Chapter 16 Electric Field

Main Points of Chapter 16

- Electric field
- Superposition
- Electric dipole
- Electric field lines
- Field of a continuous distribution of charge
- Motion of a charge in a field
- Electric dipole in external electric field

16-1 Electric Field

1. Electric Field

Earlyan action-at-a-distance force

Later Faraday introduced Electric Field

E field

charge

Charge

Electric field

2. Definition of Electric field

Field Point

Size is small enough

Test charge

Charge is very small

source charge

Test charge must be small enough that it does not

affect field

The electric field is defined

Some values of Electric fields

atmosphere at earths surface in clear weather

100-200 N/C

value sufficient to cause electrical breakdown

in dry air

For a point charge

Therefore, the electric field of a point charge

is

The field points outwards from a positive charge

and inwards to a negative one

Comparison of Electric Force with Electric Field

- Electric Force (F) - the actual force felt by a

charge at some location. - Electric Field (E) - found for a location only

tells what the electric force would be if a

charge were located there

A useful concept

3. Principle of Superposition of Electric Fields

Since electric forces add by superposition, the

electric field does as well.

For a group of point charges

- Act What is the electric field at the origin due

to - this set of charges?

a

a

Solution

q

q

a

a

Notice that the fields from the top-right and

bottom left cancel at the origin

a

q

4. Electric Dipoles and Their Fields

An electric dipole is defined as equal and

opposite charges a distance l apart

electric dipole moment is defined as

and points from the negative charge towards the

positive one.

Electric dipoles are often found in nature.

Charged objects can induce electric dipoles

(left) molecules may have permanent electric

dipole moments due to their structure (right).

- Example Find the electric field of the electric

dipole - at point a and point b.

b

Solution

r

a

O

x

Similarly, for point b,

b

r

- ACT The electric field of a dipole at distance

large compared to the charge separation ________ - A) decreases linearly with increasing distance.
- B) remains constant as the distance increases.
- C) decreases inversely with the cube of the

distance. - D) decreases inversely with the square of the

distance. - E) cannot be determined.

16-2 Electric Field Lines

Electric field lines are a useful aid to

visualizing the electric field. There are two

rules to drawing these lines

- The electric field is tangent to the field line

at every point. - 2. The density of electric field lines is an

indicator of relative field strength.

The next slide shows field lines for a point

charge, including the decrease in density as one

moves farther from the charge.

Field lines of a positive point charge

Left two equal, same-sign charges Right an

electric dipole

Important note We always draw only a few sample

field lines otherwise the sketch would be solid

color.

Field Lines of two equal, same-sign charges

- There is a zero halfway between the two charges

- r gtgt a looks like the field of point charge

(2q)

Electric field lines have certain properties

which should be carefully noted

- Lines leave () charges and return to (-)

charges, - never discontinue in empty space
- Tangent of line direction of E
- Local density of field lines µ local magnitude

of E

- No two field lines ever cross, even when

multiple charges are present.

- ACT What are the signs of the charges whose

electric fields are shown at right?

Electric field lines originate on positive

charges and terminate on negative charges.

Which of the charges has the greater magnitude?

The red one

16-3 The Field of a Continuous Distribution

To find the field of a continuous distribution of

charge, treat it as a collection of near-point

charges

Summing over the infinitesimal fields

Finally, making the charges infinitesimally small

and integrating rather than summing

? linear charge density

? surface charge density

? volume charge density

- Example An infinitely long wire is uniformly

charged. - The charge density is l .Find the Electric

field at point - P on the x-axis at xx0

y

Solution

P

x

x0

The Electric Field produced by an infinite line

of charge is

- everywhere perpendicular to the line - is

proportional to the charge density

next lecture Gauss Law makes this trivial!!

- ACT A long line of charge with charge per unit

length ?1 is located on the x-axis and another

long line of charge with charge per unit length

?2 is located on the y-axis with their centers

crossing at the origin. In what direction is the

electric field at point z a on the positive

z-axis if ?1 and ?2 are positive? - A) the positive z-direction
- B) halfway between the x-direction and the

y-direction - C) the negative z-direction
- D) all directions are possible parallel to the

x-y plane - E) cannot be determined

- Act The figure here shows three nonconducting

rods, one circular and two straight. Each has a

uniform charge of magnitude Q along its top half

and another along its bottom half. For each rod,

what is the direction of the net electric field

at point P?

