Why the electric field is identical in all the slabs?
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The slab parallel to the xx-zz plane, and is thus perpendicular to the yy-axis, contained between y=−dy=−d and y=dy=d but reaching infinitely into the xx and zz directions.
The book I'm reading utilizes Gauss's law (using a "Gaussian pillbox" circling around the yy-axis), but I was a bit confused by the method they used, and when doing the problem I instead thought like this:
If we place a test charge on the yy-axis at y=ay=a, then the charge experiences a positive force (pointing the positive yy direction) due to the volume charge behind it (from y=−dy=−dto y=ay=a) and experiences a negative force (gets "pushed backwards") due to the volume charge in front of it (from y=ay=a to y=dy=d). So for |y|<d|y|<d, the electric field would be the total field made up of a bunch of infinitesimally-thin charged planes behind the test charge minus the field made up of a bunch of infinitesimally-thin charged places in front of the test charge, or:
The book I'm reading utilizes Gauss's law (using a "Gaussian pillbox" circling around the yy-axis), but I was a bit confused by the method they used, and when doing the problem I instead thought like this:
If we place a test charge on the yy-axis at y=ay=a, then the charge experiences a positive force (pointing the positive yy direction) due to the volume charge behind it (from y=−dy=−dto y=ay=a) and experiences a negative force (gets "pushed backwards") due to the volume charge in front of it (from y=ay=a to y=dy=d). So for |y|<d|y|<d, the electric field would be the total field made up of a bunch of infinitesimally-thin charged planes behind the test charge minus the field made up of a bunch of infinitesimally-thin charged places in front of the test charge, or:
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