90 Week 2: Continuous Charge and Gauss’s Law
Note well thatall of these properties are for equilibrium only!As we will shortly learn, conductors
that carry current arenotin equilibrium anddohave nonzero electric fields inside thatareparallel
to the surfaces. I often ask questions that test whether or notyou understand this on exams, so be
careful!
Example 2.4.1: Field and Charge Distribution of a Blob of Conductor
E = 0+ +
+
+
+++
++
+
− − −−−
−++−Figure 23: A conductor with an arbitrary shape near an external charge rearranges its charge into a
surface charge that cancels the field inside and causes the field near the surface to be perpendicular
to the surface.
Suppose we have anarbitrary shapeof conducting material. As usual, we’ll visualize this as an
amoeboid blob of metal with no particular symmetry or shape so thatwe aren’t tempted to use any
“special” property of a regular shape like a sphere or cylinder in our analysis. It isat restin the
field produced by a number of nearby fixed point charges (in the plane of the figure) of either or
both signs, and has been for some time.
What can we tell about the field inside the conductor, the charge distribution of the conductor,
and so on usingjust the principles enumerated above? The following are possiblequestionsyou
might be asked on a quiz or exam, with an explanation of the answers.
- Where is the field inside strongest? (The field inside iszero everywhere, trick question.)
- Given the conductor and the charges, can we sketch a guesstimate of the field in the plane
of the figure? (Yes, done for you above. Note the use of the rule that the field lines enter
or leave the surface of the conductor at right angles. Of course inreality the conductor and
location of external charge could/would be three dimensional and everything could be more
complicated...) - Is the entire conductor electrically neutral? (No, charge on thesurface onlyhas rearranged,
with negative electrons being attracted to the positive charges and getting as “close as they
can” to them (while still remaining as far apart as possible from each other, in competition)
and leaving behind positive charges on the atoms as “close as possible” to the nearby negative
charges ditto. The + and - signs on the figure represent a possible visualization of this surface
charge, which is related to the field outside by:
E⊥= 4πkeσ