### Magnetic Fields

```The Definition of B:
We can define a magnetic field, B, by firing a charged particle through
the point at which is to be defined, using various directions and speeds
for the particle and determining the force that acts on the particle at
that point. B is then defined to be a vector quantity that is directed
along the zero-force axis.
The magnetic force on the charged particle, FB, is defined to be:
Here q is the charge of the particle, v is its velocity, and B the magnetic
field in the region. The magnitude of this force is then:
Here f is the angle between vectors v and B.
Finding the Magnetic Force on a Particle:
The Definition of B:
The SI unit for B that follows is
newton per coulomb-meter per
second. For convenience, this is
called the tesla (T):
An earlier (non-SI) unit for B is the
gauss (G), and
Magnetic Force on a
Current-Carrying Wire:
Here L is a length vector that has
magnitude L and is directed along the
wire segment in the direction of the
(conventional) current.
If a wire is not straight or the field is not uniform, we can imagine the wire broken up into
small straight segments . The force on the wire as a whole is then the vector sum of all
the forces on the segments that make it up. In the differential limit, we can write
and we can find the resultant force on any given arrangement of currents
by integrating over that arrangement.
Torque on a Current Loop:
To define the orientation of the loop in the magnetic field, we use a normal
vector n that is perpendicular to the plane of the loop.
For side 2 the magnitude of the force acting on this side is
F2=ibB sin(90°-q)=ibB cosq =F4.
F2 and F4 cancel out exactly.
Forces F1 and F3 have the common magnitude iaB. As Fig. 28-19c shows,
these two forces do not share the same line of action; so they produce a net
torque.
is:
For N loops, when A=ab, the area of the loop, the total torque
TOP VIEW
B
F
q
n
q
Fsinq
x
Fsinq
q
F
F=IaB
TOP VIEW
B
F
q
n
x
q
Fsinq
b
Fsinq
q
= 2*(b/2)Fsinq
= IbaBsinq
= IABsinq
F
F=IaB
Torque on a Current Loop
= IABsinq
If multiple (N) Loops
= NIABsinq
If multiple Circular Loops
=
2
NIpr Bsinq
The Magnetic Dipole Moment, m:
Definition:
Here, N is the number of turns in the coil, i is the current through
the coil, and A is the area enclosed by each turn of the coil.
Direction: The direction of m is that of the normal vector to the
plane of the coil.
The definition of torque can be rewritten as:
Just as in the electric case, the magnetic dipole in an
external magnetic field has an energy that depends on
the dipole’s orientation in the field:
A magnetic dipole has its lowest energy (-mB cos 0=-mB)
when its dipole moment m is lined up with the magnetic field.
It has its highest energy (-mB cos 180°=+mB) when m is
directed opposite the field.
ConcepTest 22.1a Magnetic Force I
A positive charge enters a
uniform magnetic field as
shown. What is the direction of
the magnetic force?
1) out of the page
2) into the page
3) downward
4) to the right
5) to the left
Using the right-hand rule, you can
x x x x x x
v
see that the magnetic
x xforce
x x xis x
directed to the left. xRemember
x x x x x
q
that the magnetic force must be
perpendicular to BOTH the B field
and the velocity.
x x x x x x
v
x x x x x x
x xFx xq x x
ConcepTest 22.1b Magnetic Force II
A positive charge enters a
uniform magnetic field as
shown. What is the direction of
the magnetic force?
1) out of the page
2) into the page
3) downward
4) upward
5) to the left
Using the right-hand rule, you can
x x x x x x
see that the magnetic force is
x x x x x x
directed upward. Rememberv that
x xq x x x x
the magnetic force must be
perpendicular to BOTH the B field
and the velocity.
x x x
F
x x x
q
x x x
x x x
x vx x
x x x
ConcepTest 22.1c Magnetic Force III
A positive charge enters a
uniform magnetic field as
shown. What is the direction of
the magnetic force?
1) out of the page
2) into the page
3) zero
4) to the right
5) to the left
Using the right-hand rule,
you can

see that the magnetic force
is

v
directed into the page. Remember
that the magnetic force
must be
q
perpendicular to BOTH
the B field
and the velocity.


v

 q
F

ConcepTest 22.2 Atomic Beams
A beam of atoms enters
x x x x x x x x x 1x x x
a magnetic field region.
x x x x x x x x x x x x
What path will the
x x x x x x x x x x x x
atoms follow?
x x x x x x x x x x x x
x x x x x x x x x x x x
4
2
3
x x x x x x x x x x x x
Atoms are neutral objects whose net charge is zero.
