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Transcript
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. 1