E & M, Infinite sheet of charge

[SpaceExplorer]

The problem I have is about a simple remark made in the book 'Berkeley Physics Course Volume 2, Electricity and Magnetism', chap. 3 figure 3.4 b. It says that if we have an infinite sheet of charge but with 'other charges' present elsewhere in the system, the only thing we can predict is that at the surface, there will be a change of 4(pi)s, where 's' is the surface charge density, in E_x and 0 in E_y. Why is that? (the book only deals in m.k.s units)

 

[Doc Al]

Why is that?

Consider the field from an infinite sheet of charge with nothing else around. What's the field on each side? Then consider the additional field that might be present because of other charges. Apply the principle of superposition.

(the book only deals in m.k.s units)

Purcell uses (if I recall) Gaussian CGS units, not S.I. (mks) units.

 

[SpaceExplorer]

Oh... I wanted to write C.G.S, but wrote M.K.S instead, thanks for pointing that out. The problem with the superposition thing is that the scenario doesn't consider a particular arrangement of external charges, but rather generalises the proposition to all the possible arrangements; logically, the change in the field at a point on the surface should be different for different arrangements, so it doesn't make sense that the end result will always have to be 4.(pi).s. Moreover there's no variable or constant in the final result that even accounts for the presence of extra charges, 's' is just the surface charge density, while 4 and pi are unrelated constants.

 

[Doc Al]

The problem with the superposition thing is that the scenario doesn't consider a particular arrangement of external charges, but rather generalises the proposition to all the possible arrangements;

That's the beauty of the superposition argument, not the problem.

logically, the change in the field at a point on the surface should be different for different arrangements,

Why is that?

The idea is this. Say there are a bunch of charges in arbitrary arrangement (but just not on the sheet) that end up creating a field E where that sheet of charge is to be located. To find the total field, you would add to that the field from the sheet of charge. So on one side you'd have $E - 2\pi\sigma$ and on the other side you'd have $E + 2\pi\sigma$, for a difference of $4\pi\sigma$.

 

[SpaceExplorer]

I think you're right.I thought of the same thing at the beginning. But I think the author has done a technical mistake because of which I was having the trouble. Purcell says 'the change 'at' the surface' must be 4(pi)s, if he said 'across' the surface, I would have been convinced by the same argument you are giving me now long before.