Thus they do not experience a magnetic force.
ConcepTest 22.3 Magnetic Field
A proton beam enters into a
magnetic field region as shown
below. What is the direction of
the magnetic field B?
1) + y
2) – y
3) + x
4) + z (out of page)
5) – z (into page)
The picture shows the force acting
in the +y direction. Applying the
y
right-hand rule leads to a B field
that points into the page. The B
field must be out of the plane
because B  v and B  F.
x
ConcepTest 22.4b Mass Spectrometer
A proton enters a uniform
magnetic field that is
perpendicular to the
proton’s velocity. What
happens to the kinetic
energy of the proton?
The velocity of the proton
changes direction but the
magnitude (speed) doesn’t
change. Thus the kinetic
energy stays the same.
1) it increases
2) it decreases
3) it stays the same
4) depends on the velocity direction
5) depends on the B field direction
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
ConcepTest 22.6a Magnetic Force on a Wire I
A horizontal wire carries a current
and is in a vertical magnetic field.
What is the direction of the force
on the wire?
Using the right-hand rule, we
see that the magnetic force
must point out of the page.
Since F must be perpendicular
to both I and B, you should
realize that F cannot be in the
plane of the page at all.
1) left
2) right
3) zero
4) into the page
5) out of the page
I
B
ConcepTest 22.6b Magnetic Force on a Wire II
A horizontal wire carries a current
and is in a vertical magnetic field.
What is the direction of the force
on the wire?
1) left
2) right
3) zero
4) into the page
5) out of the page
I
When the current is parallel to
the magnetic field lines, the force
on the wire is zero.
B
ConcepTest 22.7a Magnetic Force on a Loop I
A rectangular current loop is
in a uniform magnetic field.
What is the direction of the
net force on the loop?
1) + x
2) + y
3) zero
4) - x
5) - y
Using the right-hand rule, we find that
each of the four wire segments will
experience a force outward from the
center of the loop. Thus, the forces of
the opposing segments cancel, so the
B
z
net force is zero.
y
x
Biot-Savart Law (today)
and Ampere’s Law
29.2: Calculating the Magnetic Field due to a Current
The magnitude of the field dB produced
at point P at distance r by a current
length element i ds turns out to be
where q is the angle between the
directions of and , a unit vector that
points from ds toward P. Symbol m0 is a
constant, called the permeability
constant, whose value is
Therefore, in vector form
29.2: Magnetic Field due to a Long Straight Wire:
ConcepTest 22.10 Current Loop
1) left
What is the direction of the
magnetic field at the center
(point P) of the square loop
of current?
2) right
3) zero
4) into the page
5) out of the page
Use the right-hand rule for each
wire segment to find that each
segment has its B field pointing
out of the page at point P.
I
P
ConcepTest 22.8a Magnetic Field of a Wire I
If the currents in these wires have
1) direction 1
the same magnitude but opposite
2) direction 2
directions, what is the direction of
the magnetic field at point P?
3) direction 3
4) direction 4
5) the B field is zero
1
P
Using the right-hand rule, we
can sketch the B fields due
them up as vectors gives a
total magnetic field pointing
downward.
4
2
3
ConcepTest 22.8b Magnetic Field of a Wire II
Each of the wires in the figures
below carry the same current,
either into or out of the page.
In which case is the magnetic
field at the center of the square
greatest?
1
2
1) arrangement 1
2) arrangement 2
3) arrangement 3
4) same for all
3
Some examples: Magnetic Field due to a Long Straight Wire
Another example: Magnetic Field due to a Current in a Circular Arc of Wire
The equation for the magnetic field of a straight, current
m0i
carrying wire is given byB 
, but the magnetic field at the center
2p R
m0i
of a single closed circular loop is given by
B
2R
. Although these
equations look similar, there is an important difference between these two
equations, other that the factor of p. What is it?
a) The µ0 factor is different for the two situations.
b) The variable R represents two different lengths.
c) The i represents two different types of current.
ConcepTest 22.9a Field and Force I
A positive charge moves parallel
1) + z (out of page)
to a wire. If a current is suddenly
2) - z (into page)
turned on, in which direction will
3) + x
the force act?