- Example A uniformly charged circular arc has
- radius R and subtends an angle 2q0.The total

charge - is q. Calculate the electric field at point P .

y

Solution

P

x

R

- Act Find the electric field at point P on the

axis - of a uniformly charged ring of total charge q.

The radius of the ring is R, the distance from P

to the center of the ring is x.

Solution

P

R

- when x 0(P is at the center of the ring)

- when xgtgtR

- Act Find the electric field at a distance x

along the axis of a uniformly charged circular

disk of radius R and charge Q

Solution

P

r

R

P

If xgtgtR

r

R

Field of a point charge

If Rgtgtx

Field of an infinite uniformly charged plane

Field of an infinite uniformly charged plane

Divide the plane into narrow straight strips

s

q1

x

x

From the electric field due to a uniform sheet of

charge, we can calculate what would happen if we

put two oppositely-charged sheets next to each

other

Summary of Electric Field Lines

- Act A large flat has a uniform charge density s

. A small - circular hole of radius R has been cut in the

middle of the - surface. Calculate the electric field at point

P(x)

Solution

Think that the configuration is composed of one

uniformly charged plane with charge density s

and a uniformly charged circular disk -s

R

. P

x

Use superposition

x

- Example Find the electric field at point P on

the central axis of the solid cone. The total

charge is q

Solution

z

P

H

R

y

O

x

16-4 Motion of a Charge in a Field

Deflection of Moving Charged Particles( electrons)

We can control the motion of a beam of charged

particles

- ACT An electron beam moving horizontally at a

speed V enters a region between two horizontally

oriented plates of length L1. When the electrons

reach a fluorescent screen located at a distance

L2 past these plates, they have been deflected a

vertical distance y from their original

direction. If the speed of the electrons is

doubled what is the new value of the deflection?

A) y/4 B) 4y C) 2y D) y E) y/2

- ACT A ring of negative, uniform charge density

is placed on the xz-plane with the center of the

ring at the origin. A positive charge moves along

the y axis toward the center of the ring. At the

moment the charge passes through the center of

the ring ________ - A) its velocity and its acceleration reach their

maximum values. - B) its velocity is maximum and its acceleration

is zero. - C) its velocity and its acceleration have

non-zero values but neither is at its maximum. - D) its velocity and its acceleration are both

equal to zero. - E) its velocity is zero and its acceleration is

maximum.

16-5 The Electric Dipole in an External Electric

Field

1. The motion of a dipole in an uniform external

electric field

stable equilibrium

unstable equilibrium

A dipole in an uniform external electric field

only rotates

2.The energy of a dipole in an external electric

field

We choose

(a) 4, 3, 1, 2

(b) 3, the 1 and 4 tie, then 2

(No Transcript)

3. The motion of a dipole in an nonuniform

external electric field

A dipole placed in an nonuniform external

electric field experiences not only a net

force but also a torque.

The motion is a combination of linear

acceleration and rotation

We can explain the phenomena that a charged rod

can attracts small pieces of paper.

- Act A neutral water molecule (H2O) in its vapor

state has an electric dipole moment of magnitude

(a) How far apart are the molecule's centers of

positive and negative charge

(b) If the molecule is placed in an electric

field of , what maximum

torque can the field exert on it? (Such a field

can easily be set up in the laboratory.)

(c) How much work must an external agent do to

turn this molecule end for end in this field,

starting from its fully aligned position, for

which q 0?

Solution

Summary of Chapter 16

- Electric field is defined as the force per unit

charge

- Force on a point charge q

- Electric field lines are very useful for

visualizing the electric field, as long as their

limitations are taken into account.

Summary of Chapter 16, cont.

- Electric field of a point charge

- Electric fields obey the superposition

principle. - Electric dipole equal and opposite charges

separated by a distance L. The electric field is

proportional to the dipole moment, which is

Summary of Chapter16, cont.

- An electric dipole in an external electric field

feels a torque, and has potential energy

- Electric field due to a continuous charge

distribution