Physics 122B Electricity and Magnetism

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Physics 122B Electricity and Magnetism
Lecture 5E-fields and Charged Particles in
E-Fields April 04, 2007

• Martin Savage

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Lecture 5 Announcements

• Lecture Homework 1 have been posted on the
  Tycho system. It is due at 10 PM tonight.

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E-Field of a Charged Sphere
Therefore, the electric field outside the
surface of a uniformly spherically charged shell
of charge Q with radius R is
(independent of R)
In other words, it has the same E-field as
that produced by a point charge Q located at the
center of the sphere.
If hQ/4pR2 is the surface charge density,
then the surface E field is Esurfh/e0, i.e.,
twice as big as the E field of an infinite plane
of charge. (Why?)
The electric field inside the spherical-shell is
zero (because the sum of the forces cancel).
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Example A Charged Sphere
A sphere with a radius R 0.1 m has a
charge Q 20.0 nC on its surface. What is
the electric field at a distance of 1.0 m from
the sphere?
Note that the electric field does not depend
on the radius of the sphere (because the E-field
of a charged sphere is the same as that of the
same charge at a point at the center of the
sphere.) Measuring the external E-field tells
you the charge but not the radius of the sphere.
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A Parallel-Plate Capacitor
The parallel plate capacitor is an important
component of electronic circuits. It is
constructed of two oppositely charged electrodes
separated by a small gap. The net charge of the
capacitor is zero.
We can model the parallel plate capacitor as
two parallel infinite charged planes placed a
distance d apart. Let the x axis go from to -.
Then we have only to superpose the fields that
we have previously calculated. We find that in
the two regions outside the gap the superposed
fields cancel to give 0, while in the gap they
add. Therefore
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Capacitor Edge Effects
Since the electrodes of a real parallel
plate capacitor are not infinite, there are edge
effects at the ends of the electrodes.
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ExampleA Parallel-Plate Capacitor
The parallel plate capacitor consists of two
circular electrodes of radius R0.1 m separated
by a gap of d1 mm, with opposite charges of
magnitude Q20 nC on the two electrodes.
What is the electric field between the plates?
R
1 e0
(Notice that the electric field does not depend
on the gap d.)
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Motion of a Charged Particlein an Electric Field
In an electric field, does a charged particle
follow the field lines? NO! Only a massless
charged particle would follow the field lines.
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Example An electron moving across a capacitor
Two 6.0 cm diameter circular electrodes are
spaced 5.0 mm apart. They are charged by
transferring 1.0 x 1011 electrons from one
electrode to the other. A electron is
released from rest at the surface of the negative
electrode. How long does it take the electron to
cross to the positive electrode? Assume the
space between the electrodes is a vacuum.
V/c a D t/c 0.1
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ExampleDeflecting an electron beam
An electron gun creates a beam of electrons
moving horizontally with a speed vx 3.34 x 107
m/s. The electrons enter a 2.0 cm long gap
between two parallel electrodes producing a
downward electric field of E 5.0x104 N/C.
In what direction and by what angle is the
electron beam deflected? (Neglect fringing)
The deflection is upward because the
negatively charged electron experiences a force
in the opposite direction from E.
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Motion in a Nonuniform Field
The motion of a charged particle in a
non-uniform field can be quite complicated.
However, one case is easy to analyze the
circular orbit of a charged particle around
another point charge, a charged sphere, or a long
straight charged wire (so that the field at some
distance r is constant). In such cases,
Example An electron in a hydrogen atom orbits
at a radius of r0.053 nm. What is the orbital
velocity of the electron?
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Question 1
Which field is responsible for the
trajectory of the proton as shown?
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Dipole in a Uniform Field
Consider a permanent electric dipole of
dipole moment pqs in a uniform external electric
field E.
If the axis of the dipole is not aligned
with the field (a), the dipole will experience a
torque. If the dipole is aligned with the field
(b) it will not experience a torque, and it is in
stable equilibrium. In either case, there is no
net force on the dipole.
When dipoles, e.g., water molecules, are
placed in an external field, they will tend to
align with the field as shown, so that pE.
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Torque on a Dipolein a Uniform Field
If the axis of the dipole is not aligned with
the field, the dipole will experience a torque t.
The dipole moment is
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Example Angular acceleration of a dipole dumbbell
Two 1.0 g spheres are connected by a 2.0 cm long
insulating rod of negligible mass. The spheres
have equal and opposite charges of 10 nC. The
rod is held in a uniform field of E 1.0x104
N/C at an angle of 300 to the field, as shown,
then released. What is the angular
acceleration a of the system?
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Dipole in a Nonuniform Field
In the non-uniform field of a point charge,
the dipole will first align with the field (pE)
and then be attracted by it. The net force
on the dipole will be attractive, independent of
whether the field is from a negative or a
positive point charge. In general, a dipole will
experience a net force toward any charged object.
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ExampleThe force on a water molecule
The water molecule H2O has a permanent dipole
moment of p 6.2x10-30 C m. A water molecule
is located 10 nm from a Na ion in a salt water
solution. What force does the ion exert on
the water molecule?
Trick Calculate force on Na ion from dipole
field.
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End of Lecture 5

• Before the next lecture on Wednesday, read
  Knight, Chapters 27.2 through 27.4
• Lecture Homework 1 should be submitted on the
  Tycho system by tonight at 10 PM .