4) - x
5) - y
Using the right-hand rule to determine the
magnetic field produced by the wire, we
find that at the position of the charge +q
(to the left of the wire) the B field points
out of the page. Applying the right-hand
rule again for the magnetic force on the
charge, we find that +q experiences a force
in the +x direction.
y
+q
x
z
I
ConcepTest 22.9b Field and Force II
Two straight wires run parallel to
each other, each carrying a
current in the direction shown
below. The two wires experience
a force in which direction?
1) toward each other
2) away from each other
3) there is no force
The current in each wire produces a magnetic
field that is felt by the current of the other
wire. Using the right-hand rule, we find that
each wire experiences a force toward the
other wire (i.e., an attractive force) when the
currents are parallel (as shown).
Force Between Two Parallel Wires:
Two parallel wires have currents that have the same
direction, but differing magnitude. The current in wire A
is i; and the current in wire B is 2i. Which one of the
following statements concerning this situation is true?
a) Wire A attracts wire B with half the force that wire B
attracts wire A.
b) Wire A attracts wire B with twice the force that wire B
attracts wire A.
c) Both wires attract each other with the same amount of
force.
d) Wire A repels wire B with half the force that wire B
attracts wire A.
e) Wire A repels wire B with twice the force that wire B
attracts wire A.
ConcepTest 22.7b Magnetic Force on a Loop II
1) move up
If there is a current in
2) move down
the loop in the direction
3) rotate clockwise
shown, the loop will:
4) rotate counterclockwise
5) both rotate and move
Look at the North Pole: here the
F
magnetic field points to the right and
the current points out of the page.
N
S
The right-hand rule says that the force
must point up. At the south pole, the
same logic leads to a downward force.
Thus the loop rotates clockwise.
F
The drawing shows two long, straight wires that are parallel to each other
and carry a current of magnitude i toward you. The wires are separated
by a distance d; and the centers of the wires are a distance d from the y
axis. Which one of the following expressions correctly gives the
magnitude of the total magnetic field at the origin of the x, y coordinate
system?
a)
m 0i
2d
b)
c)
d)
e)
m 0i
2d
m 0i
2p d
m 0i
pd
zero tesla
Ampere’s Law
Ampere’s Law:
Ampere’s Law, Magnetic Field Outside a Long Straight Wire
Carrying Current:
A hollow cylindrical conductor (inner radius = a, outer
cross section. Which graph below correctly gives B as a
function of the distance r from the center of the cylinder?
Magnetic Field Inside a Long Straight Wire Carrying Current:
Solenoids and Toroids:
Solenoids:
Here n is the number of turns per unit length of the solenoid
A copper cylinder has an outer radius 2R and an inner radius of R and
carries a current i. Which one of the following statements
concerning the magnetic field in the hollow region of the cylinder
is true?
a) The magnetic field within the hollow region may be represented as
concentric circles with the direction of the field being the same as
that outside the cylinder.
b) The magnetic field within the hollow region may be represented as
concentric circles with the direction of the field being the opposite
as that outside the cylinder.
c) The magnetic field within the hollow region is parallel to the axis of
the cylinder and is directed in the same direction as the current.
d) The magnetic field within the hollow region is parallel to the axis of
the cylinder and is directed in the opposite direction as the current.
e) The magnetic field within the hollow region is equal to zero tesla.
The drawing shows a rectangular wire loop that has one side passing
through the center of a solenoid. Which one of the following
statements describes the force, if any, that acts on the
rectangular loop when a current is passing through the solenoid.
a) The magnetic force causes the loop
to move upward.
b) The magnetic force causes the loop
to move downward.
c) The magnetic force causes the loop
to move to the right.
d) The magnetic force causes the loop to move to the left.
e) The loop is not affected by the current passing through the
solenoid or the magnetic field resulting from it.
A solenoid carries current I as shown in the figure. If the observer could
“see” the magnetic field inside the solenoid, how would it appear?
The diagrams show three circuits consisting of concentric circular arcs
(either half or quarter circles of radii r, 2r, and 3r) and radial lengths. The
circuits carry the same current. Rank them according to the magnitudes of the
magnetic fields they produce at C, least to greatest.
A) 1, 2, 3
B) 3, 2, 1
C) 1, 3, 2
D) 2, 3, 1
E) 2, 1, 3

•
When the sign of the charge which carries current
changes, the magnetic force FB does not change.
Thus, measuring
Conductor in B-field
BI
VH 
,
nqt
the Hall voltage
difference between the
top (c) and bottom (a)
of the conducting slab,
reveals the sign of the
charge carrier q and its
density n.
VH
```