3. Electric Field
a. Electric Field: In the case of the electric force, we can do a similar
thing. Let’s suppose that in Coulomb’s law we let q1 = Q, a large
electric charge, and q2 = qo, a small positive electric charge that we
will call a test charge. If we examine the force on qo, we can write
G
1 Q
 2  qo , and we can introduce the electric field E ,
4πε o  r 
G
G F
defined by E = . Units for E are given from this equation as N/C,
qo
that F =
with no special name. For a point charge Q, we can find the electric
field it generates directly from Coulomb’s law
E=
1 Qq0
1 Q
=
2
4πε o r q0 4πε o r 2
What have we gained in introducing E over F? Why bother?
b. Significance of E: When discussing forces, we have to have in mind
the force on a specific charge at a specific location. To be specific,
let’s take the case of just two charges. Since the charges are not in
contact with each other, they exert forces from a distance – this
“action at a distance” leaves us wondering how each charge knows
about the presence of the other charge. Moreover, if one of the
charges moves, how does the other charge learn of this motion – does
it learn instantly or is there some delay time during which the
information travels from the moving charge to the other one? The
concept of electric field provides answers to these questions. We will
see that each charge produces an E field in all space – introducing the
notion of a vector field. Thus the electric field of a charge represents
an infinite set of vectors – one for each point in space – and the vector
G
field E ( x, y, z, t ) in general will give a unique vector at a point (x,y,z)
at time t. We will study this in some detail. It turns out that the
electric field has real physical significance – it is not just a
convenience as an intermediary between the source charge and the
force on another charge. In general, the electric field can carry
energy, momentum and angular momentum and is a real, physically
measurable quantity. It also travels at the speed of light and so when
a charge moves, the information about this motion travels outward at
the speed of light to other charges. Thus “action at a distance” occurs
through the interaction of the electric field (produced by some other
charge) with the charge experiencing a force.
c. Visualizing Electric fields (vector fields in general): One method of
viewing vector fields is to simply draw representative vectors at a set
of points in space with the convention that the vector “tails” are
located at the point {x,y,z} and the length and direction of the vector
represent its magnitude and direction at that point. A second, more
common method for visualizing vector fields is called a field line
mapping. For electric fields this mapping has the following
properties:
i. Field lines originate on positive charges and terminate on either
negative charges or at infinity
ii. The direction of the field line at a point is given by imagining a
positive test charge placed at that point
and finding the direction of
G
G
the force on that charge – since E =
F
this is the same direction as
qo
E.
iii. The strength or magnitude of E is proportional to the density of
field lines (# of field lines/unit area across a surface perpendicular
to the field lines).
iv. The number of field lines leaving or entering a charge is
proportional to the magnitude of the charge.
Note: field lines can never cross – why?
Note: charges do not move along E field lines in general. Why?
The direction of E, tangent to a field line, represents the
direction of the force on a positive charge, and thus also the direction
of its acceleration – not its velocity or its displacement – so the charge
will not follow a field line – unless in the special case when the field
lines are straight.
Examples: Field lines for point charge:
Since the surface area of a sphere is 4πr2 the
density of lines crossing the surface (or the
magnitude of E) varies as fixed #/4πr2 or E
∝ 1/r2. Note this is a positive charge, a
negative charge has lines towards it.
Note: We will have our second lab on
electric field mapping.
d. Electric field of a dipole: Use superposition principle, just as for
forces. Worked example of field along perpendicular bisector. Along
the dipole direction is worked out in the text on p. 526.
e. Example problem Three point charges at corners of a square – to find
the E field at the center of the square.