92 Week 2: Continuous Charge and Gauss’s Law
each surface charge layer goall he way to infinity, where the total field is the vector sum of the
two fields, one from the upper layer +σ, the other from the lower layer−σ.
As you can see in the figure, above +σthe up field from the upper layer and the down field from
the lower layercancel, making the field zero (as desired) everywhere in the metal plate above +σ.
The same is true below the lower layer−σ. In between the plates, though, the field from the upper
layer is down, the field from the lower layer is downalsoand hence the total field is:
Etot=Eu+El=σ
2 ǫ 0 +σ
2 ǫ 0 =σ
ǫ 0down. The field runs from the positive surface layer to the negative surface layer and is zero
everywere inside the bulk conductor and for that matter in the air above and below the plates!
This is an important example as finding this field in terms ofσ=Q/Ais a required step for
finding first the potential difference between the two plates (nextchapter) and then the capacitance
of this arrangement of conductors (the chapter after that).
Note well! The charges spread out on these surfacemust be equal and opposite! This is true
even if one putsdifferentcharges on the two plates! You will work some examples for spherical
conducting shells for homework and should pay attention to this happening there as well, and for
the same reasons.
Creating Charged Objects
As noted at the beginning of week 1, the ability to demonstrate things like Coulomb’s Law revolves
around several things. One is the ability to accurately measure very small forces – this Coulomb
was able to do with his personally invented torsional balance. The other was the ability to create
controlled amounts of charge and place it on isolated conductors onhis balance.
This section is intended to give yousomeidea of how one can generate charge (by means of friction
or induction) and how one can then use it to generate like amounts ofcharge for experiments. The
primary two means for the latter are charging by induction and charge transfer.
Charging by induction is illustrated below:Q = 0
−q
+++
++
++−q
+++
++
++−−
− −−−+
++
+++
+Q = 0
(a) (b) (c) (d)
Figure 25: Charging by induction in four steps.In the first panel (a), a neutral, spherical conductor is connected to “ground”, which can be
thought of as areally, really big conductor, a reservoir of charge that generates essentially no
additional field no matter how much charge you pull from it or deliver to it. Note well the symbol
used for ground.