Magnetic Force Acting on a
Current-Carrying Conductor
• (a) Magnetic field lines
coming out of the paper
are indicated by dots,
representing the tips of
arrows coming outward.
• (b) Magnetic field lines
going into the paper are
indicated by crosses,
representing the
feathers of arrows
going inward.
• A segment of a currentcarrying wire in a
magnetic field B. The
magnetic force exerted
on each charge making
up the current is q vd x
B and the net force on
the segment of length L
is I L x B.
• (a) A wire suspended vertically
between the poles of a
magnet. (b) The setup shown
in part (a) as seen looking at
the south pole of the magnet,
so that the magnetic field
(blue crosses) is directed into
the page. When there is no
current in the wire, it remains
vertical. (c) When the current
is upward, the wire deflects to
the left. (d) When the current
is downward, the wire deflects
to the right
A wire segment of
arbitrary shape carrying a current I
in a magnetic field B experiences a
magnetic force. The magnetic
force on any segment ds is I ds x B
and is directed out of the page. You
should use the right-hand rule to
confirm this force direction.
• The magnetic force
exerted on a small
segment of vector
length ds in the
presence of a field B is
• (a) A curved wire carrying a
current I in a uniform
magnetic field. The total
magnetic force acting on
the wire is equivalent to the
force on a straight wire of
length L( running between
the ends of the curved wire.
(b) A current-carrying loop
of arbitrary shape in a
uniform magnetic field. The
net magnetic force on the
loop is zero.
Conclusion
• the magnetic force on a curved currentcarrying wire in a uniform magnetic field is
equal to that on a straight wire connecting the
end points and carrying the same current
• the net magnetic force acting on any closed
current loop in a uniform magnetic field is
zero
Quick Quiz
• The four wires shown in
Figure, all carry the same
current from point A to
point B through the same
magnetic field. In all four
parts of the figure, the
points A and B are 10 cm
apart. Rank the wires
according to the
magnitude of the
magnetic force exerted on
them, from greatest to
least.
Motion of a Charged Particle in a
Uniform Magnetic Field
• When the velocity of a
charged particle is
perpendicular to a
uniform magnetic field,
the particle moves in a
circular path in a plane
perpendicular to B. The
magnetic force FB
acting on the charge is
always directed toward
the center of the circle.
• These results show that the
angular speed of the particle
and the period of the circular
motion do not depend on the
linear speed of the particle or
on the radius of the orbit.
• The angular speed ω is often
referred to as the cyclotron
frequency because charged
particles circulate at this
angular frequency in the type
of accelerator called a
cyclotron.
• If a charged particle moves in
a uniform magnetic field with
its velocity at some arbitrary
angle with respect to B, its
path is a helix
• if the field is directed in the x
direction, as shown in Figure,
there is no component of force
in the x direction. As a result,
ax =0, and the x component of
velocity remains constant.
• However, the magnetic force q
v x B causes the components
v and v to change in time, and
the resulting motion is a helix
whose axis is parallel to the
magnetic field.
• The projection of the path onto
the yz plane (viewed along the
x axis) is a circle
y
z
Example
• A proton is moving in a
circular orbit of radius
14 cm in a uniform
0.35-T magnetic field
perpendicular to the
velocity of the proton.
Find the linear speed of
the proton
• Solution
The Biot–Savart Law
The magnetic field
dB at a point due to the current I
through a length element ds is
given by the Biot–Savart law. The
direction of the field is out of the
page at P and into the page at P%
The Magnetic Force Between
Two Parallel Conductors
parallel conductors carrying currents in the same direction
attract each other, and parallel conductors carrying currents in
opposite directions
repel each other.
• magnitude in terms of the force per unit
length
Ampère’s Law
• (a) When no current is present in
the wire, all compass needles
point in the same direction
(toward the Earth’s north pole).
• (b) When the wire carries a
strong current, the compass
needles deflect in a direction
tangent to the circle, which is the
direction of the magnetic field
created by the current
• Ampère’s law describes
the creation of
magnetic fields by all
continuous current
configurations
Example
The Magnetic Field of a Solenoid
• (a) Magnetic field lines for a
tightly wound solenoid of finite
length, carrying a steady current.
The field in the interior space is
strong and nearly uniform. Note
that the field lines resemble
those of a bar magnet, meaning
that the solenoid effectively has
north and south poles.
• (b) The magnetic field pattern of
a bar magnet, displayed with
small iron filings on a sheet of
paper
Magnetic Flux
• Magnetic flux through a
plane lying in a magnetic
field. (a) The flux through
the plane is zero when the
magnetic field is parallel to
the plane surface. (b) The
flux through the plane is a
maximum when the
magnetic field is
perpendicular to the plane.
Example
The Magnetic Field of the Earth
• the Earth’s south magnetic pole is
located near the north
geographic pole, and the Earth’s
north magnetic pole is located
near the south geographic pole
• south magnetic pole is near the
north geographic pole, and a
north magnetic pole is near the
south geographic pole.