induction is to understand the idea of
magnetic flux. Flux is a general term associated with a FIELD that
is bound by a certain AREA. So MAGNETIC FLUX is any AREA that has a
MAGNETIC FIELD passing through it. A B We generally define an AREA
vector as one that is perpendicular to the surface of the material.
Therefore, you can see in the figure that the AREA vector and the
Magnetic Field vector are PARALLEL. This then produces a DOT
PRODUCT between the 2 variables that then define flux.
Slide 76
Magnetic Flux How could we CHANGE the flux over a period of
time? We could move the magnet away or towards (or the wire). We
could increase or decrease the area. We could ROTATE the wire along
an axis that is PERPENDICULAR to the field thus changing the angle
between the area and magnetic field vectors.
Slide 77
Magnetic flux is used in the calculation of the induced
emf.
Slide 78
Slide 79
Slide 80
Magnetic flux lines are continuous and closed. Direction is
that of the B vector at any point. Flux lines are NOT in direction
of force but . When area A is perpendicular to flux: The unit of
flux density is the Weber per square meter.
Slide 81
Calculating Flux Density When Area is Not Perpendicular The
flux penetrating the area A when the normal vector n makes an angle
of with the B-field is: The angle is the complement of the angle a
that the plane of the area makes with the B field. (Cos = Sin
Slide 82
Example
Slide 83
Slide 84
You can change the magnetic flux through a given surface by
A.changing the magnetic field. B.changing the surface area over
which the magnetic field is distributed. C.changing the angle
between the magnetic field and surface in question. D.any
combination of a through c. E.none of these strategies.
Example
Slide 85
The figure shows a flat surface with area 3cm 2 in a uniform
B-field. If the magnetic flux through this area is 0.9mWb, Find the
magnitude of B-field. Example
Slide 86
Suppose you double the magnetic field in a given region and
quadruple the area through which this magnetic field exists. The
effect on the flux through this area would be to A.leave it
unchanged. B.double it. C.quadruple it. D.increase it by a factor
of six. E.increase it by a factor of eight. Example
Slide 87
What is the flux density in a rectangular core that is 8 mm by
10 mm if the flux is 4 mWb? Example
Slide 88
Calculate the B field due to a current using Biot-Savart Law
Permiability constant: B due to long straight wire: circular arc:
complete loop: Force between two parallel currents Another way to
calculate B is using Amperes Law (integrate B around closed
Amperian loops): = permeability constant exactly Summary