Download Bonus page #2

Bonus page #2
So in problem 2.43 we want to calculate the force on the charge in the northern hemisphere
due to the charge in the southern hemisphere. The prescription for doing this is well
defined: we first calculate the electric field of the southern hemisphere, then we calculate
the force on the northern hemisphere with the following integral:
F~ =
Z
N
~ S dτ
ρE
Where the “N” on the integral indicates the integral is over the northern hemisphere
only, and the “S” on the field indicates the field is due to the southern hemisphere alone.
But note that this is very difficult because the electric field of the southern hemisphere
alone is not at all easy to calculate! You can NOT use Gauss’ law, you must compute the
integral over the charge distribution instead (once you cut the sphere in half, you loose
the spherical symmetry and Gauss’ law is useless in trying to find the field). Instead of
doing this, we do the following (the reason for doing this will be clear in a moment); we
add the force of the northern hemisphere on itself to both sides:
F~ +
Z
N
~ N dτ =
ρE
Z
N
~ tot dτ
ρE
Where Etot = ES + EN is the electric field of the sphere as a whole. So why in the heck do
we do this?? The reason is that the electric field of the sphere as a whole is incredibly easy
to calculate – we use Gauss’ law here now that we have spherical symmetry. But now we
have this extra term on the left, which represents the force of the northern hemisphere on
itself, which at first glance seems just as daunting as the original problem. But of course
there is a trick and the trick is that we can easily argue that this extra term has to be zero.
The reason is not limited to the electromagnetic force; in general the net force on an object
can ONLY be due to EXTERNAL forces, meaning forces not generated within the object
in question. This comes from Newton’s third law – if chunk A of the northern hemisphere
acts on chunk B with a certain force, chunk B acts on A with an equal and opposite force.
The two forces cancel, and the same happens everywhere in the northern hemisphere. (A
good example I like to think of is: If I stand in a bucket and pull on the handle, can I lift
myself up? The answer is, of course, no.) So, with this argument, we can ignore the extra
term on the left hand side, and easily get the answer to the problem.
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