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AP C - UNIT 9
Magnetic Fields and Forces
Magnetic Materials
• All magnetic materials
have two poles
– Labeled: North and
South poles
– Like poles repel
each other and
opposite poles
attract.
Magnetic Monopoles
Unlike electrostatics,
magnetic monopoles
have never been
detected.
Modern View of Magnetism
Magnetism is associated with
charges in motion (currents):
–microscopic currents in the atoms
of magnetic materials.
–macroscopic currents in the
windings of an electromagnet.
Magnetic Field
Similar to E-field, but where
attraction and repulsion is
occurring due to
magnetism, either magnets
or moving charges.
Magnetic Field Lines
The Earth’s Magnetic Field
The spinning iron
core of the earth
produces a
magnetic field.
Magnetic Fields
Magnetic field is a vector. It has direction
and can be cancelled by another field.
Magnetic Field Units
The SI unit of magnetic field is the Tesla (T).
Dimensional analysis:
1 T = 1 N·s/(C·m) = 1 V·s /m2
Sometimes we use a unit called a Gauss (G):
1 T = 104 G
The earth’s magnetic field is about 0.5 G.
Magnetic Force on Moving Charges
A charged particle in a static magnetic field will
experience a magnetic force only if the particle is
moving.
If a charge q with velocity v
moves in a magnetic field B
and v makes an angle q w.r.t.
B, then the magnitude of the
force on the charge is:
Finding the Direction of Magnetic Force
• The direction of the magnetic
force is always perpendicular
to both v and B
• Fmax occurs when v is
perpendicular to B
• F = 0 when v is parallel to B
Notation
To represent the z-axis on an xy plane of paper.
• • •
• • •
B out of the page
  
  
B into the page
Right Hand Rule
• Draw vectors v and B with
their tails at the location
of the charge q.
• Point fingers of right hand
along velocity vector v.
• Curl fingers towards
Magnetic field vector B.
• Thumb points in direction
of magnetic force F on q,
perpendicular to both v
and B.
• Use RHR to find F on
moving charged
particle for each
situation a through f
Motion of Charges in B Fields
If a charged particle is moving
perpendicular to a uniform magnetic
field, its trajectory will be a circle
because the force F=qvB is always
perpendicular to the motion, and
therefore centripetal.
If path of charge is not perpendicular to field where it
enters field at some angle, charge will follow a spiral
path (helix) which will spiral around the B-field.
The component of velocity that is parallel
to the magnetic field is unaffected. Its
circlular motion will drift at a constant
speed along the magnetic field
The perp component of the velocity
to B-field causes the particle to
executes uniform circular motion
perpendicular to the magnetic field.
Velocity Selector
A particle is accelerated through a potential difference where it
then enters a magnetic and electric field. If the forces are
balanced, the particle will move only horizontally between the
plates (assuming negligible gravity)
Mass Spectrometer
The mass spectrometer utilizes the
velocity selector and is an instrument
which can measure the masses and
relative concentrations of atoms and
molecules. The radius of turn yields
the mass of the particles.
An ion is accelerated through a voltage of 1000V with a
charge of 1.6x10-19C after which it enters a chamber with a
uniform B-field equal to 80mT. The ion follows a semicircle path and strikes a photographic plate 1.62m from
where it entered. Find its mass.
Hall Effect
If an electric current flows through a conductor in a magnetic field,
the magnetic field exerts a force on the moving charge carriers
which tends to push them to one side of the conductor. A buildup
of charge at the sides of the conductor will balance this magnetic
influence w/ E-field, producing a measurable voltage between the
two sides of the conductor. The charge carries will then just pass
through in a straight line.
Force on a Wire
Similar to the force on a moving
charge in a B field, we have for a
conductor of length L carrying a
conventional current of I in a B field.
The force experienced by the
conductor is:
B
I
53.1o
‘P’
FS
FM
A thin, massless, uniform rod with
length 0.20m is attached to a frictionless
hinge at point ‘P’. A horizontal spring (k
= 4.8 N/m) connects the other end of the
rod to a vertical wall. A uniform B-field
equal to 0.34T is shown and a 6.5A
current exists in the rod directed
towards the hinge. How much energy is
stored in the spring?
Torque on a current loop
a
A wire loop is freely pivoting in
a uniform B-field (+x) as
shown. Each side is length ‘a’.
B
a
Electric Motor
• An electric motor converts
electrical energy to
mechanical energy
– The mechanical energy is
in the form of rotational
kinetic energy
• When area vector of loop is
parallel to field there is no
torque
After the loop moves a ½ turn, the current needs to switch
direction to keep it rotating. Inertia will carry it past the edge
(top of rotation) and at that moment if the current is switched,
the loop keeps going. If the current isn’t switched and it
passes the edge, it will rotate the other way and get nowhere
Biot Savart Law
This law is seen as the magnetic equivalent of Coulomb's Law
(brute force way of doing it vs Gauss) . This finds B at a point
P, a distance r from the differential element of current Ids.
m0 = 4p  10-7 Tm/A, magnetic
permeability of free space
Current makes
B-fields
Biot-Savart Law – Set-Up
The magnetic field dB is at some point P
The length element is ds
The wire is carrying a steady current of I
• dB 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 ds
– The integral is over the entire
current distribution
B-Field due to a long straight wire along y-axis

