Physics 122B Electricity and Magnetism

Lecture 5E-fields and Charged Particles in

E-Fields April 04, 2007

- Martin Savage

Lecture 5 Announcements

- Lecture Homework 1 have been posted on the

Tycho system. It is due at 10 PM tonight.

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).

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.

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

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.

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.)

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.

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

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.

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?

Question 1

Which field is responsible for the

trajectory of the proton as shown?

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.

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

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?

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.

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.

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 .