Chapter 19 Magnetic Field
We have discussed how a charged plastic rod produces a vector field-the electric field E -at all points in the space around it. Similarly, a magnet produces a vector field-the magnetic field B -at all points in the space around it.
19.1 The Definition of B
1. We determined the electric field
E at a point by putting a test particle of charge q at rest at that point and measuring the electric force
E F acting on the particle. We then defined E as q F E E =.
2. To define the magnetic field
B , we can fire a charged particle through the point where
B is to be defined, using various directions and speeds for the particle and determining the force B F that acts on the particle at that point.
(1) After many such trials we would find that when the
particle ’s velocity v is along a particular axis through the
point, force B F is zero .
(2) For all other directions of
v , the magnitude of B F is always proportional to φsin v , where φ is the angle between
the zero-force axis and the direction of v .
(3) Furthermore, the direction B F is always perpendicular to
the direction of v (These results suggest that a cross product
is involved).
3. We can then define a magnetic field
B to be a vector quantity that is directed along the zero-force axis . We can next measure the magnitude of B F when v is directed perpendicular to
that axis and then define the magnitude of
B in terms of that force magnitude: v
q F B B =, where q is the charge of the particle. 4. We can summarize all these results with the following vector equation: B v q F B ?=.
5. Finding the magnetic force on a particle: The force
B F acting on a charged particle moving with velocity
v through a magnetic field B is always perpendicular to
v and B . 6. The SI unit for B is the netown per coulomb-meter per
second. For convenience, that is called the tesla (T). An earlier (non-SI) unit for B , still in common use, is the gauss (G), and 1 tesla = 104 gauss.
7. Right table lists the
magnetic fields that
occur in a few
situations.
8. We can represent magnetic fields with field lines , just as we
did for electric field. Similar rules apply. That is, (1) the direction of the tangent to a magnetic field line at any point gives the direction B at that point, and (2) the spacing of the lines represents the magnitude of B -the magnetic field is stronger where the lines are closer together, and conversely. 9.Figures show the magnetic field lines for magnet with different shapes. We find that the lines all pass through the magnet, and they form closed loops(even those that are not
shown closed in the figures). The closed field lines enter one end of a magnet and exit the other end. The end of a magnet from which the field lines emerge is called the north pole of the magnet; the other end, where field lines enter the magnet, is called the south pole.
10.W hen two magnets are moved to each other, we find that opposite magnetic pole attract each other, and like magnetic pole repel each other.
11.E arth has a magnetic field that is produced in its core by still
unknown mechanisms.
19.2 Crossed Field
Both an electric field E and a magnetic field B can produce a force on a charged particle. When the two fields are perpendicular to each other, they are said to be crossed fields. 1.Discovery of the electron: Figure shows a modern version of J.
J. Thomson’s apparatus for measuring the ratio of mass to charge for the electron.
2.The Hall effect: Hall
effect allows us to find
out whether the charge
carriers in a conductor
or semiconductor are
positively or
negatively charged.
Beyond that, we can measure the number of such carriers per
unit volume.
(1) Figure shows a strip of copper carrying a current i is immersed in a magnetic field.
(2) A hall potential difference is the difference of the electric filed potential between to edges of the strip. The magnitude of that potential difference is Ed V =.
(3) When the electric and magnetic force are in balance , we have B ev eE d =, in which the drift speed of electrons is neA i v d =
, in which n is the number density of charge carries.
(4) From above three equations, we have
Vle
Bi n =, in which l (=A/d) is the thickness of the strip.
19.3 A Circulating Charged Particle
1. If a particle moves in a circle at constant speed, we can be sure that the net force acting on the particle is constant in magnitude and points toward the center of the circle, always perpendicular to the particle ’s
velocity.
2. Figure shows that a beam of
electrons is projected into a
chamber by an electron gun G .
The electrons enter in the
plane of the page with velocity v and move in a region of
uniform magnetic field B directed out of the plane of the
figure. As a result, a magnetic force continually deflects the electrons, and because v and B are perpendicular to each
other, this deflection causes the electrons to follow a circular path.
3. From Newton ’s law applied to uniform circular motion, we have r
mv qvB 2=. (1) So the radius of the circular path is qB
mv r =. (2) The period (the time for one full revolution) is equal to the circumference
divided by the speed:
qB
m v r T ππ22==. (3) The frequency is m
qB T f π21==. (4) The angular frequency of the
motion is then m
qB f ==πω2. (5) They are do not depend
on the speed of the particle.
4. Helical paths : If the velocity
of a charged particle has a
component parallel to the (uniform) magnetic field , the
particle will move in helical path about the direction of the field vector, as shown in figure (a) and (b). The parallel component of the velocity determines the pitch of the helix, that is, the distance between adjacent turn. The perpendicular component determines the radius of the helix.
5.Figure (c) shows the charged particle spiraling in a non-uniform magnetic field. The more closely spaced field lines at the left and right sides indicate that the magnetic field is stronger there. When the field at end is strong enough, the particle “reflects”from that end. If the particle reflects from both ends, it is said to be trapped in a magnetic bottle. Electrons and protons are trapped in this way by the terrestrial magnetic field, forming the Van Alen radiation belts, which loop well above Earth’s atmosphere, between Earth’s north and south geomagnetic poles. The trapped particles bounce back and forth, from end to end of the magnetic bottle, within a few seconds.
19.4 Cyclotrons and Synchrotrons
1.The Cyclotron is a device for
giving high speed to charged
particles by magnetic and electric
fields. See the figure.
2. The proton synchrotron:
19.5 Magnetic Force on a Current-Carrying Wire
1. See right figure, in which the
current-carrying wire is perpendicular to
the magnetic field, the magnetic force is
iLB F B =.
2. If the magnetic is not perpendicular to
the wire, as in right figure, the
magnetic force is given by
B L i F B ?=. Here L is a length
vector that points along the wire
segment in the direction of the conventional current.
3. If a wire is not straight , we can imagine it broken up into small straight segments and apply above equation to each segment. The force on the wire as a whole is then the vector sum of all the force on the segments that make it up. In the differential limit, we can write B L id F d B ?=, and we can find the resultant force on any given arrangement of currents by integrating it over that arrangement .
19.6 Torque on a Current Loop and the Magnetic dipole
1. Figure shows a simple
motor, consisting of a
single current-carrying
loop immersed in a magnetic field B .
2. The magnitude of the torque is θθτsin )(sin B iA iabB ==.
3. We can describe the current-carrying coil with a single vector μ , its magnetic dipole moment , we take the direction of μ to
be that of the normal vector n to the plane of the coil, as
figure (c). We define the magnitude of
μ as NiA =μ, in which N is the loop of the coil; A is the area enclosed by the coil. Then the torque can be written as B
?=μτ 4. While an external magnetic field is exerting a torque on a magnetic dipole, work must be done to change the orientation of the dipole. The magnetic dipole must then have a magnetic
potential energy . We can write it as B U ?-=μθ)(.