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Electric Field – Continuous Charge Distribution As the average separation between source

charges is smaller than the distance between the charges and a point of interest, the system of charges can be modeled as continuous.

The system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or through some volume.

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Charge Densities Volume charge density – when a charge is

distributed evenly throughout a volume = Q / V

Surface charge density – when a charge is distributed evenly over a surface area = Q / A

Linear charge density – when a charge is distributed along a line = Q / l

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Electric Field – Continuous Charge Distribution

Procedure: Divide the charge distribution

into small elements, each of which contains q

Calculate the electric field due to one of these elements at point P

Evaluate the total field by summing the contributions of all the charge elements

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Electric Field – Continuous Charge Distribution, equations For the individual charge elements

Because the charge distribution is continuous

2ˆi

i e ii

qkr

E r

lim0 2 2

ˆ ˆi

iq e i e

i i

q dqk k

r r

E r r

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19.6 Electric Field Lines Field lines give us a means of representing

the electric field pictorially The electric field vector is tangent to the

electric field line at each point The line has a direction that is the same as that of

the electric field vector The number of lines per unit area through a

surface perpendicular to the lines is proportional to the magnitude of the electric field in that region

E

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Electric Field Lines, General The density of lines through

surface A is greater than through surface B

The magnitude of the electric field is greater on surface A than on surface B

The lines at different locations point in different directions

This indicates the field is non-uniform

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Electric Field Lines – Rules for Drawing The lines must begin on a positive charge

and terminate on a negative charge In the case of an excess of one type of charge,

some lines will begin or end infinitely far away The number of lines drawn leaving a positive

charge or approaching a negative charge is proportional to the magnitude of the charge

Field lines cannot intersect

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Electric Field Lines, Positive Point Charge

The field lines radiate outward in all directions

In three dimensions, the distribution is spherical

The lines are directed away from the source charge

A positive test charge would be repelled away from the positive source charge

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Electric Field Lines, Negative Point Charge

The field lines radiate inward in all directions

The lines are directed toward the source charge A positive test charge

would be attracted toward the negative source charge

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Electric Field Lines – Dipole A dipole: two equal and

opposite charges separated at a distance.

The number of field lines leaving the positive charge equals the number of lines terminating on the negative charge

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Electric Field Lines – Like Charges

Two equal and like charges separated at a distance

The same number of lines leave each charge since they are equal in magnitude

At a great distance, the field is approximately equal to that of a single charge of 2q

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Electric Field Lines, Unequal Charges

The positive charge is twice the magnitude of the negative charge

Two lines leave the positive charge for each line that terminate on the negative charge

At a great distance, the field would be approximately the same as that due to a single charge of +q

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19.7 Motion of Charged Particles When a charged particle is placed in an

electric field, it experiences an electrical force

If this is the only force on the particle, it must be the net force

The net force will cause the particle to accelerate according to Newton’s Second Law

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Motion of Particles, cont If E is uniform, then a is constant If the particle has a positive charge, its

acceleration is in the direction of the field If the particle has a negative charge, its

acceleration is in the direction opposite the electric field

Since the acceleration is constant, the kinematic equations can be used

e q m F E a

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Electron in a Uniform Field, Example

The electron is projected horizontally into a uniform electric field

The electron undergoes a downward acceleration

The charge is negative, so the acceleration is opposite the field

Its motion is parabolic while between the plates

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Cathode Ray Tube (CRT) A CRT is a vacuum tube, which is

commonly used to obtain a visual display of electronic information in oscilloscopes, radar systems, televisions, etc

In a CRT, a beam of electrons is accelerated and deflected under the influence of electric or magnetic fields

The electrons are deflected in various directions by two sets of plates

The charge on the plates create the electric field between the plates and allows the beam to be steered

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19.8 Electric Flux Electric flux is the

product of the magnitude of the electric field and the surface area, A, perpendicular to the field

E = E A

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Electric Flux, General Area The electric flux is

proportional to the number of electric field lines penetrating some surface

The field lines may make some angle with the perpendicular to the surface

Then E = E A cos

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Electric Flux, Interpreting the Equation The flux is a maximum when the

surface is perpendicular to the field The flux is zero when the surface is

parallel to the field If the field varies over the surface, =

E A cos is valid for only a small element of the area

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Electric Flux, General In the more general

case, look at a small area element

In general, this becomes

cosE i i i i iE A E A

lim

0i

E i i

surfaceA

d

E A E A

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Electric Flux, final The surface integral means the integral

must be evaluated over the surface in question

In general, the value of the flux will depend both on the field pattern and on the surface

The units of electric flux will be N.m2/C

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Electric Flux, Closed Surface Assume a closed

surface The vectors point in

different directions At each point, they are

perpendicular to the surface

By convention, they point outward

iA

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Flux Through Closed Surface, cont

At (1), the field lines are crossing the surface from the inside to the outside; <90o, is positive

At (2), the field lines graze the surface; =90o, = 0 At (3), the field lines are crossing the surface from the

outside to the inside;180o > >90o, is negative

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Flux Through Closed Surface, final The net flux through the surface is

proportional to the net number of lines leaving the surface This net number of lines is the number of

lines leaving the volume surrounding the surface minus the number entering the volume

If En is the component of E perpendicular to the surface, then

E nd E dA E A

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