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General Physics II
Magnetic Fields and Forces
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Magnetic Force on a Moving Charge
• A charged particle moving within a magnetic field will in
general experience a force that we call a “magnetic force.”
This magnetic force has the following properties:
• If the charged particle is at rest, there is no force.
• If the charged particle moves parallel or antiparallel to the
magnetic field, there is no force.
• The magnitude of the force is maximum when the
magnetic field and the velocity of the charged particle are
perpendicular.
• The force is always perpendicular to the plane containing
the magnetic field vector and the particle’s velocity vector.
Thus, the force is perpendicular to both the velocity and
magnetic field.
• The greater the charge, or speed of the particle, or
strength of the magnetic field, the greater the force.
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Magnetic Force on a Moving Charge
• The magnitude of the magnetic force is given by
F = q vBsinα ,
where q is the charge, v is the magnitude of the velocity, B
is the magnetic field strength, and α is the angle between
the velocity and the field.
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Magnetic Force on a Moving Charge
• The direction of the magnetic force on a positively charged
particle is given by the right hand rule for forces.
• The direction of the magnetic force on a negatively charged
particle is opposite to that for a positively charged particle.
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Workbook: Chapter 24, Questions 16, 17, 22
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Paths of Charged Particles in a Magnetic
Field
• Recall that in circular motion, there must be a centripetal
force, which is directed toward the center of the circle. For a
satellite in a circular orbit around the Earth, the centripetal
force is gravity.
• In uniform (constant speed) circular motion, the net force is
the centripetal force. Thus, this net force is always
perpendicular to the velocity.
• Note that for a charged particle moving in a magnetic field,
the magnetic force is always perpendicular to the velocity.
Thus, if a charged particle moves perpendicular to a uniform
magnetic field, it will execute uniform circular motion. (We
assume that there are no other forces acting on the
particle.)
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Paths of Charged Particles in a
Magnetic Field
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Circular Path of a Charged Particle
The magnitude of the centripetal force is given by
2
mv
Fcent = r .
This must be equal to the magnitude of the magnetic force,
which is responsible for the circular motion:
2
Fcent = mv
r = q vB.
Note that the velocity and the magnetic field are
perpendicular in this motion, so α = 90° and so sinα = 1.
Solving for the radius gives
r = mv .
qB
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Paths of Charged Particles in a
Magnetic Field
• If the velocity of a charged
particle has a component
parallel to the magnetic field,
that component remains
unchanged.
• The velocity component
perpendicular to the
magnetic field results in
circular motion.
Parallel
component
Perpendicular
component
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The Aurora
The net result is that the particle moves in a spiral or helix
around the field lines, in the direction of the velocity
component parallel to the field lines.
Spiral
Charged particles high above the Earth spiral around the Earth’s
magnetic field lines. Particles coming at the poles can penetrate into
the atmosphere where they ionize air molecules, which causes the 10
emission of light.
Workbook: Chapter 24, Questions 18, 21
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Magnetic Forces on Currents
• Since electric currents are
moving charges, a current that is
within a magnetic field will
generally experience a magnetic
force. The right hand rule
applies in the same way as for
moving charges (thumb in the
direction of the current).
• The magnitude of the force on a
straight current-carrying wire in
a uniform magnetic field is given
by
Fwire = ILBsinα ,
where L is the length of the wire,
I is the current, B is the
magnetic field strength and α is
the angle between the current
and the field.
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Magnetic Forces Between Currents
• A current-carrying wire creates a
magnetic field. If another currentcarrying wire is within this
magnetic field, it experiences a
magnetic force.
• Consider the case of two parallel
current-carrying wires with
currents in the same direction.
The lower wire produces a
magnetic field pointing out of the
page above it, where the upper
wire is located. This causes a
downward force on the upper
wire according to the right hand
rule.
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Magnetic Forces Between Currents
• The upper wire produces a
magnetic field pointing into the
page below it, where the lower
wire is located. This causes an
upward force on the lower wire
according to the right hand rule.
• Thus, parallel currents in the
same direction attract each
other.
• One can show that parallel
currents in opposite directions
repel each other.
• These conclusions agree with
experiments.
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Magnetic Forces Between Parallel Currents
• The magnetic field at a distance
d from the lower wire is
μ0I2
B2 =
.
2π d
• The current in the upper wire
is perpendicular to this field.
Thus, the magnitude of the
force on the upper wire due to
the magnetic field of the lower
wire is
μ0I2
. So
F = I1LB2 = I1L
2π d
μ0I1I2L
F=
.
2π d
• Both wires experience a force of this magnitude,
consistent with Newton’s third law.
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Workbook: Chapter 24, Questions 23, 25
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Magnetic Torque on a Current Loop
• A current loop is a magnetic dipole,
like a compass or bar magnet.
Therefore, a current loop will
experience a torque (twist) when it
is within an external magnetic field.
If free to rotate, the magnetic
dipole will rotate until its own
magnetic field along its central axis
is aligned with the external
magnetic field.
A current loop behaves like
a magnet. The N and S
poles of the equivalent bar
magnet are shown. The field
lines shown are those due to
the loop itself.
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Current Loop in a Uniform Field
• Applying the right hand rule for
force to each side of the loop
gives forces in the direction
shown. The forces on opposite
sides are equal in magnitude
and opposite in direction, so the
net force on the loop is zero.
• The forces on the front and back
sides have lines of action
through the center of the loop
and will not cause it to turn, i.e.,
they exert no torque. However,
the forces on the top and bottom
sides do exert a net torque,
tending to rotate the loop about
the x axis.
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Current Loop in a Uniform Field
• The magnitude of the torque of
a force = force×moment arm.
(Moment arm is the same as
lever arm, which is the
perpendicular distance from the
line along which the force acts
to the axis of rotation.)
• The total torque on the current
loop is given by
τ = IABsinθ ,
where I is the current, A is the
area of the loop, B is the
magnetic field strength, and θ is
the angle between the magnetic
field and the direction of the
loop’s North pole.
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Current Loop in a Uniform Field
Current (or magnetic poles) should be in opposite direction!
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Workbook: Chapter 24, Questions 28, 30
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Chapter 24, Problem 42
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The Electric Motor
The electric motor is a coil consisting of many loops that uses
magnetic torque for rotation. A commutator reverses the
current every half cycle to keep the coil rotating in the same
direction.
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