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Magnetic Fields
AP Physics C
Montwood High School
R. Casao
Definition & Properties of the Magnetic Field
• The electric field E at a point in space is
defined at the electric force per unit
charge acting on a test charge placed at
that point.
• The gravitational field g at a point in
space is the gravitational force per unit
mass acting on a test mass.
• The magnetic field B at some point in
space is the magnetic force that would
be exerted on a charged particle moving
with a velocity v.
Properties of the Magnetic Force on a Charge
Moving in a Magnetic Field B
• The magnetic force Fmag is proportional to the
charge q and the speed v of the particle.
• The magnitude and direction of the magnetic
force depend on the velocity of the particle and
on the magnitude and direction of the magnetic
field.
• When a charged particle moves in a direction
parallel to the magnetic field vector, the
magnetic force Fmag on the charge is zero.
Properties of the Magnetic Force on a Charge
Moving in a Magnetic Field B
• When the velocity vector makes an angle q with
the magnetic field, the magnetic force acts in a
direction perpendicular to both v and B; that is,
Fmag is perpendicular to the plane formed by v and
B.
• The magnetic force on a positive charge is in the
direction opposite the direction of the force on a
negative charge moving in the same direction.
• If the velocity vector makes an angle q with the
magnetic field, the magnitude of the magnetic
force is proportional to sin q.
Properties of the Magnetic Field
• Equation:
Fmag
 
 q  v x B
– where the direction
of the magnetic
force is in the
direction of v x B,
which by definition
of the cross product,
is perpendicular to
both v and B.
Right-Hand Rule
• The right-hand rule is used to
determine the direction of the cross
product v X B.
• Figure a: the fingers of the right
hand are placed in the direction of
the velocity v. The palm of the right
hand is placed in the direction of the
magnetic field B. The fingers of the
right hand are curled through angle f
from v towards B. The thumb points
in the direction of the resultant vector
v x B, or the magnetic force Fmag.
Right-Hand Rule
• Figure b: If charge q is positive, then
the direction of the resultant vector
v x B, or the magnetic force Fmag, is
in the direction of the thumb of the
right hand (upwards as in the figure).
• Figure c: If charge q is negative,
then the direction of the resultant
vector v x B, or the magnetic force
Fmag, is in the oppposite direction of
the thumb of the right hand
(downwards).
Cross-Products
• Whenever you see a cross-product, such
as v X B:
– the fingers of the hand will always point in the
direction of the first variable.
– the palm of the hand will always point in the
direction of the second variable.
– the thumb will always point in the direction of
the resultant vector.
• All three vectors are at right angles to
each other.
Casao’s Version of the Right-Hand Rule
• I use the right hand as described for
positive charges. No change here.
• I use the left hand for negative charges.
– For the v X B cross product using a negative
charge:
– Point the fingers of the left hand in the
direction of v.
– Point the palm of the left hand in the direction
of B.
– The thumb of the left hand points in the
direction of the resultant vector, v X B, or the
magnetic force Fmag.
Casao’s Version of the Right-Hand Rule
Equation
• The magnitude of the magnetic force:
Fmag = q·v·B·sin q
where q is the angle between v and B.
• Fmag = 0 N when v is parallel to B (sin 0 = 0).
• Fmag is maximum when v is perpendicular to B (sin
90 = 1).
Differences Between E and B
• The magnetic field is defined in terms of a
sideways force acting on a moving
charged particle.
• Differences between electric field E and
the magnetic field B:
– E is always in the direction of the electric field;
B is perpendicular to the magnetic field.
– E acts on a charged particle independent of
the particle’s velocity (the charge can be
stationary); B acts on a charged particle only
when the particle is in motion.
Differences Between E and B
– E does work in displacing a charged particle
because the direction of motion is parallel to
the direction of the force; B does NO WORK
when a particle is displaced because the
direction of the motion is perpendicular to the
direction of the force.
– Work is a dot product of force and
displacement; F•dx = F•x•cos q; cos 90 = 0
– Since no work is done on the moving charged
particle, the kinetic energy of the charged
particle cannot be changed by a magnetic field
alone.
Differences Between E and B
• When a charged particle with a velocity v
moves in a magnetic field, the magnetic
field can change the direction of the
velocity vector, but in cannot change the
speed of the charged particle.
• Unit of the magnetic field: Tesla, T.
– Tesla is a weber per square meter (Wb/m2).
– English unit is the Gauss (G); 1 T = 1 x 104 G.
Wb
N
N
T 2 

m
Am
Cm
s
Example 29.1 Proton Moving in a Magnetic Field
• A proton moves with a speed of 8 x 106 m/s
along the x-axis. It enters a region where there
is a field of magnitude 2.5 T, directed at an angle
of 60° to the x-axis and lying in the xy plane.
Calculate the initial magnetic force and
acceleration of the proton.
 
Fmag  q  v  B  sin θ
Fmag  1.602x10 19 C  8x106 m/s  2.5T  sin60
Fmag  2.7747x1012 N
Example 29.1 Proton Moving in a Magnetic Field
• If the particle had been an electron, do not use the
negative sign of the charge in the calculation. We will
continue to let the direction of the vector determine the
sign of the vectors associated with magnetic fields.
• Use the right-hand rule to determine the direction of
Fmag: point the fingers of the right hand in the direction
of v; the palm of the right hand is in the direction of the
magnetic field B; and the thumb points up in the
direction of Fmag (the positive z direction).
• If you would rather work with vectors at right angles to
each other, you can resolve the velocity vector into a
component that is parallel to B and a component that
is perpendicular to B.
Example 29.1 Proton Moving in a Magnetic Field
• The velocity component parallel to B will
have an Fmag = 0 N. The velocity component
perpendicular to B will have an Fmag =
q·v·B·sin 90.
• Acceleration:
12
F 2.7747 x 10 N
15
2
a


1.6615
x
10
m/s
m
1.67 x 10 27 kg
Magnetic Field Lines
• Magnetic fields can be represented with
field lines just as we used for electric
fields.
• The direction of the tangent ot a magnetic
field line at any point give the direction of
B at that point.
• The spacing of the lines represents the
magnitude of B, the magnetic field is
stronger where the lines are closer
together.
Magnetic Field Lines
• For the bar magnet: the
magnetic field lines all pass
through the magnet and form
closed loops.
• The external magnetic effects
are strongest near the ends
where the field lines are most
closely spaced.
• Because the magnetic field
has direction, the field lines
enter one end of a magnet
and exit the other end.
Magnetic Field Lines
• The end of a magnet from which the
field lines emerge is called the North
pole; the end of the magnet where the
field lines enter the magnet is the
South pole.
• Magnetic field lines always begin on
the North pole and end on the South
pole of a magnet.
• Opposite magnetic poles attract each
other; like magnetic poles repel each
other.
Magnetic Field Lines
Magnetic Field Lines
• Magnetic field lines
that emerge from a
plane are illustrated
by a dot (the
arrowhead)
• Magnetic field lines
that go into a plane
are illustrated by a
cross (the tail of the
arrow).
Magnetic Field Lines
Earth’s Magnetic Field
• The south pole of Earth’s magnetic field is
the direction the north pole of a compass
points towards, so we call it the north pole.
– The magnetic field lines of Earth generally
point down into Earth.
• The north pole of Earth’s magnetic field is
the direction the south pole of a compass
points towards, so we call it the south
pole.
– The magnetic field lines of Earth generally
point away from the Earth.