0
ds
Example
b
P
I
a
Consider wire bent into the shape shown above.
Find B at ‘P’ if current is flowing clockwise in wire.
Magnetic Field from a Wire, RHR #3
The magnetic field lines from a current form circles around a
straight wire with the direction given by another right hand
rule.
The magnitude of the magnetic
field a distance r from a straight
wire is given by (just proven with
rigorous proof)
Force between 2 current-carrying wires
What happens when current as shown in 2 parallel wires?
I1
a
I2
Force between two parallel wires
a
Find BNET at point P
I2 = I1
P
R
I1
45o
45o
d
I2
Loop of
Current

m o Ids  rˆ
dB 
2
4p r
Consider a coil of radius R with current
flowing CW. Find B at center of coil.
X
A loop of Current is RHR #4
Fingers are current direction and thumb is magnetic
north pole
A current I flows in the positive y
direction in an infinite wire; a current I
also flows in the loop as shown in the
diagram.
I
I
What is the direction of the net force
on the loop?
(a) F = -x
(c) F = +x
(b) F = 0
(d) F = +y
(e) F = -y
d
B-field due to a current loop a distance ‘x’ away
ds
frontal view of
loop of current
Ampere’s Law
Ampere’s Law is to magnetic fields as Gauss’ Law was to electric
fields. Both are used for high symmetry problems.
Similar to drawing a
Gaussian surface
NOW apply Ampere’s Law to find B surrounding long
straight current carrying wire (already did this using Biot Savart)
Consider long wire with
current I into page.
r

ds
Magnetic field inside a wire
Find B inside the wire with uniform
current a distance r from the center
where r < R
B-field of a solenoid
Solenoid is a series of
loops of current. One end
acts like a N & the other like
a S-pole. Ideal solenoid is
approached when turns are
closely spaced and length is
much greater than radius of
turns.
Current would flow from left
to right across top of solenoid
Derivation of B inside solenoid
Cross sectional view of solenoid
showing current into page on
top, out of page on bottom
Derivation of B inside solenoid
2
3
1
4
Toroid - Solenoid shaped like a doughnut
A current I flows in an infinite straight wire in the +z
direction as shown. A concentric infinite cylinder of radius
R carries current 2I in the -z direction.
a) What is the magnetic field Bx at point ‘a’,
just outside the cylinder as shown? Take
CW as positive for B.
a) Bx(a) < 0
b) Bx(a) = 0
y
c) Bx(a) > 0
a
x
x
x
x
b
x 2I
I
x
x
x
x
b) What is the magnetic field Bx at point ‘b’, just inside the
cylinder as shown?
(a) Bx(b) < 0
b) Bx(b) = 0
c) Bx(b) > 0
Two cylindrical conductors each carry current I into the page as
shown. The conductor on the left is solid and has radius R=3a. The
conductor on the right has a hole in the middle and carries current
only between R=a and R=3a.
I
3a
a
3a
I
What is the relation
between the magnetic
field at R = 6a for the two
cases (L=left, R=right)?
a) BL(6a)< BR(6a) b) BL(6a)= BR(6a) c) BL(6a)> BR(6a)
Two cylindrical conductors each carry current I into the page as
shown. The conductor on the left is solid and has radius R=3a. The
conductor on the right has a hole in the middle and carries current
only between R=a and R=3a.
I
3a
a
3a
I
What is the relation
between the magnetic
field at R = 2a for the two
cases (L=left, R=right)?
a) BL(2a)< BR(2a) b) BL(2a)= BR(2a) c) BL(2a)> BR(2a)
Consider a thin, infinitely large sheet of current that carries
a linear current density, λ. The current is out of the page.
Find B near the sheet.
Recall a similar problem with infinitely charged sheet of charge density, σ.