1. The Electric Field
We know that the electric force between charges is transmitted by force carriers, known as “photons”, more technically known as “virtual photons” since they are not freely traveling through space.
But even single, isolated charged particles are surrounded by clouds of virtual photons. A charged particle is said to have an “electric field” extending into space in all directions. The presence of this field can be tested by bringing another charged particle nearby (a “test charge”), to see if a force appears along the line between their centers (see Coulomb’s Law).
How is the electric field defined mathematically? It is the electric force per unit charge. If we think of a test charge q0 as a “field measuring device”, and place it at some position r where there is a field, then:
Note: The electric field is there, even when the test charge is not!
Force on the test charge
Electric field
Units
Magnitude of test charge

2. Electric field of a single charge, Q
If we bring a test charge to some distance r from a single charge Q, we can solve for the electric field at that position using Coulomb’s Law:
If we carry the test charge to many locations to map out the electric field, we will get the following picture:
Electric field of a single charge Q

3. Voila! Formidable! The “other” electric constant,
Before we go any further, you should be aware that there is another constant, e0 (the “permittivity of free space”), that we can use as an alternate to k in our expressions for electric force or electric field (or anywhere else it appears). This new constant will be handy in some problems involving electric fields, but often it clutters up expressions because of extra constants appearing with it.
Voila!
For example, Coulomb’s Law and the electric field of a single charge would appear as:
Formidable!

4. Electric field lines of a point charge, Q
There is an alternate, “easier to read”, method for graphing electric fields. Instead of graphing individual E field vectors at a selection of locations, we connect the vectors of the field into continuous lines, called “field lines”. + +
In this representation, the lines show only the direction of E. Field lines start on positive charges and end on negative charges.
The line length no longer represents the field magnitude. Now, the field is stronger where the lines are more densely packed!

5. Like electric force vectors, electric field vectors from different charges add. (Superposition principle.) + q2 + q1 r2 r1 r3 q3 + Q E

6.  Sketch electric field.
Electric dipole field y
Dipole charges are equal and opposite, and equal distances from the x axis. r+ E +Q r- x -Q
 Sketch electric field.

7. Electric field for symmetric charge pairs
(dipole)

8. Continuous charge distributions
Up to this point we have been analyzing electric forces and fields
produced by one or more point charges. Now we turn to the problem of charge distributions that have so many elementary charges in a given region that they may be considered smooth, or “continuous”.
Line charge
Surface charge
Volume charge dq dq dq
Q,L ds
Q,A dA
Q,V dV

9. Calculating E for continuous charge distributions
As implied by the infinitesimals such as dq, to calculate the electric field (seen at point “P ”) due to continuous charge distributions, we must be prepared to integrate over all the charge in the problem.
Line charge
Surface charge
Volume charge P P P r r dq dq dq r ds dA dV
We are using superposition here in integral form, “adding” the contributions to E from each dq. In general, these vector integrals will be hard to do.

10. Ring of uniformly distributed charge
We are restricting this calculation to the x axis. This allows us to use symmetry to greatly simplify the calculation.
Since Q is uniform around the ring:
 Solve for E by integration.

11. Ring of charge: sketch and graph of E along x axis
 Sketch full field. Also find expressions for field in the limits of small x and large x.

12. Disk of uniformly distributed surface charge
We will again find the electric field along the x axis, using the results from the ring-charge calculation. This time we must integrate over r. Use integral tables in appendix at back of book.
Since Q is a uniform surface charge:
Binomial expansion needed for large x limit
 Solve for E by integration. Also, sketch. Find E in the limits of small x and large x.

13. Infinite plane of uniformly distributed surface charge
We have the answer for this already! What is it?

14. Two planes of opposite charge: “parallel plate capacitor”
Superposition principle at work!
 Sketch the E field everywhere. Take the large d and small d limits “graphically”. What is the result for infinite planes?

15. Line of uniformly distributed charge
We will find the electric field along the x axis (at mid-line) by integrating dE caused by the line charge elements dQ along the y axis. Again, symmetry makes this calculation “easier”.
Since Q is a uniform line charge:
 Solve for E at mid-line by integration, and sketch full field. Also, find expressions for field in the limits of small x and large x.

16. Summary: E fields for various charge distributions
Ring charge
small x
large x
Disk charge
small x
large x
Line charge
small x
large x
Notice dependence on r when symmetry is 1D, 2D, or 3D.

17. Charged particles motion in an electric field
A positive charged particle in an electric field, E, will feel a force, F, in the direction of the field.
F = qE
In a uniform electric field, the particle trajectories will, in general, be parabolic. Where else do we see this shape of trajectory? “Why?” What motion occurs if we release the particle from rest in this field?
Electrons or protons in this picture?

18. Electric dipoles in a uniform electric field
A system that has equal and opposite electric charges at a fixed distance, d, from each other is called a “dipole” (Discuss H2O.)
An electric dipole in a uniform electric field, E, will feel a torque, t, perpendicular to E. But it feels no net force! Why?
The torque will try to align the dipole as shown. This this equilibrium stable or unstable? What about the opposite orientation?

19. Electric dipole definition, and behavior in a uniform electric field
The “dipole moment” (not momentum)
Torque: vector product, and magnitude
Potential energy: dot product and magnitude
 Derivations

20. Dipoles aligned in a non-uniform electric field experience a net force
Dipoles aligned in a non-uniform electric field experience a net force. Why? What direction?

21. Continuous charge distributions