1 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.

2 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

3 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

4 Electric Field – Continuous Charge Distribution, equations For the individual charge elements Because the charge distribution is continuous

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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

16 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

17 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

18 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

19 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

20 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

21 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

22 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

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

24 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

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27 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

28 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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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

34 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 

35 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

36 Electric Flux, General In the more general case, look at a small area element In general, this becomes

37 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. m 2 /C

38 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

39 Flux Through Closed Surface, cont At (1), the field lines are crossing the surface from the inside to the outside;  <90 o,  is positive At (2), the field lines graze the surface;  =90 o,  = 0 At (3), the field lines are crossing the surface from the outside to the inside;180 o >  >90 o,  is negative

40 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 E n is the component of E perpendicular to the surface, then

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