AP Physics C: E&M
8.6 Gauss’s Law
8.4 Electric Fields of Charge Distributions
A solid sphere of radius \(R\), an infinitely long solid cylinder of radius \(R\), and an infinite slab of thickness \(2R\) each carry a uniform positive volume charge density \(\rho_0\).
The distance \(r\) is measured from the center of the sphere, from the central symmetry axis of the cylinder, and from the central midplane of the slab, respectively.
Which row in the table correctly identifies the functional dependence of the electric field magnitude \(E(r)\) on \(r\) for all three configurations in the region \(r < R\) and for each individual configuration in the region \(r > R\)?
The distance \(r\) is measured from the center of the sphere, from the central symmetry axis of the cylinder, and from the central midplane of the slab, respectively.
| Row | Inside all three (\(r < R\)) | Outside sphere (\(r > R\)) | Outside cylinder (\(r > R\)) | Outside slab (\(r > R\)) |
|---|---|---|---|---|
| A | Zero | Proportional to \(1/r^2\) | Proportional to \(1/r\) | Independent of \(r\) |
| B | Proportional to \(r\) | Proportional to \(1/r\) | Proportional to \(1/r^2\) | Independent of \(r\) |
| C | Proportional to \(r\) | Proportional to \(1/r^2\) | Proportional to \(1/r\) | Independent of \(r\) |
| D | Proportional to \(r\) | Proportional to \(1/r^2\) | Independent of \(r\) | Proportional to \(1/r\) |
Which row in the table correctly identifies the functional dependence of the electric field magnitude \(E(r)\) on \(r\) for all three configurations in the region \(r < R\) and for each individual configuration in the region \(r > R\)?
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