Chapter 29: Magnetic Fields
Learning Goals
- Understand the properties of magnets and magnetic fields.
- Understand forces on current-carrying wires.
- Understand torques on current loops.
- Understand the motion of a charged particle in a uniform magnetic field.
- Understand the Hall effect.
Magnetic Force and the Magnetic Field
As with electric fields, the magnetic field
B is defined from a force - in this case the
magnetic force:
where q is the particle’s charge and v is the
particle velocity. From the definition of cross products,
the magnetic force has magnitude
where q
is the angle between the velocity and the external magnetic field.
Notice that only moving charges experience magnetic forces,
and then only if the velocity has a non-zero component
transverse to the magnetic field B.
Magnetic Forces on Current-Carrying Wires
Moving charge carriers within a conducting wire can
experience a magnetic force if that wire is placed in
an external magnetic field. If the wire carries a current,
the charge carriers transmit a net force to the wire.
For a straight wire segment of length L, this force is given by
where L is a vector with magnitude equal to the
length of the wire segment and direction given by the
direction of current flow in the wire. Magnetic forces on
wires with arbitrary shape may be calculated by using the
above formula with an infinitesimal length element.
The net force on a current loop in a uniform magnetic
field is zero.
Magnetic Moments and Magnetic Torques
The magnetic moment of a current loop
with N turns and current I is
Current loops in a magnetic field can experience a
torque. For a uniform field B, this torque is given
by
where the direction of m
is given by the right hand rule.
Motion of Charged Particles in a Uniform Magnetic Field
Charged particles whose velocity v is at right angles
to an external magnetic field B move in a circle of radius
The angular frequency of the rotational motion is
